Barrier-induced dielectric counterion relaxation at super-low frequencies in salt-free polyelectrolyte solutions

Based on the thermally activated diffusion of counterions over the barrier of the electrostatic binding potential, we construct a scaling theory for the slow dielectric response in dilute and semi-dilute polyelectrolyte solutions. The theory is based on an analytic evaluation of the mean-escape time of a single counterion from the surface of a polyelectrolyte chain and uses a variational expression for the electrostatic potential of a charged cylinder including counterion condensation. This mean-escape time shows a range of characteristic power-law dependencies on the polyelectrolyte length and the polyelectrolyte monomer concentration. The existence of this novel dielectric mode at super-low frequencies reflects the wide spectrum of experimental findings for the super-low-frequency dielectric relaxation mode and thereby helps to reconcile conflicting interpretations of experimental data in terms of conventional scaling laws. We also devise a scaling theory for the counterion condensation of finite-length polyelectrolyte chains at finite concentration, which allows us to include polyelectrolyte charge renormalization in dilute as well as semi-dilute solutions in a unified theoretical framework