Maxwellian double layer forces: From infinity to contact

A linear Maxwellian model of dissociative electrical double layers (DEDL) is formalized and extended to cover separation distances from infinity to contact in planar geometry. The entire interaction range is covered by three electrostatic models: the low potential (LP) model, the co-ion exclusion (CX) model, and the high potential (HP) model. The three models are contiguous variations of the linear Poisson-Boltzmann equation, which is consistent under all physicochemical conditions, including high potentials. Infinitely separated (single) double layers are classified into the LP double layers and the CX double layers. The LP double layers consist of the Debye-Huckel (DH) ionic cloud only. The CX double layers consist of the DH ionic cloud, which is separated from the counterion ionic cloud by a co-ion exclusion surface. The single LP and CX double layers become identical under a singular physicochemical condition. The transitions between the LP, CX, and HP double layers are discussed in detail as a function of separation. Examples of repulsive pressures are calculated from infinity to contact. The DH exponential decay of repulsive forces is recovered at large separations; at closer separations, the CX model predicts positive deviations from the DH limiting slope, which are similar to the predictions of the nonlinear Poisson-Boltzmann equation. In the contact limit, the HP interaction model predicts forces that can be much larger than those predicted by the nonlinear Poisson-Boltzmann. equation. These limiting contact forces decrease according to the inverse square of the separation. The previously discovered Lubetkin-Middleton-Ottewill law is incorporated into the DEDL theory to predict new ionic strength effects similar to the classical DH effects on bulk ionic equilibria.