Finite element approximations to a fourth-order modified Poisson-Fermi equation for electrostatic correlations in concentrated electrolytes

Abstract

In this paper, both the standard finite element method (FEM) and the mixed FEM are developed for the fourth-order modified Poisson-Fermi equation resulted from the Bazant-Storey-Kornyshev (BSK) theory to account for electrostatic correlations in concentrated electrolytes. Optimal convergence properties are obtained for both FEMs in their respective norms, additionally, the mixed FEM can produce one-order higher approximation accuracy for the electric field in L2 norm than that of the standard FEM, resulting in more accurate approximations to the electrostatic stress and the force of interaction in concentrated electrolytes. All attained theoretical results are validated by numerical experiments. Furthermore, a practical example is studied to validate the necessity of introducing the fourth-order modified Poisson-Fermi equation to describe the electrostatic potential field due to correlation effects by comparing with the classical Poisson equation that accounts for the classical mean-field Poisson-Boltzmann (PB) theory, illustrating that numerical trends obtained from the modified model are all consistent with experimental results under the circumstance of electrostatic correlations in concentrated electrolytes, which however cannot be predicted by the classical model.