Electrical Potential Distribution for Multiple Charged Surfaces under a General Boundary Condition

The electrical potential distribution of a system containing multiple charged surfaces with a general boundary condition is investigated theoretically. Here, a surface can assume a constant potential/charge density, or an arbitrary combination of the two, i.e., a mixed boundary condition; the latter is of particular significance in practice. Typical example includes surfaces containing various ionizable functional groups, charge-regulated surfaces, dynamic surface conditions, and patchwise charged surfaces. A systematic iterative method is proposed for the resolution of the linearized Poisson-Boltzmann equation governing the electrical potential distribution of the system under consideration. The sufficient and necessary condition under which the method proposed is applicable is discussed. Since the coefficients in the expression for the boundary condition at surface can be an arbitrary function, the present problem is a generalized Robin problem. The conventional constant potential (Dirichlet) problem and constant surface charge (Neumann) problem can be recovered as special cases of the present model. A criterion is proposed to decide whether the separation distances between particles is appropriate for various approximate procedures, e.g., pairwise addition and linear superposition. We show that a system containing a large number of surfaces can be simulated by one which has relatively few surfaces.