  In this paper we consider a two-dimensional model of a copolymer consisting
of a random concatenation of hydrophilic and hydrophobic monomers, immersed in
a micro-emulsion of random droplets of oil and water. The copolymer interacts
with the micro-emulsion through an interaction Hamiltonian that favors matches
and disfavors mismatches between the monomers and the solvents, in such a way
that the interaction with the oil is stronger than with the water.
  The configurations of the copolymers are directed self-avoiding paths in
which only steps up, down and right are allowed. The configurations of the
micro-emulsion are square blocks with oil and water arranged in
percolation-type fashion. The only restriction imposed on the path is that in
every column of blocks its vertical displacement on the block scale is bounded.
The way in which the copolymer enters and exits successive columns of blocks is
a directed self-avoiding path as well, but on the block scale. We refer to this
path as the coarse-grained self-avoiding path. We are interested in the limit
as the copolymer and the blocks become large, in such a way that the copolymer
spends a long time in each block yet visits many blocks. This is a
coarse-graining limit in which the space-time scales of the copolymer and of
the micro-emulsion become separated.
  We derive a variational formula for the quenched free energy per monomer,
where quenched means that the disorder in the copolymer and the disorder in the
micro-emulsion are both frozen. In a sequel paper we will analyze this
variational formula and identify the phase diagram. It turns out that there are
two regimes, supercritical and subcritical, depending on whether the oil blocks
percolate or not along the coarse-grained self-avoiding path. The phase
diagrams in the two regimes turn out to be completely different.
