DOUBLE-LAYER INTERACTIONS BETWEEN CHARGE-REGULATED COLLOIDAL SURFACES - PAIR POTENTIALS FOR SPHERICAL-PARTICLES BEARING IONOGENIC SURFACE GROUPS

The variational approach of Reiner and Radke (1990) is employed to investigate the effect of surface charge regulation upon the double layer interaction free energy V(e) of pairs of colloidal particles immersed in an electrolyte. A model for dissociating surface groups that permits the consideration of an arbitrary number of ion-complexation reactions is introduced. The variational method is then used to derive (in the Poisson-Boltzmann approximation) the configurational free energy functional OMEGA of an ensemble of particles bearing such groups. The Debye-Huckel (DH) linearization process is applied to this functional, and ensuing consistency issues are examined. The DH free energy is extremized for a configuration of two interacting flat plates, and Derjaguin's (1934 and 1939) method is used to obtain an approximate analytical form for V(e) for two different-sized spherical particles bearing different surface groups. This second problem is next considered from the perspective of Levine's (1934, 1939b) exact multipole expansion of the electrostatic potential surrounding two axisymmetric particles. It is shown that the linear superposition approximation (LSA) for V(e) developed by Levine (1939c) and Verwey and Overbeek (1948) emerges rigorously from this formulation in the limit of large interparticle separations. The interaction free energy from Levine's expansion is calculated to a six digit accuracy for identical spheres over the range of regulated behavior from fixed surface charge density q(s) to fixed surface potential psi(s) for surface-surface separation h to Debye length lambda ratios from 0 to 2 and ratios of the particle radius a to lambda of 0.1, 1, and 10. These results are compared to those obtained from Derjaguin's method and the linear superposition approximation. Derjaguin's method is only quantitatively accurate (in error by less than 10%) for the largest value of a/lambda and becomes progressively less so as the boundary is changed from perfectly regulating (constant psi(s)) to unregulated (constant q(s)). Agreement of the LSA with the exact V(e) is good over a wide range of parameters, but worsens for large a/lambda and small h/lambda. Appendices present extensions of our approach to surfaces bearing more than one type of complexing group and to the consideration of Stern layer formation at the particle-electrolyte boundary in the context of a standard model for metal oxide-aqueous interfaces.