Electrostatic Correlations: From Debye-Huckel Ionic Atmospheres to Counterion Condensation

We study electrostatic correlations by quantifying the relative contributions of linear-response ionic atmospheres (as first described by Debye and Huckel) and counterion condensation (CC). Even for aqueous solutions of highly charged polyelectrolytes, CC may contribute less than half of the (electrostatic) osmotic coefficient drop. We further show that after CC sets in, its relative contribution to the osmotic coefficient actually decreases with increasing concentration. We quantify the strength of CC in terms of an electrostatic binding constant that captures both direct interactions between condensing charges as well as non-local effects of screening and chain connectivity. Using this binding constant we elucidate the (anti)-cooperative nature of CC and the importance of chain conformations. Critically, we retain the discrete nature of the polyelectrolyte backbone and counterions -- this preserves "residual" electrostatic fluctuations of condensed charge pairs. Depending on the strength of residual charge fluctuations, CC can either stabilize or destabilize polyelectrolyte solutions against phase separation.