ABSTRACT
I. INTRODUCTION Counterion condensation theory has evolved through several stages. At first, the goal was to formulate a Debye-Hiickel theory for the thermodynamic properties of polyelectrolyte solutions, like activities, osmotic pressure, and heats of dilution [1-12], and an Onsager theory for dynamic quantities, like conductivity and mobilities [11,13-26]. It was clear that the DebyeHiickel-Onsager limiting laws, which are leading-order terms in salt con centration expansions, could not simply be adapted to polymer geometries, since polyionic chains usually link many charged groups in a high-density configuration. Unalloyed Debye-Hiickel theory would be applicable only to sufficiently weakly charged chains. It was known, however, that a system of continuously charged zero-radius lines and independently mobile oppo sitely charged point ions is unstable if the charge density of the lines exceeds a precisely determined critical value (L. Onsager, personal communication, 1968). The system becomes stable only if sufficiently many point ions merge their charge with the lines to lower the net “renormalized” line charge den sity to the critical value. The new system of renormalized lines and depleted supply of oppositely charged point ions is stable, and should follow the rules of Debye-Hiickel theory to leading order in concentration. If the original lines have charge densities below the critical value, then Debye-Hiickel theory should be applicable with no charge renormalization required (to leading order in a concentration expansion). With the assumption that on the relevant scale of a Debye length, a polyionic chain can be modeled as a continuously charged line, there then appears a transparent path to a theory of polyelectrolyte solutions (at low polymer concentrations, so that devia tions from ideality are caused predominantly by the electrostatic interactions
between a single chain and the small ions that surround it). The number of counterions that condense on the polymer to renormalize its charge is known; as part of the chain, these ions are not independently active, either thermodynamically or kinetically (the latter depending on the degree of mo lecular involvement with the chain); and all that is then needed is application of Debye-Huckel-Onsager theory to the interactions between the free coun terions (and coions) and the electric field set up by the net charge of polymer and condensed counterions [1,13].
