Existence and Uniqueness of the 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. The Poisson-Boltzmann (PB) problem is central to understanding electrostatic interactions in biological and soft matter systems, such as cellular membranes, DNA, and colloidal suspensions. These interactions govern phenomena like like-charge attraction, ion distribution, and molecular self-assembly, which are critical for biological processes and nanotechnology applications. Progress in understanding these interactions is crucial for the advancement in fields like biophysics and engineering (e.g., drug delivery systems, biosensors, etc). In this work, we consider an ionic solution containing several types of spherical ions between two uniformly charged parallel planes separated by a distance D. 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 concentration minimizing the Gibbs free energy of the system. In this work, we minimize the Gibbs free energy directly, so ading the complexities of traditional variational approaches and offering a more efficient numerical framework to study ion equilibrium concentrations. The Gibbs free energy of a general thermodynamic system is, essentially, the maximum amount of work that can be performed by the system. The resulting minimization problem is infinite-dimensional, and so it is discretized into a finite-dimensional constrained nonlinear programming problem, using either a pseudo-spectral or a finite-element approach. In the pseudo-spectral method, the ion concentrations are approximated by Lagrange interpolation polynomials at Chebyshev nodes, ensuring high accuracy for smooth solutions. Alternatively, the finite-element method divides the domain into subintervals and uses piecewise polynomial basis functions offering flexibility for non-uniform geometries. We solve the finite-dimensional discrete minimization problem arising from the discretization of the Gibbs free energy functional through interior point methods. Finally, we perform a convergence analysis of the discrete minimum to the continuous minimum.