Chemistry Reference
In-Depth Information
0.25
0.2
SCFT [N = 128]
N = 48
N = 64
N = 96
N = 128
0.15
0.1
0.05
0
0
5
10
n
AB
/N
1/2
Fig. 24 Cluster size distribution
Pðn
AB
Þ
at the CMC plotted versus the scaled aggregation number,
n
AB
=
p
. The
symbols
represent the simulation data (all taken for
eN
2) for different choices
of
N
, as indicated. The
solid lines
show fits with a Gaussian function. For the largest chain length,
N
128, the SCFT result (using the equivalent choice of incompatibility between A and B,
wN
100) is shown as a
dashed line
. The normalization of the SCFT result was chosen such that it
agrees with the micellar peak of the MC simulation. From Cavallo et al. [
125
]
19
:
this scaling does not yet hold. One sees that the SCFT distribution is very close to
the MC result for
N
128, but this is probably a coincidence with no deep meaning
(when one studies radial concentration distributions of micelles at the CMC, the
data for
N
¼
128 have not yet converged to the SCFT result). The reasons for the
strong deviations from this simple scaling implied by SCFT are not really clear
[
125
]. Thus, we conclude that the equation of state of block copolymer solutions at
the CMC, where micelles form, is far from being understood. Of course, there are
many other interesting questions to ask. For example, when one goes beyond the
CMC, the micelles can either grow, cluster together in cylindrical objects (some
simulation evidence for cylindrical micelles has been seen, e.g., by Viduna et al.
[
303
]), or form micellar mesophase lattices. The non-Gaussian character of
P
¼
n
AB
Þ
ð
for
dm ¼
100) is also an indication
of some very large, nonspherical micelles in the system at these conditions.
7
:
0 in Fig.
23
(e.g., the tail extending to
n
AB
¼
5 Conclusions and Outlook
This article gives a brief review of the state of the art of the modeling of the
equation of state of the solutions of (short) polymer chains by MC and MD
simulations. Emphasis has been on the type of results that can be obtained, although
