Phase diagram for a copolymer in a micro-emulsion

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) is an element of R-2: alpha >= vertical bar beta vertical bar} 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 polymer moves through the micro-emulsion. The analysis of the phase diagram is based on three hypotheses about the possible frequencies at which the oil blocks and the water blocks can be visited. We show that these three hypotheses are plausible, but do not provide a proof.