Manning-Oosawa counterion condensation

Counterion condensation is a basic feature of 2D electrostatics exhibited by highly charged rodlike polymers such as DNA. In the framework of the Poisson Boltzmann equation with salt, we show that such a polymer of radius a attracts a condensate of thickness R-M=A(axi)(1/2) where xi is the Debye length and A depends weakly on the polymer charge density q(0). To leading order in 1/ln(xi/a), we derive the condensate structure and show that free ions follow universal density profiles independent of a and q(0). Generalizing this approach we calculate ion profiles for finite concentration solutions.