= Cylindrical counterion-only Poisson-Boltzmann profile
{title2=$\Phi=2\log[1+(\eta-1)u]$}
For $u=\log(r/a)$ and $\Phi=\beta q\phi-2u$, a counterion-only <Poisson-Boltzmann equation> reduces to $\Phi''=-Ce^{-\Phi}$. Conditions $\Phi(0)=0$, $\Phi\to\infty$ and $\Phi'\to0$ give $\Phi=2\log(1+bu)$, $b=\sqrt{C/2}>0$. Matching surface <Gauss's law> gives $b=\eta-1$, so this branch requires the <Manning parameter> $\eta>1$. Its number density is $n(r)=(\eta-1)^2(a/r)^2/[2\pi\ell_Ba^2(1+(\eta-1)\log(r/a))^2]$. It leaves residual line-charge magnitude $q/\ell_B$.
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