Osmotic stress in a symmetric electric double layer ($\Pi =k_{B}T(n_{+}+n_{-}-2n_0)-\epsilon E^2/2$)

For a symmetric electrolyte with mobile charges $\pm q$, bulk density $n_0$ of each sign, and constant permittivity $\epsilon$, the Poisson-Boltzmann equation gives $n_{\pm }=n_0{e}^{\mp q\psi /(k_{B}T)}$. In planar geometry let $E=-{\psi }^{\prime }$ and $\rho =q(n_{+}-n_{-})$. Then $k_{B}T(n_{+}+n_{-}{)}^{\prime }=\rho E$ and $\epsilon E{E}^{\prime }=\rho E$ 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 $\Pi =2n_0{k}_{B}T[\cosh (q{\psi }_{\mathrm{mid}}/(k_{B}T))-1]$. Its ionic osmotic pressure and electrostatic contributions together transmit the force; bare surface-charge repulsion alone omits essential terms.