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# Annals of Mathematical Sciences and Applications

## Volume 1 (2016)

### Number 1

### The Poisson–Boltzmann equation and the charge separation phenomenon at the silica-water interface: A holistic approach

Pages: 217 – 249

DOI: http://dx.doi.org/10.4310/AMSA.2016.v1.n1.a7

#### Authors

#### Abstract

The Poisson–Boltzmann (PB) equation is well 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 PB equation essentially accounts for only half of the electrical double layer, with the other half—the surface charge layer—being beyond the PB equation’s 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 (PNP) equations are used as the rigorous basis for our methodology. This holistic approach has the inherent 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. These are usually known as the charge regulation phenomena. We enumerate the “difficulty” of the holistic approach that leads to the introduction of a surface potential trap as the single physical input to drive the charge separation within the computational domain. As the electrical double layer must be overall neutral, we use this constraint to derive both the form of the static limit of the PNP equations, as well as a global chemical potential $\mu$ that is shown to replace the classical zeta potential (with a minus sign) as the boundary value for the PB equation, which can be re-derived from our formalism. In contrast to the zeta potential, however, $\mu$ is a calculated quantity whose value contains information about the surface charge density, salt concentration, etc. By using the Smoulochowski velocity, we define a generalized zeta potential that can better reflect the electrokinetic activity in nano-sized liquid channels. We also present several predictions of our theory that are beyond the framework of the PB equation alone: (1) the surface capacitance and the so-called pK and pL values that reflects the surface reactivity, (2) the isoelectronic point at which the surface charge layer is neutralized, in conjunction with the surface charge variation as a function of the solution acidity (pH), and (3) 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 experimental observations.

#### Keywords

Poisson–Boltzmann equation, interfacial charge separation phenomenon, electrical double layer, holistic approach, chemical potential, zeta potential, Donnan potential