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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.

Analytical Solutions to the Steady State Poisson-Nernst-Planck Equations in Electrobiochemical Systems