= Grahame equation
{c}
{title2=$\sigma=2\varepsilon\kappa(k_BT/e)\sinh[e\psi_0/(2k_BT)]$}
For a planar surface contacting a symmetric monovalent ideal electrolyte on one side, the <Poisson-Boltzmann equation> is $\psi''=(2en_\infty/\varepsilon)\sinh(e\psi/(k_BT))$. Multiplication by $\psi'$ and integration using the zero field and zero potential in the bulk gives
$$
(\psi')^2=\frac{8n_\infty k_BT}{\varepsilon}\sinh^2\left(\frac{e\psi}{2k_BT}\right).
$$
The solution decays towards the bulk, and the boundary condition $\sigma=-\varepsilon\psi'(0^+)$ therefore yields the displayed charge-potential relation, with $\kappa^2=2e^2n_\infty/(\varepsilon k_BT)$. Its small-potential limit is the <planar Debye-Hückel double layer>. It is nonlinear but still a mean-field model with point-like <ions> and constant <permittivity>; it does not include a compact interfacial layer or finite-ion-size saturation.
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