# Interdisciplinary Applied Mathematics

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w

1

ee0kBT

(7.1)

where n is the concentration, kB is the Boltzmann constant, e is the electron charge, z is the valence, T is the temperature, eo is the dielectric permittivity of vacuum, and e is the dielectric constant of the solvent. The subscript i indicates the ith species. Using equation (7.1) and assuming symmetric electrolyte at no = 1.0 x 10~2,1.0 x 10~5, and 1.0 x 10~6M(= Mole/liter), we obtain Debye lengths of Ad = 3 nm, 100 nm and 300 nm, respectively.

Ion distribution due to the EDL can be characterized using an electrokinetic (electric) potential ф. Since the oppositely charged ions in the Stern layer shield some of the electric charges on the surface, the electrokinetic potential drops    rapidly    across    the    Stern    layer.    The    value    of ф    at    the edge

of the Stern    layer    is known    as    the    zeta    potential    (Z). For    most    practical

cases, we can employ the zeta potential to describe electrokinetic flows rather than the wall potential ф0 (See Figure 7.1). Ion distribution in the diffuse layer results in a net electric charge, which can be related to the electrokinetic potential as follows:

VV = — •    (7.2)

eeo

The electric charge density pe is given by

Pe = F^^,    (7.3)

where F is Faraday’s constant. Here we emphasize that:

• The net electric charge contained in the diffuse layer is the primary reason for electrokinetic effects, where charged ions or particles can be mobilized by externally applied electric fields.

If we assume a symmetric electrolyte of equal valence that is in equilibrium with the charged surface, equation (7.3) leads to a Boltzmann distribution, resulting in the Poisson—Boltzmann equation

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