An Algorithm for the Calculation of the Electrical Potential Distribution in a Charged Capillary with General...

Knowledge of the electrical potential distribution is an essential basis for analyzing the flow behavior of electrolytes in a charged capillary, such as electroviscous effects. The cylindrical Poisson-Boltzmann equation (PBE) governs the distribution in the capillary. The PBE is a differential equation that is difficult to solve analytically, especially when the capillary is filled with general electrolytes. In this paper we propose a numerical algorithm to obtain the electrical potential distribution in a charged capillary filled with arbitrary electrolytes. First, we introduce a Poisson-Boltzmann integral equation (PBIE) governing the potential distribution and derived from the physical principles for electrostatic fields and thermodynamic systems. Then we solve the PBIE numerically by iteration. In numerical calculation only the discrete potential is used, and the potential differentials of the first and higher orders are not required. This algorithm essentially removes the difficulty caused by very steep variation of the potential near the wall of the capillary and is easily extended to cover the more general case of arbitrary electrolytes. The results of the examples given in the paper show that the algorithm proposed here is correct, effective, accurate (the relative errors are less than 0.01% when normalized surface potential xi