Numerical Solution to the Poisson-Boltzmann Equation for Spherical Polyelectrolyte Molecules

Abstract

The Poisson‐Boltzmann equation for a penetrable spherical polyelectrolyte ion in the presence of smaller ions has been solved precisely by numerical integration methods using a high‐speed computer. The polyelectrolyte ion was assumed in one case to be immersed in a sea of simple electrolyte and in another case to be surrounded by an appropriate volume of solution containing only the polyelectrolyte's counter ions. Since the absolute value of the electrostatic potential energy of a small ion in the center of a polyelectrolyte ion is found to be several times kT, the nonlinear differential equation cannot justifiably be rendered linear by the Debye‐Hückel type of approximation. The fraction of the counter ions held within the sphere of the polyelectrolyte ion is calculated from the solution to the differential equation. The extent of such ion association is found, for reasonable values of the molecular parameters, to range between 50 and 75% in substantial agreement with experimental observations based on transference studies. The results of the numerical computations are for the most part presented graphically.