Like likes like: counterion-mediated attraction in macroionic and colloidal interaction

The colloidal and macroionic interaction is discussed within the mean-field approach. Bound pairs of latex particles are photographed at a low particle volume fraction of 10−4. The effective pair-potential obtained therefrom has an attractive tail for highly charged samples, while no attraction is detected for a low-charge sample. This attraction also manifests itself in the reversible aggregation of polystyrene sulfonate or DNA double strands by multivalent counterions. It is furthermore found in intra-macroionic interaction, affecting the conformation of flexible macroions. The reason why the DLVO theory predicts only repulsion is discussed. The Fowler-Guggenheim-McQaurrie analysis of the Debye–Hückel theory indicates that electrostatic Helmholtz free energy Fel is generally not equal to Gibbs free energy Gel. The difference (Gel − Fel)/V (V: system volume) corresponds to the electrostatic osmotic pressure pel, which increases with increasing charge number. This consideration hints that (Gel − Fel) might not be negligible for highly charged macroions and colloidal particles. On the other hand, the DLVO approach is based on the assumption of Gel = Fel. Using a mean-field approach, Sogami showed that the pair-potential is purely repulsive at the level of Fel, in conformity with the DLVO theory, whereas it contains a short-range repulsion and a long-range attraction at the level of Gel. The prevailing view that the interparticle interaction is purely repulsive in the mean-field approach, is not necessarily correct: it originates from the assumption that Gel = Fel. The DLVO theory is correct for low charge samples while the like-likes-like attraction appears for highly charged ones.