Bethe approximation for a hydrophobic-polar random copolymer

A previous analysis of the configurational statistics of a lattice self-avoiding walk, based on the pair approximation of the cluster variation method (Bethe approximation), is generalized to describe a random copolymer in dilute solution, in which monomers interact with one another and with the solvent. The phase diagram is worked out numerically and the coil-globule transition is obtained. Entropy and internal energy in the coil phase, as well as the Theta point are evaluated analytically, by means of a Landau expansion of the variational free energy. A detailed analysis is carried out for the case in which the only interaction present is an attraction between monomers of one species. Such a model has been proposed in the literature to mimic a random copolymer made up of hydrophobic and polar monomers in water solution, a physical system which is believed to be relevant to understanding the behavior of proteins.