The Poisson Boltzmann equation and the charge separation phenomenon at   the silica-water interface: A holistic approach
  The Poisson Boltzmann equation is known for its success in describing the Debye layer that arises from the charge separation phenomenon at the silica/water interface. However, by treating only the mobile ionic charges in the liquid, the Poisson Boltzmann equation accounts for only half of the electrical double layer, with the other half, the surface charge layer, being beyond its computational domain. In this work, we take a holistic approach to the charge separation phenomenon at the silica/water interface by treating, within a single computational domain, the electrical double layer that comprises both the mobile ions in the liquid and the surface charge density. The Poisson Nernst Planck equations are used as the rigorous basis for our methodology. The holistic approach has the advantage of being able to predict surface charge variations that arise either from the addition of salt and acid to the liquid, or from the decrease of the liquid channel width to below twice the Debye length. As the electrical double layer must be overall neutral, we use this constraint to derive both the form of the static limit of the Poisson Nernst Planck equations, as well as a global chemical potential that replaces the classical zeta potential as the boundary value for the PB equation, which can be re-derived from our formalism. We present several predictions of our theory that are beyond the framework of the PB equation alone, e.g., the surface capacitance and the so-called pK and pL values, the isoelectronic point at which the surface charge layer is neutralized, and the appearance of a Donnan potential that arises from the formation of an electrical double layer at the inlet regions of a nano-channel connected to the bulk reservoir. All theory predictions are shown to be in good agreement with the experimental observations. 
Maijia Liao, Li Wan, Shixin Xu, Chun Liu, Ping Sheng
cond-mat.soft physics.bio-ph physics.chem-ph