Thermodynamic Theory of Counterion Association in Rigid Polyelectrolytes

(Received 1 July 1996, received in final form 23 September 1996, accepted 7 October 1996)

Abstract
We present a new theory of rodlike polyelectrolyte solutions. It is found that at low densities, and below temperature , the counterions associate with the polyions forming clusters consisting of one polyion and n counterions. The distribution of clusters sizes is Gaussian, leading to strong logarithmic corrections to the limiting laws obtained by Manning. In the mathematical limit of infinite dilution the distribution of the cluster sizes approaches a delta function centered on the value postulated by Manning. Above the solution consists of free (unassociated) counterions and polyions, and the limiting laws remain unaffected. Finally, unlike some recent suggestions, we demonstrate that the counterion condensation is distinct from the Kosterlitz-Thouless transition. In fact, the association phenomena inside a polyelectrolyte solution can be compared to a formation of micelles in amphiphilic systems. There is, however, an important difference. While the amphiphilic micelles can be diluted away (broken apart) by increasing the amount of solvent, the strong electrostatic interactions keep the polyion-counterions clusters from dissociating, even as the concentration of polyelectrolyte is decreased all the way down to zero.