Surface charge and electrostatics {#surfacecharge}
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A solid surface in contact with an electrolyte solution will in general acquire a net charge, due for example to the ionization of surface groups, ion adsorption and/or dissolution. This surface charge is a thermodynamic property of the solid-electrolyte pair and the reader is referred to [@saville; @israelachvili] for detailed presentations of the phenomenon. The equilibrium surface potential is called the zeta potential $\zeta$.

Such surface charges are screened by a diffusive cloud of counter-ions in the solution. At equilibrium, the electrostatic potential $\psi$ in the electrolyte satisfies the Poisson-Boltzmann equation which quantifies the balance between purely electrostatic interactions and diffusion [@saville], $$\label{PB}
\nabla^2 \psi =\frac{2en_0}{\e}\sinh
\left(\frac{e\psi}{k_BT}\right),$$ where we consider here for simplification only the case of monovalent 1:1 ions, e.g. Na$^+$ and Cl$^-$ or OH$^-$ and H$^+$.

A convenient approximation usually made to solve is the Debye-Hückel approximation [@saville; @rice; @hunter; @probstein] of small field strength, $|e\psi| \ll k_BT$, in which case the equation simplifies to the linearized Poisson-Boltzmann equation $$\label{debye}
\nabla^2 \psi =\kappa^2\psi, \quad \kappa^{-1}=\left(\frac{\e
k_BT}{2e^2n_0}\right)^{1/2},$$ where $\kappa^{-1}$ is the Debye screening length: it is the typical length scale in the solution over which counter-ions screen the charged solid surface, and beyond 