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

An absolute upper bound of Debye-Hückel form, $U\left(r_s\right)\frac{e^{-r}}{r}$, 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 $r_s\ll 1$, $U\left(r_s\right)\propto r_s\ln \left(\frac{1}{r_s}\right)$. Thus, as $r_s\to 0$, $U\left(r_s\right)\to 0$ 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.