  This is the second in a series of papers in which we derive a
$\Gamma$-expansion for the two-dimensional non-local Ginzburg-Landau energy
with Coulomb repulsion known as the Ohta-Kawasaki model in connection with
diblock copolymer systems. In this model, two phases appear, which interact via
a nonlocal Coulomb type energy. Here we focus on the sharp interface version of
this energy in the regime where one of the phases has very small volume
fraction, thus creating small "droplets" of the minority phase in a "sea" of
the majority phase. In our previous paper, we computed the $\Gamma$-limit of
the leading order energy, which yields the averaged behavior for almost
minimizers, namely that the density of droplets should be uniform. Here we go
to the next order and derive a next order $\Gamma$-limit energy, which is
exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a
limiting interaction energy for vortices in the magnetic Ginzburg-Landau model.
The derivation is based on the abstract scheme of Sandier-Serfaty that serves
to obtain lower bounds for 2-scale energies and express them through some
probabilities on patterns via the multiparameter ergodic theorem. Without thus
appealing to the Euler-Lagrange equation, we establish for all configurations
which have "almost minimal energy" the asymptotic roundness and radius of the
droplets, and the fact that they asymptotically shrink to points whose
arrangement minimizes the renormalized energy in some averaged sense. Via a
kind of $\Gamma$-equivalence, the obtained results also yield an expansion of
the minimal energy for the original Ohta-Kawasaki energy. This leads to
expecting to see triangular lattices of droplets as energy minimizers.
