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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 $Γ_0$, where the degree of dissociation $α$ is determined by the proton density in water close to the film. We show how $α$ depends on the density $n_0$ of NaOH in water and the surface charge density $σ_m$ on the electrode. For $σ_m>0$, the protons are expelled away from the film, leading to an increase in $α$. In particular, in the range $0<σ_m<eΓ_0$, self-regulation occurs to realize $α\cong σ_m/eΓ_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 $κ^{-1}$, where a crossover occurs at the Gouy-Chapman length. In particular, when $σ_m \sim eΓ_0$ and $H\ll κ^{-1}$, the surface charges remain only partially screened by ions, leading to an electric field in the interior.