### 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 radius a is large compared to κ^{-1} and κ̄^{-1}, the respective Debye screening lengths for the exterior and interior of the drop. Using δ = (aκ̄)^{-1} as a perturbation parameter, we obtain a matched-asymptotic solution that adds corrections through O(δ^{3}) to the flat-plate and Debye-Huckel solutions of the Poisson-Boltzmann equation. In the process, we recover expressions for the drop exterior that constitute an O(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% for aκ as low as 5 with surface potentials as high as 250 mV. Accuracy improves for larger values of aκ̄, with a maximum relative error below 0.1% for aκ̄ > 15. The asymptotic techniques are also used to obtain expressions for the surface charge density, with equally satisfactory results.

Original language | English (US) |
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Pages (from-to) | 76-88 |

Number of pages | 13 |

Journal | Journal of Colloid And Interface Science |

Volume | 179 |

Issue number | 1 |

DOIs | |

State | Published - Apr 15 1996 |

### Keywords

- Poisson-Boltzmann equation
- aqueous two-phase systems
- colloidal dispersions
- electrical double layer
- perturbation methods

### ASJC Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Biomaterials
- Surfaces, Coatings and Films
- Colloid and Surface Chemistry