Fate of the Coulomb singularity in nonlinear Poisson-Boltzmann theory: The point charge as an electrically invisible object

Abstract

An absolute upper bound of Debye-H\"uckel form, $U({r}_{s})\frac{{e}^{\ensuremath{-}r}}{r}$, is derived for solutions to the nonlinear, spherical, radial Poisson-Boltzmann equation for the problem of an isolated charged sphere, dimensionless radius ${r}_{s}$, in an infinite volume of electrolyte. For ${r}_{s}\ensuremath{\ll}1$, $U({r}_{s})\ensuremath{\propto}{r}_{s}\mathrm{ln}(\frac{1}{{r}_{s}})$. Thus, as ${r}_{s}\ensuremath{\rightarrow}0$, $U({r}_{s})\ensuremath{\rightarrow}0$ and the shielding cloud around a point charge shrinks to zero radius. From any finite distance away, however small, the point charge is electrically invisible.