Bethe approximation for a model of polymer solvation

The phase diagram of a recently proposed model for the solvation of monomers and polymers in water is studied in the homopolymer case, and several thermodynamic quantities are computed by means of pair approximation of the cluster variation method, i.e., the Bethe approximation. The model takes into account the water degrees of freedom in a simplified way, so that they can be integrated out analytically. The resulting effective Hamiltonian contains only a temperature-dependent water-monomer interaction and its phase diagram can be easily studied thanks to the simplicity of the Bethe approximation and exhibits, for a hydrophobic polymer, both cold and warm unfolding transitions in a wide region of the parameter space. This suggests that the present one might be a toy-model description of the phase behavior observed experimentally in water solutions of hydrophobic polymers, such as poly-N-isopropylacrylamide (PNIPAM), as well as a step to understand the mechanism of cold unfolding in proteins.