An iterated three-layer model of the double layer with permanent dipoles

Abstract

There does not exist a theory of the ionic double layer at a completely blocking metal electrode in liquid electrolytes which is adequate in the charge/potential region where ions and solvent molecules begin to approach saturated conditions. Under these conditions, a continuum theory, such as that of Gouy and Chapman (GC), becomes entirely inadequate. Here the problem is attacked in a semi-discrete way by first partitioning the space charge region into layers parallel to the planar blocking electrode. Each layer is part of a cubic lattice with lattice-site spacing determined by the pure solvent concentration. Lattice sites may be occupied by ions of either sign or by solvent molecules, taken as spheres having a permanent dipole moment. The solvent molecule finite-length dipoles are then approximated by slabs of constant point-dipole polarization. Thus each of the planes parallel to the electrode is a locus of ion centers, and the polarization is accounted for by equal and opposite charge layers equidistant on either side of an ionic charge layer. The mean polarization and ionic concentration in each three-layer region are determined self-consistently by free energy minimization, and electrostatic equations are employed to couple the electrical conditions in one layer to those adjacent. This ion-dipole model (IDM) is solved self-consistently for arbitrary molarity in two regimes: the weak-field situation where the electrode charge approaches zero, and the arbitrary field-strength regime. In the first case, an, exact, closed-form solution is obtained which reduces to that of GC in the appropriate limit, but numerical analysis is required in the second situation. The present treatment provides a more realistic account of the electrical effects of discrete solvent dipoles than do those treatments, such as the GC model, which represent them entirely by a background, non-saturable, or even saturable, bulk dielectric constant. Here polarization saturation enters naturally in a fully self-consistent way Thus although dipoles line up with the field in high-field regions, they tend to be displaced by ions of a given sign in the layers immediately adjacent to the blocking, electrode, reducing the net polarization. A simpler model, more directly appropriate for single crystals than for liquids, the layered lattice gas model (LLM), retains layering but represents the permanent dipolar polarization by a non-saturable continuum bulk dielectric constant; it is thus intermediate between the IDM and the GCM. Predictions of the three, models are compared with Grahame's experimental differential capacitance results for NaF in the low-field region. The IDM is found to be much superior to either the LMM or the GCM. Many results are presented for the three models in the arbitrary field region. One of the most striking is that the IDM alone yields a strong oscillation in potential versus distance away from the blocking electrode, as first predicted by Kirkwood and Poirier for layered ionic structures.