Generalized Poisson-Boltzmann Equation In Fractional Dimension

Abstract

Recently, fractional dimensional calculus is widely applied in areas including nonlinear dynamics, electron transport, infectious disease spread, and economies theory, to capture the non-locality or spatial complexity of the physical system. This has motivated us to study the electrostatic potential distribution of an electrical-double-layer (EDL) in fractional dimensional space. In this work, we employ Stillinger’s non-integer Laplacian operator [1] to generalize a modified Poisson-Boltzmann (MPB) model by Borukhov [2] into fractional dimension. The fractional dimension provides a simplified effective model of non-uniformity caused by impurities or inhomogeneity of the electrolyte solution. Numerical method has been developed to solve the fractional MPB model, and a modified Grahame equation in fractional dimension is derived to link the potential profile to surface charge density. It is found that the potential profile decays slowly as compared to its full-dimensional counterpart, which suggests a wider saturated layer under the same surface potential. Under constant surface charge density, the surface potential is reduced in fractional space while the distribution of the counter-ions does not show significant changes. Such findings offer a potential theoretical framework to model the electrostatic interaction of ions in realistic EDLs which contain spatial complexities that might distort the distribution of the ions in any systems governed by the coupled Poisson-Boltzmann equation.