Electrostatic interaction in macro-ionic solutions and gels

Abstract

The Coulombic attraction theory of colloid stability proposed by Sogami and Ise is applied to the one-dimensional problem of the electrostatic interaction of plate macro-ions. The total electrostatic energy of the macro-ionic solution is calculated by solving the Poisson-Boltzmann equation with the linearisation approximation. It is proved that the Helmholtz free energy of interaction derived from this quantity is identical to the effective repulsive potential obtained in DLVO theory for the interaction of two flat double layers, in the limit where the Debye approximation holds. It is demonstrated that Overbeek's criticism of the Sogami theory violates the fundamental thermodynamic relations and that the Gibbs pair potential that is appropriate for describing the interaction of macro-ionic particles under isobaric conditions is equal to the sum of the pair term in the electrostatic energy and the Helmholtz pair potential. The Gibbs free energy of interaction between plate macro-ions is calculated as an analytic function of their separation X, the inverse Debye screening length κ and the plate thickness 2a. This pair potential leads to a repulsion at small interparticle distances and to an attraction at large interparticle separations, creating a ‘secondary’ minimum without the need for a van der Waals attraction. It is shown that this allows for the thermodynamic stability of lyophobic colloids, in contrast with DLVO theory. The position of the secondary minimum X min is calculated as an analytic function of κ and a. For dilute solutions the term containing a is negligible and the Coulombic attraction theory reduces to the simple equation κX min = 4, in contrast with the rapid variation of X min with κ predicted by DLVO theory. These predictions are compared with the observed interplate separations in n-butylammonium vermiculite gels. There is no agreement with DLVO theory, even when the Hamaker constant is treated as an adjustable parameter to fit the data. There is qualitative agreement with the Coulombic attraction theory, but the new theory predicts spacings systematically lower than those observed.