On the electrostatic interaction in macroionic solutions

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Abstract
A general theory on the electrostatic interaction in macroionic solution is developed. It is assumed that the motion of macroions is adiabatically cut off from that of simple ions and the distribution of simple ions is determined by the Boltzmann distribution. A generalized Poisson–Boltzmann equation is solved with the linearization approximation. The total electrostatic energy of the solution is obtained, from which the Helmholtz free energy of interaction is calculated. The Gibbs free energy is derived by the use of its additivity relation with respect to the number of the ionic species. The two free energies are demonstrated not to be equal to each other for highly charged macroions, in contrast with widely accepted view. It is shown that the electrostatic potential for a pair of macroions predicts strong attraction between the macroions, as was the case with the Levine–Dube treatment, and the Helmholtz pair potential results in purely repulsive interaction, in accordance with the DLVO theory. However, the Gibbs pair potential leads us to repulsion at small interparticle separations and attraction at large distances, creating a ‘‘secondary’’ minimum (with a potential valley deeper than the thermal energy for spherical colloidal particles). Without taking refuge in a van der Waals attraction the theory evidently substantiates the experimental fact recently reported, namely the existence of Coulombic intermacroion attraction through the intermediary of counterions. Thus, the ordering of macroions (synthetic and biological) and charged latex particles in solutions can be accounted for in terms of the present theory: By taking the lattice sum of the pair potential, it is shown that the bcc structure for polymer latex particles is favored in the small κa region (κ−1: the Debye radius, a: the radius of particle), whereas close‐packed structures dominate in the intermediate and large κa regions. In the whole regions, the simple cubic structure is disfavored. The interparticle distance (the position of the secondary minimum) is shown to become smaller with increasing κa, while the potential valley takes the deepest value at κa≊1. The interparticle distance is found to decrease with latex concentration and temperature. These theoretical outcomes are compared with experimental data currently available.