Ionization at a solid-water interface in an applied electric field: Charge regulation

We investigate ionization at a solid-water interface in applied electric field. We attach an electrode to a dielectric film bearing silanol or carboxyl groups with an areal density $\Gamma_0$, where the degree of dissociation $\alpha$ is determined by the proton density in water close to the film. We show how $\alpha$ depends on the density $n_0$ of NaOH in water and the surface charge density $\sigma_m$ on the electrode. For $\sigma_m>0$, the protons are expelled away from the film, leading to an increase in $\alpha$. In particular, in the range $0<\sigma_m<e\Gamma_0$, self-regulation occurs to realize $\alpha \cong \sigma_m/e\Gamma_0$ for $n_0\ll n_c$, where $n_c$ is $0.01$ mol$/$L for silica surfaces and is $2\times 10^{-5}$ mol$/$L for carboxyl-bearing surfaces. We also examine the charge regulation with decreasing the cell thickness $H$ below the Debye length $\kappa^{-1}$, where a crossover occurs at the Gouy-Chapman length. In particular, when $\sigma_m \sim e\Gamma_0$ and $H\ll \kappa^{-1}$, the surface charges remain only partially screened by ions, leading to an electric field in the interior.