Solution of the Poisson–Boltzmann Equation about a Cylindrical Particle

Abstract

An analytical approximation is developed to the solution of the Poisson–Boltzmann equation about a charged cylindrical particle in an electrolyte solution. A symmetrical electrolyte is treated specifically, but the method applies also to nonsymmetrical ones. The analysis preserves the mathematical character of the exact problem and (as is shown by comparison with machine solutions) yields results with errors which are for most purposes negligibly small. The complete solution involves five separate cases depending on the value of ${\lim }_{\mathrm{R\to \infty }}{\mathrm{[\Psi \hspace{0.17em}/\hspace{0.17em}K}}_0\mathrm{(R)]}$ , where $R$ is the radial coordinate and $\Psi$ the electrostatic potential, both in dimensionless form. In two cases the singularity in $\Psi$ is at $\text{R\hspace{0.17em}=\hspace{0.17em}0}$ , and the Debye–Hückel solution is a uniformly valid, though quantitatively poor, approximation. In the other three cases the (physically relevant) singularity is away from the origin. Similar singular behavior would account for the limited success of iterative and regular perturbation schemes for the solution about the sphere.