(untitled) (Communications in Computational Physics, 2022)
In this paper we propose a computational framework for the investigation of the correlated motion between positive and negative ions exposed to the attraction of a bubble surface that mimics the (oscillating) cell membrane. Specifically we aim to investigate the role of surface traps with substances freely diffusing around the cell. The physical system we want to model is an anchored gas drop submitted to a diffusive flow of charged surfactants (ions). When the diffusing surfactants meet the surface of the bubble, they are reversibly adsorbed and their local concentration is accurately measured. The correlated diffusion of surfactants is described by a Poisson-Nernst-Planck (PNP) system, in which the drift term is given by the gradient of a potential which includes both the effect of the bubble and the Coulomb interaction between the carriers. The latter term is obtained from the solution of a self-consistent Poisson equation. For very short Debye lengths one can adopt the so called Quasi-Neutral limit which drastically simplifies the system, thus allowing for much faster numerical simulations. The paper has four main objectives. The first one is to present a PNP model that describes ion charges in presence of a trap. The second one is to provide benchmark tests for the validation of simplified multiscale models under current development [1]. The third one is to explore the relevance of the term describing the interaction among the apolar tails of the anions. The last one is to quantitatively explore the validity of the Quasi-Neutral limit by comparison with detailed numerical simulation for smaller and smaller Debye lengths. In order to reach these goals, we propose a simple and efficient Alternate Direction Implicit method for the numerical solution of the non-linear PNP system, which guarantees second order accuracy both in space and time, without requiring solution of nonlinear equation at each time step. New semi-implicit scheme for a simplified PNP system near quasi neutrality is also proposed.