Charged micelles in salt-free dilute solutions

We study micelles in salt-free dilute aqueous solutions of diblock copolymers made of long charged A chains and short Hydrophobie B chains. The number density of aggregates with p chains (n_p) and the number of condensed counterions in a p-aggregated chain (N_c(p)) are obtained equating the chemical potentials of the p-aggregated chains (p = 2, 3, ...) with those of the free chains (p = 1) and equating the chemical potentials of the condensed counterions in the aggregates (p = 1,2, ...) with those of the free counterions. The critical micelle concentration (cmc) and the degree of polydispersity are large in charged diblock copolymer solutions. Since N_c(p) increases as p increases, micelles with p ≫1 are nearly neutral. The most populated micelles in experimetally accessible systems, however, have a small number of chains and their effective charge is not strongly reduced. As the concentration of copolymer increases above the cmc, the concentration of free chains decreases.