Electrostatic potentials and fields in the vicinity of engineered nanostructures

Abstract

We have developed a method for calculating the electrostatic potentials and fields in the vicinity of geometrically complex engineered nanostructures composed of varying materials in electrolytes of arbitrary pH and ionic strength. The method involves direct summation of charged Debye–Hückel spheres composing the nanostructural surfaces and, by including charge redistribution on the surface of conducting materials held at constant potential, is applicable to mixed boundary conditions. The method is validated by comparison to analytical solutions for an infinite plane (Gouy–Chapman), an infinite cylinder (Bessel functions), and an infinite plane which contains a hole and which is held at constant potential. Excellent agreement between the potentials obtained by our numerical method and the closed form solutions is found for these conditions. The method is applied to the calculation of the electric field enhancement in the vicinity of a nanomembrane whose pore wall is held at constant charge and whose membrane surfaces are held at constant potential. The electric field is found to be enhanced by the charge buildup in the rim of the hole of the nanomembrane; the buildup results from the potential being held constant in the conducting region. Ion concentrations are also calculated. Positive ion rejection is found to be enhanced by this charge buildup in the region of the rim when a constant positive potential is applied.