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 ${\mathrm{\Gamma }}_0$, 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 $0<{\sigma }_m<e{\mathrm{\Gamma }}_0$, 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$\mathrm{/}$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 ${\sigma }_m\sim e{\mathrm{\Gamma }}_0$ and $H\ll \kappa^{-1}$, the surface charges remain only partially screened by ions, leading to an electric field in the interior.