  Confinement or geometric frustration is known to alter the structure of soft
matter, including copolymeric melts, and can consequently be used to tune
structure and properties. Here we investigate the self-assembly of ABC and ABB
3-miktoarm star copolymers confined to a shell using coarse-grained Dissipative
Particle Dynamics simulations. In bulk and flat geometries the ABC stars form
hexagonal tilings, but this is topologically prohibited in a spherical geometry
which normally is alleviated by forming pentagonal tiles. However, the
molecular architecture of the ABC stars implies an additional 'color
constraint' which only allows even tilings (where all polygons have an even
number of edges) and we study the effect of these simultaneous constraints. We
find that both ABC and ABB systems form spherical tiling patterns, the type of
which depends on the radius of the spherical substrate. For small spherical
substrates, all solutions correspond to patterns solving the Thomson problem of
placing mobile repulsive electric charges on a sphere. In ABC systems we find
three coexisting, possibly different tilings, one in each color, each of them
solving the Thomson problem simultaneously. For all except the smallest
substrates, we find competing solutions with seemingly degenerate free energies
that occur with different probabilities. Statistically, an observer who is
blind to the differences between B and C can tell from the structure of the A
domains if the system is an ABC or an ABB star copolymer system.
