Theory of counterion association in rod-like polyelectrolytes

A new theory of a polyelectrolyte solution at low densities is proposed. It is shown that below temperature T-cl, counterions associate with polyions forming clusters consisting of a polyion and of n counterions. The distribution of cluster sizes is Gaussian, and leads to a strong logarithmic correction to the limiting laws obtained by Manning. Above T-cl the limiting laws remain unchanged. Furthermore, unlike some recent suggestions it is demonstrated that counterion condensation is distinct from the Kosterlitz-Thouless phase transition.