Thermodynamic Theory of Counterion Association in Rigid Polyelectrolytes

We present a new theory of rodlike polyelectrolyte solutions. It is found that at low densities, and below teInperature T~i, the counterions associate with the polyions forIning clusters consisting of one polyion and n counterions. The distribution of du8ters sizes is Gaussian, leading to strong logarithmic corrections to the liIniting laws obtained by Manning. In the Inathelnatical limit of infinite dilution the distribution of the cluster sizes approaches a delta function centered on the value postulated by Manning. Above Tci the solution consists of free (unassociated) counterions and polyions, and the liIniting laws reInain unaffected. Finally, unlike soIne recent suggestions, we denlonstrate that the counterion condensation is distinct fron1the Kosterlitz-Thouless transition. In fact, the association phenoInena inside a polyelectrolyte solution can be coInpared to a fornlation of nlicelles in anlphiphilic systenls. There is, however, an iInportant difference. While the aInphiphilic rnicelles can be diluted