= Planar counterion-only double layer
{title2=$n(z)=[2\pi\ell_B(z+b)^2]^{-1}$}
A negatively charged plane with monovalent positive <counterions> in $z>0$ and no salt reservoir has the mean-field profile displayed above, where $b=e/(2\pi\ell_B|\sigma|)$ and <Bjerrum length> $\ell_B=e^2/(4\pi\varepsilon k_BT)$. Its potential relative to the wall is $2k_BT\log(1+z/b)/e$. The positive-ion <charge density> integrates to $|\sigma|$, neutralizing the plane; the field decays algebraically. The potential has no finite zero-at-infinity reference, unlike the screened symmetric-reservoir problem.
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