  In this paper we study a model describing a copolymer in a micro-emulsion.
The copolymer consists of a random concatenation of hydrophobic and hydrophilic
monomers, the micro-emulsion consists of large blocks of oil and water arranged
in a percolation-type fashion. The interaction Hamiltonian assigns energy
$-\alpha$ to hydrophobic monomers in oil and energy $-\beta$ to hydrophilic
monomers in water, where $\alpha,\beta$ are parameters that without loss of
generality are taken to lie in the cone $\{(\alpha,\beta)
\in\mathbb{R}^2\colon\,\alpha \geq |\beta|\}$. Depending on the values of these
parameters, the copolymer either stays close to the oil-water interface
(localization) or wanders off into the oil and/or the water (delocalization).
Based on an assumption about the strict concavity of the free energy of a
copolymer near a linear interface, we derive a variational formula for the
quenched free energy per monomer that is column-based, i.e., captures what the
copolymer does in columns of different type. We subsequently transform this
into a variational formula that is slope-based, i.e., captures what the polymer
does as it travels at different slopes, and we use the latter to identify the
phase diagram in the $(\alpha,\beta)$-cone. There are two regimes:
supercritical (the oil blocks percolate) and subcritical (the oil blocks do not
percolate). The supercritical and the subcritical phase diagram each have two
localized phases and two delocalized phases, separated by four critical curves
meeting at a quadruple critical point. The different phases correspond to the
different ways in which the copolymer can move through the micro-emulsion. The
analysis of the phase diagram is based on three hypotheses of percolation-type
on the blocks. We show that these three hypotheses are plausible, but do not
provide a proof.
