Counterion-condensation-induced collapse of highly charged polyelectrolytes

Dilute or semidilute solutions of polyelectrolytes show below a critical temperature the phenomenon of counterion condensation. This effect leads (a) to a decrease of the effective total charge of the macroion and (b) to a counterion-induced attraction between different parts of the chain. In this paper we provide simple scaling arguments that allow one to calculate the chain size and the effective total charge as a function of temperature, solvent quality etc. Especially we show that effect (a) alone leads to a continuous reduction of the chain size with decreasing temperature. Due to a feedback-type mechanism effect (b) amplifies this collapse process in such a way that it is reminiscent of a first-order phase transition.