Abstract
The Poisson-Nernst-Planck equations are relevant in numerous electrobiochemical applications. In this paper, we provide analytical solutions to the steady state Poisson-Nernst-Planck (PNP) systems of equations for situations relevant to applications involving bioelectric dressings and bandages. The PNP system of equations is analyzed for two ionic species (one positively charged and the other negatively charged) both in the one-dimensional and two dimensional cases. The equations were formulated, non-dimensionalized, and an order of magnitude analysis was performed. Additionally, the method of singular perturbations was utilized in the two dimensional case. In the one-dimensional case, an exact solution is obtained while in the two-dimensional case an asymptotic solution is obtained. Both analytical solutions are compared with numerical solutions of the equations, and exhibit good agreement. The analytical solutions for the benchmark problems presented here are useful for verifying numerical solutions to more complex problems, and may also enable simple interpretation of experimental data for electrobiochemical systems.
Highlights
In many electrochemical and electrobiochemical applications, the continuum governing equations describing the concentrations of ionic species and their transport under the action of electric fields are the Poisson-Nernst-Planck (PNP) Equations
Despite the fact that these equations have been well known and studied for over a century, they remain of current interest because of applications ranging from batteries (Torabi & Aliakbar, 2012; Venkatraman & Van Zee, 2007), bioelectric dressings used in wound healing (Banerjee et al, 2014), diffusion of charged species through ion channels in cell membranes and ion selective membranes (Coalson & Kurnikova, 2005; Fíla & Bouzek, 2003), electro-osmosis in micro and nano-channel systems (Hrdlička, Červenka, Přibyl, & Šnita, 2010), transport of charge carriers in semiconductors (Markowich, 1986), ionic current rectification through charged micro and nano channels (Chein & Chung, 2013) to as diverse a field as deterioration of reinforced concrete structures due to diffusion and attack by the chloride ion
Analytical solutions for the steady state Poisson-Nernst-Planck equations in one and two dimensions have been given for model problems in the area of electrobiochemical systems such as bioelectric dressings used in would healing
Summary
In many electrochemical and electrobiochemical applications, the continuum governing equations describing the concentrations of ionic species and their transport under the action of electric fields are the Poisson-Nernst-Planck (PNP) Equations. We consider a benchmark problem pertaining to electrobiochemical systems, bioelectric dressings used in wound healing (Banerjee et al, 2014), which involve the two-species, steady-state PNP equations in 1-D and 2D. Approximate solutions to these equations are obtained using order of magnitude analysis and the singular perturbation method for the 2-D case. The following section discusses formulation of a model 1-D, steady-state benchmark problem with two ionic species and its analytical solution.
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