The Equilibrium Electric Potential and Surface Charge Density of Spherical Emulsion Drops with Thin Double Layers

Abstract

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 radiusais large compared to κ−1and κ−1, the respective Debye screening lengths for the exterior and interior of the drop. Using δ = (aκ)−1as a perturbation parameter, we obtain a matched-asymptotic solution that adds corrections throughO(δ3) to the flat-plate and Debye–Hückel solutions of the Poisson–Boltzmann equation. In the process, we recover expressions for the dropexteriorthat constitute anO(δ) 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 flat-plate problem. Each technique yields accurate solutions. For example, the maximum relative error over the drop interior is on the order of 1% foraκ as low as 5 with surface potentials as high as 250 mV. Accuracy improves for larger values ofaκ, with a maximum relative error below 0.1% foraκ > 15. The asymptotic techniques are also used to obtain expressions for the surface charge density, with equally satisfactory results.