  In this paper we consider a two-dimensional copolymer consisting of a random
concatenation of hydrophobic and hydrophilic monomers near a linear interface
separating oil and water acting as solvents. The configurations of the
copolymer are directed paths that can move above and below the interface. The
interaction
  Hamiltonian, which rewards matches and penalizes mismatches of the monomers
and the solvents, depends on two parameters: the interaction strength
$\beta\geq 0$ and the interaction bias $h \geq 0$. The quenched excess free
energy per monomer $(\beta,h) \mapsto g^\mathrm{que} (\beta,h)$ has a phase
transition along a quenched critical curve $\beta \mapsto
h^\mathrm{que}_c(\beta)$ separating a localized phase, where the copolymer
stays close to the interface, from a delocalized phase, where the copolymer
wanders away from the interface.
  We derive a variational expression for $g^\mathrm{que}(\beta,h)$ by applying
the quenched large deviation principle for the empirical process of words cut
out from a random letter sequence according to a random renewal process. We
compare this variational expression with its annealed analogue, describing the
annealed excess free energy $(\beta,h) \mapsto g^\mathrm{ann}(\beta,h)$, which
has a phase transition along an annealed critical curve $\beta \mapsto
h^\mathrm{ann}_c(\beta)$. Our results extend to a general class of disorder
distributions and directed paths. We show that
$g^\mathrm{que}(\beta,h)<g^\mathrm{ann}(\beta,h)$ for all $(\beta,h)$ in the
localized phase of the annealed model. We also show that
$h^\mathrm{ann}_c(\beta\alpha)<h^\mathrm{que}_c
(\beta)<h^\mathrm{ann}_c(\beta)$ for all $\beta>0$ when $\alpha>1$. This gap
vanished when $\alpha=1$.
