Counterion phase transitions in dilute polyelectrolyte solutions

In dilute solutions of rodlike polyelectrolytes some counterions are distri buted far from polyions while others are located in their vicinity in the r egions of cylindrical symmetry of the electrostatic potential. For these cy lindrical regions around rodlike polyelectrolytes we find an exact solution of the nonlinear Poisson-Boltzmann equation for the case of nonzero net ch arge in these regions. This exact solution implies three qualitatively diff erent phases of counterion distribution around the polyions with Second ord er phase transitions between these phases.