A New Poisson-Nernst-Planck Equation (PNP-FS-IF) for Charge Inversion Near Walls

Abstract

The plasmas of biology are interacting mixtures of ions - often charged spheres - that do not behave like the ideal solutions of textbooks. Interactions are always present because of strong electrical forces. Flows are usually present. Life without flow is death. We analyze ionic solutions as complex fluids with an approach that has successfully analyzed complex systems like liquid crystals that are dominated by interactions between composite components. The finite size of ions is particularly important in biology in crowded environments like channels, active sites of enzymes, or charged surfaces. We here deal with surfaces and try to capture the essential features of charge inversion (layering) near a charged wall. Charge inversion (layering) near walls is a characteristic phenomenon resulting from the electrostatic interactions in systems with charged walls. The mathematical model is derived by the energy variational approach (EnVarA) _ J.Chem.Phys. (2010) 133:104104 _ that combines the action of conservative (Hamiltonian) systems and the dissipation of Onsager and Rayleigh. Both are written in the same laboratory coordinates after variational derivatives of variables are taken. The generalized energy and dissipation include entropic and electrostatic components, and repulsion between spheres. An interfacial electroneutrality constraint between bulk and charged wall captures some essential features of charge inversion. Taking variational derivatives yields a field theory of partial differential equations and boundary conditions that are appropriate for life's solutions _ that interact and flow _ as well as thermodynamic equilibrium. The new equations, PNP-FS-IF, include (1) a nonlocal contribution of finite size (FS) and (2) an interfacial constraint (IF) of electroneutrality. PNP-FS-IF produces charge inversion near walls. We compare the charge inversion seen with PNP-FS-IF and Monte-Carlo simulations.