(552h) Analytic Solutions of the Poisson-Boltzmann Equation for Nanochannels and Confined Spaces

The
electrostatic potential between charged interfaces is very important for a wide
range of phenomena related to chemistry and physics of solutions. Examples
include theory of electrolytes, colloid stability, and electrokinetic
phenomena. The electrostatic potential in electrolyte solutions is given by the
nonlinear Poisson-Boltzmann equation. This equation can be linearized
for low potentials (_26 mV at room temperature.)

The
case of high potentials and small separations between the charged surfaces is
particularly difficult to simplify. In this paper we suggest a linearization
procedure that is applicable to arbitrarily high potentials and works best in
the case of small separations. This makes our approach particularly suitable
for modeling nanofluidic channels. It allows us to
obtain simple expressions for the potential distribution for a variety of channel
shapes. We show that the analytical results can be presented in a mathematical
form that is invariant for different system geometries. Figure 1 shows the mean
error for the potential distribution in a cylindrical capillary. The error
increases with the capillary radius, or alternatively the electrolyte
concentration (inverse screening length k.) For large radii or high electrolyte
concentrations,an
alternative solution is derived that is based using the method of matched
asymptotic expansions.