Past exam of the quantum condensed matter course of the University of Cambridge / 2006 / Section A / A3 / c

Distinguish the fixed surface charge density ${\sigma }_s$ from the mobile-ion charge magnitude $q=Ze>0$. Let $n_0$ be the bulk number density of each species in a symmetric $Z:Z$ electrolyte, choose bulk potential zero, and write $\epsilon ={\epsilon }_0{\epsilon }_r$. This convention removes the ambiguity between the surface charge and ion valence in the wording.

Constancy of each ionic electrochemical potential gives the Boltzmann distribution in the local electric potential:Positive potential repels positive ions and attracts negative counterions. The coordinate $s$ measures distance from a locally planar surface; the same local relations hold for a spherical potential with its appropriate radial coordinate.

Multiplying each ion number density by its signed charge gives the local charge densityThe fixed surface charge is supplied separately as a boundary condition; it is not part of this mobile charge density in the liquid.

Substitution into Poisson's equation, ${\nabla }^2\psi =-\rho /\epsilon$, gives the Poisson-Boltzmann equationFor a plane, ${\nabla }^2\psi =d^2\psi /d{s}^2$; for a sphere, it is $r^{-2}d(r^2{\psi }^{\prime })/dr$. The bulk conditions are $\psi \to 0$ and $n_{\pm }\to n_0$. At a planar wall with negligible internal field and normal pointing into the liquid, Gauss's law imposes $-\epsilon {\psi }^{\prime }(0)={\sigma }_s$; alternatively, a prescribed surface potential gives $\psi (0)={\psi }_0$.