Modeling of transport phenomena of ions and polarizable molecules: A generalized Poisson–Nernst–Planck theory

A continuum theory of studying transport phenomena of ions and polarizable molecules in multi-component systems in the presence of electromagnetic fields is developed in this article. Physical effects of mass diffusion, heat conduction, mechanical motion, polarization and polarization relaxation in the mixture of the multi-component system, including their coupling effects, are formulated in accordance with the basic laws in non-equilibrium thermodynamics and continuum electrodynamics. A generalized Poisson–Nernst–Planck theory is introduced as a special case where thermo-mechanical effects are negligible. It is shown that electrodiffusion processes of mobile ions and polarizable molecules are generally coupled among diffusion fluxes as well as with the polarization of the molecules. For time-varying fields, the polarization relaxation of the molecules may also affect the electrodiffusion process. It is shown that the classical Poisson–Nernst–Planck theory can be recovered if these coupling effects are ignored in cases where such a simplification is justified. The generalized PNP theory formulated in the article may therefore offer a theoretical means to investigate the polarization–diffusion coupling effects of potential importance in the electrodiffusion processes of mobile ions and polarizable molecules at electrostatic cases as well as in time-varying fields, which could be of particular interest for revealing possible effects of exogenous time-varying fields on the ion transport properties and related functions of living cells. In general, the formulated theory may also be used to analyze some transport phenomena of nano-drug carriers in drug delivery systems as well as other molecular transport problems in engineering applications. While the theory is generally nonlinear, the linearization of the theory is possible in some cases. Illustratively, a coupled electrodiffusion wave problem is analyzed with a linearized model and its solution is discussed.