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

We investigate ionization at a solid-water interface in an applied electric field. We attach an electrode to a dielectric film bearing silanol or carboxyl groups with an areal density Γ₀, 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 n₀ 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 < eΓ₀, self-regulation occurs to realize α ≅ σ_m/eΓ₀ for n₀ n_c, where n_c is 0.01 mol/L for silica surfaces and is 2 × 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 ∼ eΓ₀ and H κ^-1, the surface charges remain only partially screened by ions, leading to a nonvanishing electric field in the interior.