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 σm on the electrode. For σm > 0, the protons are expelled away from the film, leading to an increase in α. In particular, in the range 0 < σm < eΓ0, self-regulation occurs to realize α ≅ σm/eΓ0 for n0 ≪ nc, where nc 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 σm ∼ eΓ0 and H ≪ κ−1, the surface charges remain only partially screened by ions, leading to a nonvanishing electric field in the interior.
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As the Stern potential drop, we find the expression , where Pb = (ε0 − 1) E(d0)/4π is the bulk polarization close to the bottom Stern layer.48 Here, P(z) can be nonvanishing even for Pb = 0 due to the molecular orientation, leading to an intrinsic potential drop.
Setting Πd = 0, we solve Eq. (41) for u < 0 and s > 0 exactly as tanh(U/4) = − Cexp[κ(z − d0)]. For A1|u| ≫ 1 we have C ≅ 1 + 1/A1|u| and s(|u| − α) ≅ α/2A1κH′. Then, s ≅ α for H′ > 1/A1ακ = ℓGC.
