Applied Science and Convergence Technology 2023; 32(3): 69-72
Published online May 30, 2023
Copyright © The Korean Vacuum Society.
School of Electrical Engineering, Chungbuk National University, Chungbuk 28644, Republic of Korea
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (https://creativecommons.org/licenses/by-nc-nd/4.0/) which permits non-commercial use, distribution and reproduction in any medium without alteration, provided that the original work is properly cited.
The Poisson-Boltzmann equation is a well-known nonlinear differential equation that is fundamental in plasma theory. However, since it has nonlinear characteristics, the conditions and situations for obtaining analytical solutions are limited. Therefore, research and development on ways to obtain solution through numerical analysis methods have been continued. In this paper, a numerical method using the Lambert W function to solve the nonlinear Poisson-Boltzmann equation is explained. To investigate the applicability of the current method, three cases were assumed: a uniform ion density, the collisionless sheath of cold ion plasma and a nonuniform ion density. Solutions can also obtained even if the density of the ions is given as a function of the potential. In this case, the iterative steps can be integrated and simplified further.
Keywords: Poisson-Boltzmann equation, Plasma, Numerical analysis, Lambert function
In plasma theory, electron density distribution is often expressed by the Boltzmann equation as
If the electron density is expressed as Eq. (1) and an ion density is given then the Poisson equation becomes the following form of the Poisson-Boltzmann equation 
The Poisson-Boltzmann equation is a nonlinear differential equation, and the situations and conditions that can obtain an analytical solution are limited . Therefore, there are many cases where the solution is obtained by transforming it into a numerical approximation formula such as a finite difference form. In particular, an analytical solution cannot be expected when the density of ions is given as only numerical values. The Poisson-Boltzmann equation, transformed into a numerical approximation, becomes a matrix equation, and the solution is usually obtained through an iteration method such as the Newton-Raphson method [5,6]. However, as is well known, the Newton-Raphson method has several failure factors such as overshoot, stational point, fore initial estimate, and so on. Of course, programs exist and used as a result of solving such problems, but research and development are continuing to find new methods.
This research proposes a method to obtain the solution of the Poisson- Boltzmann equation using the Lambert
In the subsequent section, details of the new method will be explained and the results applied to several situations will be analyzed and discussed.
The Poisson-Boltzmann equation, Eq. (2), can be normalized by thermal potential
In order to deal with Eq. (3) numerically, an approximated equation will be obtained through various numerical methods such as the finite difference method (FDM), the finite element method, the finite volume method, and so on. The present method is not applicable only to the FDM. However, it will be explained based on the FDM for simplicity.
The finite difference form of Eq. (3) becomes
As the next step, the Newton-Raphson method can be adopted for linearization of iteration process. And then, a matrix inversion or a method for linear systems will be used to obtain the solution of Eq. (7).
In this study, we develop a semi-implicit method that assumes only the center value as the implicit term as
The values in the next step are then determined through the Lambert function
The current semi-implicit Lambert function method is very simple and easy to deal with because it does not require a matrix inversion or a numerical solution of linear system. Furthermore, the linearization approximation, which is used in the Newton-Raphson method, is not necessary.
The initial values of ϕ is determined on the assumption that the electron density equals the ion density as
Also, as can be seen in Eq. (9), an overflow may occur due to an argument in which the Lambert function contains an exponential function term. In that case, it can be solved through the following asymptotic form of the Lambert function.
Three cases were assumed to investigate the applicability of the current method: an uniform ion density, the collisionless sheath of cold ion plasma and a nonuniform ion density.
Figure 1 shows the results of assuming a spatially uniform ion density. The assumed conditions in this calculation are as follows:
Analyzing the numerical stability conditions of the current method in a general range is not easy because of the term of the Lambert function of the exponential function which can be seen in Eq. (9). However, the solution was stably obtained under wide range of condition variables.
The boundary condition inside the plasma body, that is, the boundary condition at
Even when the electrode is positively biased, mathematical solutions can be obtained stably. However, in such a case, the electron flux and electron density on the electrode surface is limited by thermal diffusion.
Within a collisionless sheath of cold ion plasma, uniform flux and ion energy conservation can be assumed . Threfore, the density of ions can be expressed as a function of the electric potential as
In this case, the ion density is given as another function of the potential rather than fixed values. Since the ion density can not be treated implicitly, two iterations are required now. If ϕ is determined by the first iteration, the next iterative processes can be attempted based on the ion density through that value. However, it is also possible to unify the two iterations into one. Therefore, the finite difference form, Eq. (8) becomes
Figure 2 shows ϕ and the normalized densities approaching a convergence and Fig. 3 is the converged densities. It shows the characteristics of a typical sheath with a reduced ion density and a very low electron density. The assumed conditions are the same as in the case of uniform ion density. The same is also true about the plasma potential
As such, the present method is not only possible if the ion density is given as some fixed values. It is possible also when the ion density is a function of electric potential. If the nonlinear dependence on the potential of ion density is not severe compared to the exponential function term of electron density, it can be treated as an explicit term. If the dependence of ion density on the potential is linear, it can be included in the process of obtaining the Lambert function result through variable substitution.
As discussed in the introduction, the Boltzmann relation of electron density means the force balance between the electric force and the pressure gradient force. So, in the case of uniform ion density, when both boundaries are grounded, ϕ becomes zero at all points. In order to generate a potential difference (or an electric field) when both boundaries are grounded, an ion density gradient must exist. Therefore, to examine such an ion density gradient effect, the following form of non-uniform ion density is assumed.
Figure 4 shows the results when different potentials were applied to the two electrodes. It can be seen that asymmetric electron densities are formed in the sheath regions. In the plasma body, ϕ becomes 0 (that is,
The floated sheath structure by an actually grounded electrode surface will be implemented under the condition that ϕ = −
A numerical method using the Lambert
In terms of the calculation speed and stability, the present method will not always be superior to other methods. In many cases, Newton’s method can ensure sufficient stability and there are methods of treating a linear system of size m with the calculation time of the order m. However, also, for certain problems, the current method may be efficient.
The present method will not only be applicable to one-dimensional problems, nor will it be based solely on the finite difference method.
Also, the Poisson-Boltzmann equation, which includes Boltzmann relation for charged particles, is not limited to cold ion plasmas, but is also used in other research areas. According to a survey by Gray and Stiles , there are more than 20 such fields in science and engineering. The current method is also expected to be applicable to certain problems in such areas.
This work was supported by a funding for the academic research program of Chungbuk National University in 2022.
The authors declare no conflicts of interest.