Analytical solution of a linear nonlocal Poisson-Boltzmann equation with multiple charges in a spherical solute...

In this paper, an analytical solution of a linear nonlocal Poisson-Boltzmann equation (NPBE) test model with multiple charges in a spherical solute region surrounded by a water spherical shell is derived as a single series of Legendre polynomials and modified spherical Bessel functions. The classic Kirkwood ball model is then shown to be a special case of the NPBE test model so that its analytical solution is regained from a double series of associated Legendre polynomials (derived by Kirkwood in 1934) to a new single series of Legendre polynomials, sharply reducing its computational cost. As an application of these series solutions, a comparison study is done to demonstrate the differences between the Kirkwood and NPBE test models.