= Osmotic stress in a symmetric electric double layer
{title2=$\Pi=k_BT(n_++n_--2n_0)-\varepsilon E^2/2$}
For a symmetric electrolyte with mobile charges $\pm q$, bulk density $n_0$ of each sign, and constant permittivity $\varepsilon$, the <Poisson-Boltzmann equation> gives $n_\pm=n_0e^{\mp q\psi/(k_BT)}$. In planar geometry let $E=-\psi'$ and $\rho=q(n_+-n_-)$. Then $k_BT(n_++n_-)'=\rho E$ and $\varepsilon EE'=\rho E$ by <Gauss's law>, proving that the displayed bulk-subtracted normal stress is constant. At the midpoint of identical equally charged plates, symmetry gives $E=0$ and $\Pi=2n_0k_BT[\cosh(q\psi_{\rm mid}/(k_BT))-1]$. Its ionic <osmotic pressure> and electrostatic contributions together transmit the force; bare surface-charge repulsion alone omits essential terms.
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