Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section A(iii) / 16 / iii / Solution

Take a positive surface potential ${\Phi }_0>0$, so positive ions are co-ions and negative ions counterions. With $u_0=e{\Phi }_0/(k_{B}T)\ll 1$,The counterion density is slightly enhanced at the surface, the co-ion density slightly depleted, and both approach $n_0$ over a Debye length. To first order their sum is $2n_0$, while their difference supplies the screening charge. Reverse the surface sign and the ionic roles interchange. These are number-density profiles, not trajectories of individual ions.