Treatment of dipole orientation in the double layer at electrodes: Analysis of conditions leading to polarization catastrophes

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Abstract
Molecular polarization contributions to the potential profile and differential capacity of electrode-solution interfaces are compared, on the basis of the primitive model for the double layer, according to three simple representations of the orientation of solvent dipoles at the electrode: the Watts–Tobin/BDM two-state model, the Fawcett three-state model, and a Debye–Langevin-type continuum-of-orientations model. The so-called Cooper–Harrison catastrophe, in which the induced polarization qdip of the inner layer is calculated to exceed the surface charge density qM of the electrode, is shown to arise for all models considered, and is therefore not, as supposed previously, a specific artifact of the two-state model with respect to which it was first noted. This polarization catastrophe is shown to result from an oversimplified but frequently employed representation of the polarizing field as E1=4πqM. When the field assumed to act on the molecules in the polarization expressions is represented as E1=Δφ1/d (where Δφ1 is the potential difference across the inner layer and d is its thickness), or as E1=4π(qM−qdip), i.e., taking into account the reactive polarization of the dipole layer, there result implicit equations for qdip and Δφ1 which provide self-consistent values of these quantities as a function of qM or of electrode/solution potential difference V. Anomalous behavior is then shown not to arise. These equations are shown to be formally consistent with the conditions of continuity of potential and dielectric displacement, applied to the interfaces assumed to be present in the given model of the double layer. The calculation procedures which are described may be readily extended to other models which have been proposed to describe the polarization of the compact double layer.