Analytical solutions to the linearized poisson—boltzmann equation in cylindrical coordinates for different ionic—strength distributions

Abstract

For the system of a capillary membrane between two electrolyte solutions, we derive the complete analytical solution to the linearized Poisson-Boltzmann equation in cylindrical coordinates. By separating the total electric potential into a part that depends only on the axial coordinate and another part that depends on both the axial and radial coordinates, i.e., ψ( z, r) = ψ 1( z) + ψ 2( z, r), we obtain two independent differential equations. The equation for ψ 2 is readily solved, but that for ψ 1 requires explicit knowledge of the axial dependence of the ionic strength. Since this cannot be determined a priori, we write a generalized distribution that can describe virtually any monotonically increasing ionic strength. Boundary conditions in the z-direction have always been problematic for systems like the one under consideration, and we develop boundary conditions for ψ 1 from a charge balance at each face of the membrane between the bulk solution and the solution within the pore. From these analytical solutions of ψ 1 and ψ 2, we also derive an equation for the total membrane potential of the system described.