= Planar Debye-Hückel double layer
{title2=$\psi(x)=\psi_0e^{-\kappa x},\quad\sigma=\varepsilon\kappa\psi_0$}
For a plane bounding a symmetric electrolyte in $x>0$, assume no displacement field on the other side, bulk potential zero, and $|ze\psi|\ll k_BT$. The <Debye–Hückel approximation> gives $\psi''=\kappa^2\psi$, whose decaying solution is $\psi_0e^{-\kappa x}$. The <electrostatic interface boundary conditions> give $\sigma=-\varepsilon\psi'(0^+)=\varepsilon\kappa\psi_0$. Equivalently, its diffuse <charge density> is $\rho_e=-\varepsilon\kappa^2\psi$ and its integral is $-\sigma$. Here $\varepsilon=\varepsilon_0\varepsilon_r$ in SI units and $\kappa^2=2z^2e^2n_\infty/(\varepsilon k_BT)$ when $n_\infty$ counts each ionic species. An isolated sheet with identical electrolyte on both sides has two diffuse layers, so the total sheet charge is $2\varepsilon\kappa\psi_0$ instead. A surface geometry and the field on its other side must therefore be specified.
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