. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 1 Preliminary concepts and literature review . . . . . . . . . . . . . . . 11 1.1 Cellular excitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.1 Membranes, ionic gradients, and transport . . . . . . . . . . . . . 12 1.1.2 Electrical signaling and action potentials . . . . . . . . . . ...

In this paper, inspired by the seminal work Caffarelli, Kohn, and Nirenberg [Comm. Pure Appl. Math., 35 (1982), pp. 771--831] on incompressible Navier--Stokes equation, we establish exis...

This work presents a few variational multiscale models for charge transport in complex physical, chemical and biological systems and engineering devices, such as fuel cells, solar cells, battery cells, nanofluidics, transistors and ion channels. An essential ingredient of the present models, introduced in an earlier paper (Bulletin of Mathematical Biology, 72, 1562-1622, 2010), is the use of di...

In 1887, Walther Hermann Nernst (1864-1941; Nobel Laureate 1920) began his highly successful career as an assistant to Wilhelm Ostwald (1853-1932; Nobel Laureate 1909) at the University of Leipzig. Ostwald strongly supported the ionic theory and had extended it, especially to the dissociation of weak electrolytes. By 1889, Nernst had established the principles of electrode potential, familiariz...

We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge transport in electrolytes of inhomogeneous dielectric environment. The model includes the ionic polarization due to the dielectric inhomogeneity and the ion-ion correlation. This is achieved by the self energy of test ions through solving a generalized Debye-Hückel (DH) equation. We develop numerical methods for the s...

This work proposes a new variational, thermodynamically consistent model to predict thermal electrokinetics in electric double layer capacitors (EDLCs) by using an energetic variational approach. The least action principle and maximum dissipation from the non-equilibrium thermodynamics are employed develop modified Nernst–Planck equations for non-isothermal ion transport with temperature inhomo...

In situations involving large potentials or surface charges, the Poisson-Boltzman (PB) equation has shortcomings because it neglects ion-ion interactions and steric effects. This has been widely recognized by the electrochemistry community, leading to the development of various alternative models resulting in different sets "modified PB equations," which have had at least qualitative success in...

By application of matched asymptotic expansions, a simplified partial differential equation (PDE) model for the dynamic electrochemical processes occurring in the vicinity of a membrane, as ions selectively permeate across it, is formally derived from the Poisson-Nernst-Planck equations of electrochemistry. It is demonstrated that this simplified model reduces itself, in the limit of a long thi...