  We systematically study the design of simple patchy sphere models that
reversibly self-assemble into monodisperse icosahedral clusters. We find that
the optimal patch width is a compromise between structural specificity (the
patches must be narrow enough to energetically select the desired clusters) and
kinetic accessibility (they must be sufficiently wide to avoid kinetic traps).
Similarly, for good yields the temperature must be low enough for the clusters
to be thermodynamically stable, but the clusters must also have enough thermal
energy to allow incorrectly formed bonds to be broken. Ordered clusters can
form through a number of different dynamic pathways, including direct
nucleation and indirect pathways involving large disordered intermediates. The
latter pathway is related to a reentrant liquid-to-gas transition that occurs
for intermediate patch widths upon lowering the temperature. We also find that
the assembly process is robust to inaccurate patch placement up to a certain
threshold, and that it is possible to replace the five discrete patches with a
single ring patch with no significant loss in yield.
