Nonlinear planar Poisson-Boltzmann profile ($u(z)=4\mathrm{artanh}[\tanh (u_0/4)e^{-{\kappa }_{D}z}]$)

For a symmetric monovalent point-ion electrolyte next to a plane, define $u=e\psi /(k_{B}T)$ and ${\kappa }_D^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$, with $n_{\infty }$ the density of each sign. The Poisson-Boltzmann equation becomes $u^{\prime \prime }={\kappa }_D^2\sinh u$. Multiplying by $u^{\prime }$ and matching zero field and potential in the bulk gives $(u^{\prime }{)}^2=2{\kappa }_D^2(\cosh u-1)$. The decaying branch integrates to the displayed profile, reducing to $u_0{e}^{-{\kappa }_{D}z}$ at weak potential. This solution assumes a salt reservoir and is distinct from a counterion-only diffuse layer.