Chemistry Reference
In-Depth Information
Fig. 22 Spherical block
copolymer micelle, assuming
strong segregation between
the A blocks (
thick lines
) that
form the micellar core and the
B blocks (
chain lines
) that
form the corona. The A B
junctions (highlighted by
large dots
) are localized at the
surface of the core, which
forms a sphere of radius
R
,
while the total micelle forms
a sphere of radius
S
. From
Milchev et al. [
124
]
many block copolymers as single chains. Equilibrium then is established via
diffusion and condensation (evaporation) of chains in (from) the micelles, such
that all the micelles in the system and the remaining solution have the same
chemical potential. Such a chemical equilibrium between the micelles and the
solution can, in practice, be established only for rather small
N
, where the scaling
concepts on the micelles are not yet applicable [
124
]. Many of the simulations of
single micelles can be found in the literature (see [
124
] for some further references),
but such work that considers a constrained equilibrium where some value of
n
AB
is
a priori imposed cannot answer the questions asked above, which address the full
equilibrium aspects of micelle formation from solution.
However, there is one special case where simulations of micelle formation for a
model containing reasonably long chains has turned out to be feasible, and this is the
case where one uses as a solvent for the
A
f
B
1
f
block copolymers B-homopolymers
of the same chain length
N
rather than small molecules [
125
]. In this case, equi-
libration is achieved by working in an extension of the semigrand canonical
ensemble, using the chemical potential difference between the block copolymers
(which we shall denote as species
C
in the following) and the B-chains acting as a
solvent as the external control variable,
dm ¼ m
C
m
B
. Choosing
N
C
¼
N
B
, trial
moves can be attempted where a block copolymer turns into a homopolymer,
C
C
. At fixed chain configuration, just a fraction
f
of
monomers needs to be relabeled as A or B in such a move. The chemical potential
difference
dm
enters the transition probability of these exchange moves in much
the same way as for the semigrand canonical algorithm for ordinary polymer blends
[
6
,
82
,
170
,
171
].
Now we will discuss a few characteristic results obtained in the MC study of
Cavallo et al. [
125
], using the bond fluctuation model for the special case
f
!
B
, or vice versa,
B
!
¼
1
=
8
