Title: From Poisson-Boltzmann Theory to Poisson-Nernst-Planck Theory
Abstract: Electrolyte (water with salt dissolved to free-to-move ions) and electrodiffusion are fundamental to our understanding of natures and to our design of technology. The Poisson-Boltzmann (PB) theory has played a crucial role in electro-chemistry (or physics-chemistry). Some of its highlights are the double-layer structure near charged surface (Gouy-Chapman theory), the charge screening (Debye-H\"uckel theory and its extensions), the charge inversion, etc. These successes of the PB theory relied on the basic mathematical theory of Poisson (elliptic) equations. For ionic flow through ion channels, the Poisson-Nernst-Planck (PNP) theory was developed but the further specifics of ion channel problems prevent direct applications of PB theory and its mathematical theory -- new mathematical theory is required to study PNP models. In this talk, we will briefly review the new mathematical approach -- a geometric singular perturbation framework together with crucial intrinsic structures of ion channel problems. A number of applications to ion channel properties will be discussed, some are not so intuitive.