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

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.


Introduction
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 (K.Krabbenhøft, & J. Krabbenhøft, 2008).In their most general form (unsteady, 3-D), the PNP equations are usually solved numerically.Nevertheless, analytical and approximate closed form solutions to benchmark problems can be useful in validating numerical solutions, and in enabling a fundamental understanding of the influence of experimental parameters on ionic diffusion.
Exact analytical solutions to the PNP equations have been obtained for various situations.In addition, approximate solutions using singular perturbation analyses and numerical solutions have been also been reported.These analytical and approximate solutions can be broadly categorized as considering (1) 1-D, steady-state, two species (Barcilon, Chen, Eisenberg, & Jerome, 1997;Golovnev & Trimper, 2010;Liu, 2005;Singer, Gillespie, Norbury, & Eisenberg, 2008), (2) 1-D, unsteady, two species (Golovnev & Trimper, 2011), (3) 1-D, unsteady, single species (Schönke, 2012), and (4) 2-D and 3-D, unsteady, single species cases (Schönke, 2012).Moreover, the analytical solutions obtained for these different cases vary in the problem formulation as far as boundary conditions, scaling, and assumptions are concerned.For instance, elegant closed form solutions have been obtained in the case of a single ionic species for the unsteady 1-D, 2-D, and 3-D PNP equations (Schönke, 2012).For the 1-D, unsteady case with a single ionic species, the PNP equations have been shown to reduce to the scalar Burger's equation (Schönke, 2012).However, the single species PNP equations lead to incorrect values for the diffusivity inferred from migration experiments because of neglect of the need for quasi-neutrality (K.Krabbenhøft, & J. Krabbenhøft, 2008).In this paper, we consider a benchmark problem pertaining to electrobiochemical systems, specifically 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.This paper is organized as follows.The following section discusses formulation of a model 1-D, steady-state benchmark problem with two ionic species and its analytical solution.The two-species PNP equations for a model two dimensional problem, associated scaling, and their approximate solution using the method of singular perturbations are given in Section 3. A summary along with the conclusions are given in Section 4.

Poisson-Nernst-Planck Equations in One-Dimension
The PNP system for two ionic species (one a singly charged positive ion and the other a singly charged negative ion) in a linear, isotropic, homogeneous medium is given by:

Formulation of Problem and Scaling
Consider the model problem of a linear, isotropic, homogeneous, isothermal medium in which are present positive ions of number density p and negative ions of number density n.Suppose there are two electrodes immersed in the medium which are not connected to an external circuit.Such a situation can arise for example under open circuit conditions in a battery (Torabi & Aliakbar, 2012;Venkatraman & Van Zee, 2007), or in a 1-D analog of a bioelectric dressing in contact with water or wound exudate (Banerjee et al., 2014).Furthermore, suppose that there is a constant generation of species n at one of the electrodes, such as the generation of OH - ions at the oxide surface of a silver oxide electrode (Torabi & Aliakbar, 2012).The governing equations (1-3) apply to this case, re-written by relating the mobilities to the ion diffusivities via the Einstein relation eZDkT µ= , where 1 Z=+ for the positive ions and 1 Z=− for the negative ions: where . For the case of a bioelectric dressing, L ref ~ 1 mm, D OH-~ 4.56 x 10 -3 mm 2 /s (Lee & Rasaiah, 2011), and t ref >> 10 3 s so that Π 1 << 1 as is Π 3 .Moreover, φ ref ~ 1 V and T ~ 300 K so that Π 2 ~ O(10).Therefore, for the problems of interest, the PNP equations reduce to their steady state form: Equations ( 10) and ( 11) may be added and subtracted separately, yielding: where P = p + n and N = pn.Note that if quasi-neutrality prevails, N=0, i.e. p=n, resulting in the familiar ambipolar diffusion equation for species concentration with zero electric field and with equations (13-15) identically satisfied.
There are two possible choices for φ ref .
If we set φ ref =kT/e, then Π 2 =1 and Π = ⁄ , where is the Debye length.It should be mentioned here that the appearance of the Debye length in this analysis agrees with previous studies that also naturally led to the rise of the Debye length quantity in non-dimensionalization (Hrdlička, Červenka, Přibyl, & Šnita, 2010).This case has already been considered in the literature for ion channels in membranes and an approximate solution using singular perturbation methods has been obtained (Liu, 2005;Singer, Gillespie, Norbury, & Eisenberg, 2008).In the present case of a bioelectric dressing however, φ ref is taken to be specified, e.g. the open circuit potential difference across the silver oxide and zinc dots in the bioelectric dressing is on the order of 0.1 V (Banerjee et al., 2014), and L ref is the spacing between the electrodes which is on the order of millimeters in the case of the bioelectric dressing.Thus, Π = ~4 ⁄ for φ ref = 0.1 V and T = 300 K, and Π = ~10 for L ref = 1 mm, ε ~ 80x10 -12 F/m, and n ref ~ 10 20 m -3 .The latter values for ε and n ref are based on the medium being physiological saline (pH~7.4)from which we obtain n OH -~10 20 m -3 and n H + ~10 19 m -3 .The solution for this case is given next.

Poisson-Nernst-Planck Equations in Two Dimensions
The PNP equations in two dimensions for two ionic species with Z=+1 and Z=-1 can be written as: where the Einstein relations relating the ionic mobilities to the diffusivities have been used and the diffusivities and temperature are in turn taken to be constant.Using the same reference quantities for non-dimensionalization as described in Section 2.1.1 with the exception of L x and L y as the new characteristic length scales in the x and y directions, the PNP equations reduce to the following non-dimensional steady state form for the bioelectric dressing: where As in the case of the 1-D PNP equations, equations ( 24) and ( 25) may be added and subtracted to yield: where P=p+n N=p-n.These equations arise in the model problem described in the following section.

Conclusions
In this paper, 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.These model problems can serve to benchmark and validate numerical solutions for more complex problems.The analytical solutions are obtained for two ionic species in both one and two dimensional systems which, to the best of our knowledge, have not been investigated in the literature.Simple closed form expressions are obtained in the one-dimensional case while an approximate solution using the method of singular perturbations is obtained for the two-dimensional model problem.The analytical solutions presented here are also shown to yield good agreement with numerical solutions of the corresponding PNP equations.The analytical solutions presented here may enable interpretation of experimental data for electrobiochemical systems.
Introducing a reference length scale L ref , reference time scale t ref , reference number density n ref , and reference potential φ ref , these equations can be non-dimensionalized to yield:

Figure 1 .
Figure 1.The numerical solution for N(x) resulting from solving the full PNP system (Equations 16-18).This solution can be compared to the analytical approximation of ≈

Figure 5 .Figure 6 .
Figure5.Numerical solution for the two dimensional PNP system of the system of equations (30-32) for the case of f(x)=x and = the error in potential as a function of x and y x absolute value of (Phi(numerical)-Phi(analytical)