  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.
