Effective charge and coil-globule transition of a polyelectrolyte chain

Considering the adsorption of counterions on an isolated polyelectrolyte (PE) chain and using a variational theory, phase boundaries and the critical point for the first-order coil-globule transition are calculated. The transition is induced cooperatively by counterion adsorption and chain conformations and the calculation is done self-consistently. The size of the PE chain is a single-valued function of charge. The discontinuous transition of the coil size is accompanied by a discontinuous transition of the charge. Phase boundaries for the coil-globule transitions induced by both Coulomb strength (inverse temperature or dielectric constant) and ionic strength (salt) show that the PE chain collapses at a substantially lower Coulomb strength in the presence of salt. In the expanded state of the coil, an analytical formula is derived for the effective charge of the chain for conditions where the coupling between chain conformations and counterion adsorption is weak. In general, the dielectric heterogeneity of the solvent close to the polymer backbone is found to play a crucial role in the charge regularization and the chain collapse.