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Soft Matter

interaction with the solvent. Nonetheless numerical results are
remarkably consistent with the experimental evidence.

4 Conclusions
We have presented a model based on a set of Cahn–Hilliard
equations to study diblock copolymers confined in small capsules.
When the components of a blend of a copolymer and a homopolymer interact one another, their energetic contributions to an
energy functional cause the mixture to settle down into a steady
confined copolymer microphase surrounded by a homopolymer
phase. The added advantage of using partial differential equations
to describe the dynamics of the components in the mixture is
that we eliminate the need of boundary conditions to define the
confining capsule of particles. An underlying variational principle
ensures that these particles are confined in approximately
spherical minimal surfaces. The values of parameters a and b in
the coupled equations will account for the interaction between
the confined copolymer and the confining surface. Resulting
morphologies depend on the parameters of the model and on
the system size as well.
The corresponding phase diagrams suggest that in mediumsize systems, stacked layers and onions are often present, whereas
in small systems additional morphologies might appear such like
tennis ball, Janus and multipods. The boundary between regions
in phase diagrams is not sharp in the sense that morphologies
change gradually across the border. For instance, a small Janus
particle can be seen as layered morphology.
Multipods are often found in phase diagrams of confined
copolymers and we have analyzed how to control the number of
holes of these complex morphologies also seen in experiments.
The proposed model seems to confirm that according with the
experimental data, the number of holes increases with D/P0. The
external surface area of PI domains in multipods decreases with
D/P0, which is also consistent with experimental findings.
Future work should characterize how the confining surface
conditions of multipods change as D/P0 increases. An increase
in the preference of confining surface for the PS domain would
help to explain the deviation of the simulation results from the
experimental data. Another area of research would address the
characterization of intermediate stages between, for instance,
layered and onion morphologies (thermal annealing) or between
multipod and onion. This would further give insight to processes
involving topological changes like the one illustrated in Fig. 7.
A characterization in terms of Betti numbers80 would be desirable.

Acknowledgements
E. A. and Y. N. gratefully acknowledge the support of Council for
Science, Technology and Innovation (CSTI), Cross-ministerial
Strategic Innovation Promotion Program (SIP), ‘‘Structural Materials
for Innovation’’ (Funding agency: JST). T. T. acknowledges the use of
the computer of the MEXT Joint Usage/Research Center ‘‘Center
for Mathematical Modeling and Applications’’, Meiji Institute
for Advanced Study of Mathematical Sciences (MIMS). This work

This journal is © The Royal Society of Chemistry 2016

Paper

was supported by JSPS KAKENHI Grant Numbers: B26310205,
15KT0100, 26708025, 26620171 and 25706006.

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