Ion distribution around a charged rod in one and two component solvents: Preferential solvation and first order ionization phase transition

In one and two component solvents, we calculate the counterion distribution around a charged rod treating the degree of ionization $\alpha$ as an annealed variable dependent on its local environment. In the one component case, $\alpha$ is determined under various conditions without and with salt. In the two component case, we take into account the preferential solvation of the counterions and the ionized monomers and the short-range interaction between the rod and the solvent without salt. It then follows a composition-dependent mass action law. Mesoscopic variations of the composition and the counterions are produced around a chraged rod, which sensitively depend on various parameters of the molecular interactions. Furthermore, we predict a first order phase transition of weak-to-strong dissociation for strong preferential solvation. It can occur in expanded states of a polymer chain. This transition line starts from a point on the solvent coexistence curve and ends at a critical point in the plane of the temperature and the solvent composition. The composition change around a charged rod is long-ranged near the solvent critical point.