Numerical solution of the Poisson–Boltzmann equation for suspended charged particles surrounded by a charged permeable membrane

The equilibrium properties of a charged spherical particle surrounded by a charged permeable membrane and immersed in electrolyte solutions have been investigated using the network simulation method. A network model for the nonlinear Poisson–Boltzmann equation describing the electrostatic potential distribution for this system has been proposed. With this model and an electric circuit simulation program, a series of functions such as the electric potential, the ion concentrations, and the charge densities inside the membrane and in the electrolyte solution can be easily obtained for arbitrary values of the particle radius, particle charge, membrane thickness, membrane charge, and ionic concentrations and charge number. The method proves to be quite general and extremely efficient, as well as applicable to a great variety of double layer compositions.