Ion condensation in salt-free dilute polyelectrolyte solutions

We study ion condensation in salt-free dilute solutions of polyelectrolyte chains of different fractal geometries, from random walks to rigid rods. In finite size chains with a discrete charge distribution, ion condensation is only observed in multiple chain systems. A thermodynamic analysis of ion condensation with linearized interactions among the condensed counterions and the monomers leads to rigid rod lowest free energy conformations, regardless of the values of the valency of the monomers (z_m), the negative value of the ratio of the valency of the counterions to the valency of the monomers (P_z), and the ratio of the Bjerrum length (l_B) to the distance between charges along the polyion (b). As z_m ²P_zl_B/b increases, however, the fraction of condensed counterions increases and the polyion concentration at which ion condensation appears decreases. Above this critical polyion concentration, ion condensation is observed even when l_B/b < 1/z_m ²P_z. When correlations are included, representing the tendency of the condensed counterions to be surrounded by opposite sign charges, the chain conformation is strongly dependent on the valency of the counterions and l_B/b. For large valency counterions and/or l_B/b values, collapsed finite chains have lower free energies than rigid rods.