= Nonlinear planar Poisson-Boltzmann profile
{title2=$u(z)=4\operatorname{artanh}[\tanh(u_0/4)e^{-\kappa_Dz}]$}
For a symmetric monovalent point-ion electrolyte next to a plane, define $u=e\psi/(k_BT)$ and $\kappa_D^2=2e^2n_\infty/(\varepsilon k_BT)$, with $n_\infty$ the <density> of each sign. The <Poisson-Boltzmann equation> becomes $u''=\kappa_D^2\sinh u$. Multiplying by $u'$ and matching zero field and potential in the bulk gives $(u')^2=2\kappa_D^2(\cosh u-1)$. The decaying branch integrates to the displayed profile, reducing to $u_0e^{-\kappa_Dz}$ at weak potential. This solution assumes a salt reservoir and is distinct from a counterion-only <diffuse layer>.
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