Electrostatic analysis of the charged surface in a solution via the finite element method: The Poisson-Boltzmann theory

Electrostatic potential as well as the local volume charge density are computed for a macromolecule by solving the Poisson-Boltzmann equation (PBE) using the finite element method (FEM). As a verification, our numerical results for a one dimensional PBE, which corresponds to an infinite-length macromolecule, are compared with the existing analytical solution and good agreement is found. As a macromolecule has a rod-like shape with a finite length, a much more real case is considered, which leads to a two dimensional PBE. Furthermore, it is demonstrated that the potential and charge density decrease as the distance from the axis of the macromolecule increases. Moreover, it is concluded that the absolute value of the electrostatic field obtained from the nonlinear PBE subject to the boundary condition with a fixed charge differs from that of the linear PBE at fixed potential by an order of magnitude in the vicinity of the finite rod-like macromolecule. On the other hand, excellent agreement is observed between the electric fields calculated from the aforementioned equations at far distances.