General solution for Poisson-Boltzmann equation in semiinfinite planar symmetry.

A unique, simple, and general analytical solution for the nonlinear Poisson-Boltzmann equation in semiinfinite planar symmetry with any unmixed electrolyte is reported. This is an exact solution for symmetrical and z-2z/2z-z asymmetrical electrolytes and an approximate solution for other asymmetrical electrolytes (z-3z/3z-z or 2z-3z/3z-2z). To evaluate the accuracy of the approximate solution, the solution was compared with the numerical results obtained from the Galerkin weighted residual finite element method. The error of the approximate solution for asymmetrical z-3z/3z-z or 2z-3z/3z-2z electrolytes is not more than 1%. This new solution unifies all the cases in semiinfinite planar symmetry with any unmixed electrolyte into one expression without increasing the mathematical complexity. The general solution can serve as a functional approximation for potential distribution of two interacting electric double layers when calculating electrostatic interaction energy if the superposition approximation is used.