The equilibrium electric potential and surface charge density of spherical emulsion drops with thin double layers

Analytic approximations are derived for the solution to the Poisson-Boltzmann equation as applied to a spherical emulsion drop containing a binary electrolyte. Particular attention is given to the drop interior and the formulas that result are easily evaluated. The approximations are obtained by two separate asymptotic methods, which are analogous to those used previously by others to describe the electric potential profile on the exterior of a spherical colloidal particle. The analyses apply to emulsion drops with thin double layers, meaning the drop radius a is large compared to kappa(-1) and <(kappa)over bar>(-1), the respective Debye screening lengths for the exterior and interior of the drop. Using delta = (<a(kappa)over bar>)(-1) as a perturbation parameter, rye obtain a matched-asymptotic solution that adds corrections through O(delta(3)) to the hat-plate and Debye-Huckel solutions of the Poisson-Boltzmann equation. In the process, we recover expressions for the drop exterior that constitute an O(delta) improvement over the previously published results. Through a nonlinear transformation of the independent variable, we also derive a uniformly valid approximation that iteratively adds a correction to the hat-plate problem, Each technique yields accurate solutions. For example, the maximum relative error over the drop interior is on the order of 1% for a kappa as low as 5 with surface potentials as high as 250 mV. Accuracy improves for larger values of <a(kappa)over bar>, with a maximum relative error below 0.1% for <a(kappa)over bar> > 15. The asymptotic techniques are also used to obtain expressions for the surface charge density, with equally satisfactory results. (C) 1996 Academic Press, Inc.