Approximate methods of determining the double-layer free energy of interaction between two charged colloidal spheres

Abstract

We investigate several approximate methods of finding the double-layer interaction free energy, according to the Derjaguin-Landau-Verwey-Overbeek theory, of two charged spheres with centers at distance R apart in electrolyte solution. The spheres may have unequal radii a 1 and a 2 and unequal surface potentials. An appropriate approximation at reasonably large values of κ(R − a 1 − a 2), where κ is the Debye-Huckel constant, is that of linear superposition (L.S.A.). The potential in the neighborhood of the plane equidistant from the two centers is equated to ψ 1(r 1) + ψ 2(r 2), where r 1 and r 2 are distances from the centers and ψ 1 and ψ 2 are the potentials due to the two spheres considered as isolated systems. By using the theoretical work of Gronwall, La Mer, and Sandved or the numerical results of Loeb, Wiersema, and Overbeek for single sphere potential distributions, the method can be applied to higher surface potentials, outside the linear Debye-Hu¨ckel range. A simple general form of the interaction energy at large separations suitable for all radii and potentials is obtained. An integral equation method, recently developed to solve the Debye-Hu¨ckel equation for two identical, charged particles, is extended to the case of two particles having both unequal radii ( a 1, a 2) and unequal potentials. This is applicable when κa 1 and κa 2 ≥ 5 and surface potentials are small. It gives the correct limiting form at large separations and also reduces to a formula which is derived from the method of Derjaguin at small separations. An extension to larger potentials is suggested. We also examine the conditions under which two spherical particles, having equal radii but unequal surface potentials of the same sign, show electric double-layer attraction in the Debye-Hu¨ckel range. Numerical values of interaction free energy and force are compared with those directly computed for pairs of spheres of equal radius a and equal potential. The proposed extension to larger potentials of the interaction formula by the integral equation method gives overall satisfactory agreement with available numerical results for κa ∼ 5. As an alternative, it is possible to combine the L.S.A. at large separations with the Derjaguin formula at small separations and so obtain good values for both force and interaction energy at all separations and potentials.