Minimization of Gibbs free energy for Poisson-Boltzmann problem

The study of the electrostatic interactions between charged structures and ions is crucial in understanding the behavior of soft matter, in particular living matter. Biological systems typically consist of a multitude of charged structures. Membranes, cellular components, globular proteins, polyelectrolytes, DNA, and polystyrene sulfonate are examples of biopolymers having charges and, consequently, an electrical behavior. These charged structures are placed in biological media (for example blood) containing free ions in addition to water molecules. In this work, we consider an ionic solution containing several types of spherical ions between two uniformly charged parallel planes. Each ion in the solution is moving with a molecular motion influenced by the electric forces of the charged planes and the other ions. It is of interest to determine the ions concentration at the electrostatic and thermal equilibrium (Poisson-Boltzmann problem). The ions concentration at the equilibrium are the concentrations minimizing the Gibbs free energy of the system. The corresponding minimization problem is infinite-dimensional and so it is discretized into a finite-dimensional constrained nonlinear programming problem, which is then solved using a standard optimization solver. A convergence analysis of the discrete minimum to the continuous minimum is performed.