When, why, and how does like like like?: Electrostatic attraction between similarly charged species

In the Derjaguin-Landau-Verwey-Overbeek (DLVO) framework of the past 60 years, colloidal interaction between similarly charged particles has been claimed to be simply repulsive, and an attraction such as the van der Waals interaction is attached to the Coulombic repulsion. Statistical-thermodynamic considerations show that the electrostatic Helmholtz free energy ΔF(el) is generally not equal to the electrostatic Gibbs free energy ΔG(el) for simple ionic solutions, and the difference ΔG(el) -ΔF(el) (corresponding to the electrostatic osmotic pressure p(el) ) becomes larger with increasing charge number. Thus, it is expected that ΔG(el) -ΔF(el) be large for highly charged macroions. In the DLVO framework, however, ΔG(el) = ΔF(el) was postulated. Sogami showed that a mean field approach reproduced repulsion at the level of ΔF(el) but resulted in (repulsion and) attraction at the level of ΔG(el) . Overbeek's critique of Sogami theory is shown to be in error. If this criticism were correct, then not only the Sogami theory but also the Debye-Hückel theory would be wrong. The attraction is thus confirmed to exist not only for multi-valent but also mono-valent counterions.