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However, if one is interested only in a rough overview of the behavior of the model,
without precise characterization of these jumps and the location of the transition,
the “sedimentation equilibrium” method (Sect.
3.2
) is a conceptually simple and
straightforward alternative (Fig.
14
).
It is still an open problem to extend the above analysis to models (such as studied
by Ivanov et al. [
122
]) where an attractive interaction between the effective mono-
mers is also present, so that variable solvent quality is implicitly modeled. Clearly it
will require a major effort to extend the techniques described for short alkanes
(Sects.
4.1
and
4.2
) to coarse-grained off-lattice models for stiff chains in explicit
solvent.
4.4 Solutions of Block Copolymers and Micelle Formation
In this subsection, we return to schematic models of flexible chains again, but
consider the extension from (monodisperse) homopolymers to diblock copolymers,
i.e., we have a block of A-type monomers (chain length
N
A
) covalently linked to a
block of B-type monomers (chain length
N
B
), such that the total block copolymer
has the composition
f
N
B
. When such block copolymers
occur in a solvent, it is natural to assume that the solubility for the two blocks is
different. Of particular interest is the case in which the shorter block (say, the
A-block, so
f
¼
N
A
=
N
where
N
¼
N
A
þ
2) is under bad solvent conditions, while the solvent is still a
good solvent for the B-block. If we then have isolated single block copolymers, the
configuration of the chain should then be a collapsed spherical A-globule, with the
A B junction on the surface, so that the B polymer is outside the globule, in a
mushroom-like configuration. However, when one considers a (dilute) solution
containing many such block copolymers in a selective solvent, one may encounter
a transition from an (almost) ideal “gas” of single block copolymer chains to a
“gas” of so-called micelles, where in each micelle a number
n
AB
of chains cluster
together such that the A-parts form a common “core” (of radius
R
) while the B-parts
form the “corona” of radius
S
, see Fig.
22
).
The theory of micelle formation of such block copolymers (and the related case
of smaller surfactant molecules) in solution, within the framework of statistical
thermodynamics, is a longstanding and challenging problem, which is still incom-
pletely understood (see, e.g., [
124
,
294 303
]). One wants to predict how the critical
micelle concentration (CMC) and the number
n
AB
of chains forming a micelle (and
also geometric properties of the micelles, such as the radii
R
and
S
, Fig.
22
) depend
on the parameters of the problem (
f
,
N
, interaction parameters
e
AA
; e
AB
and
e
BB
,
chain stiffness, etc.). Depending on these parameters, the solvent is partially or
completely expelled from the micellar core, and the A B interface between core
and corona may be sharp (as hypothesized in Fig.
22
) or diffuse, etc. Thus, a variety
of scaling-type predictions exist (see [
124
] for a brief review), but it is very hard to
test them because simulations need to equilibrate large enough systems where many
micelles occur and are in equilibrium with a surrounding solution that still contains
<
1
=
