An Exact Nonlinear Solution of the Poisson-Boltzmann Equation and Its Applications to Bi-directional Electroosmotic Flow

Abstract

Electric potential due to an electric double layer (EDL) is governed by the Poisson-Boltzmann (P-B) equation. For a system described by the Gouy-Chapman model with a single symmetric electrolyte, exact analytic solutions for electric potential profiles presently exist only for two cases: (i) an EDL over a flat plate in a semi-infinite domain; (ii) EDLs within a parallel-plate channel with identical charges on the walls. For investigations involving more complex configurations or potential distributions, researchers have been obliged to employ numerical methods or the common linearization scheme - Debye-Hückel (D-H) approximation. In this talk, we present a new exact solution obtained by solving the fully nonlinear P-B equation directly using the Hirota bilinear method, without invoking the D-H approximation. This new solution is anti-symmetric about the centreline of two parallel boundaries, representing the case of a microchannel with oppositely charged walls. The electric potentials and velocity fields derived from both the complete and linearized P-B equations are compared. Significant deviations are revealed, in particular for cases with high zeta potential. We further demonstrate that the unique bi-directional flow profile of the electroosmotic flow generated under this anti-symmetric wall potential can be modified by varying the boundary slip on channel walls. These results will be useful in characterizing bi-directional electroosmotic flows in various novel applications.