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This is the first in a series of two papers in which we derive a$\Gamma$-expansion for a two-dimensional non-local Ginzburg-Landau energy withCoulomb repulsion, also known as the Ohta-Kawasaki model in connection withdiblock copolymer systems. In that model, two phases appear, which interact viaa nonlocal Coulomb type energy. We focus on the regime where one of the phaseshas very small volume fraction, thus creating small "droplets" of the minorityphase in a "sea" of the majority phase. In this paper we show that anappropriate setting for $\Gamma$-convergence in the considered parameter regimeis via weak convergence of the suitably normalized charge density in the senseof measures. We prove that, after a suitable rescaling, the Ohta-Kawasakienergy functional $\Gamma$-converges to a quadratic energy functional of thelimit charge density generated by the screened Coulomb kernel. A consequence ofour results is that minimizers (or almost minimizers) of the energy havedroplets which are almost all asymptotically round, have the same radius andare uniformly distributed in the domain. The proof relies mainly on theanalysis of the sharp interface version of the energy, with the connection tothe original diffuse interface model obtained via matching upper and lowerbounds for the energy. We thus also obtain a characterization of the limitcharge density for the energy minimizers in the diffuse interface model.

The $\mathbf \Gamma$-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density