# A Boltzmann Electron Drift Diffusion Model for Atmospheric Pressure Non-Thermal Plasma Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Ions and Neutrals—Drift Diffusion Approach

#### 2.2. Finite Volume Discretization

#### 2.3. Electrons—Poisson–Boltzmann Problem

#### 2.4. Charge Conservation

Algorithm 1: Non-linear Poisson solver with global charge conservation |

Algorithm 2: Iterative search of reference electric potential |

## 3. Simulation Results

#### 3.1. Simulation Settings

^{2}surface. The configuration is powered by a 15 $\mathrm{k}$$\mathrm{Hz}$ sinusoidal voltage with amplitude $4.8$ $\mathrm{k}$$\mathrm{V}$.

_{3}formation chain; see Table A1. The considered heavy species are N

_{2}

^{+}, O

_{2}

^{+}, O

_{2}

^{−}, O, O

^{−}, and O

_{3}. As introduced in Section 2, electrons may or may not be accounted for in the drift diffusion model, depending on the electron model chosen by the user. A numerical validation of the implemented semi-implicit approach for the source term time integration is provided in Appendix A.

_{2}

^{+}and O

_{2}

^{+}is set to 3:1. The O

_{2}

^{−}is set to a 1:8 ratio with respect to N

_{2}

^{+}and the electron density is selected to ensure overall electric neutrality. The macroscopic transport parameters for the considered species have been taken from [54].

#### 3.2. Electron Models Comparison

_{2}

^{+}number density at the right edge of the domain (cathodic side) are ${{\mathsf{N}}_{2}}_{\mathrm{FDD}}^{+}=2.43\times {10}^{18}{\mathrm{m}}^{-3}$ and ${{\mathsf{N}}_{2}}_{\mathrm{BDD}}^{+}=2.24\times {10}^{18}{\mathrm{m}}^{-3}$. Similarly, the electron number density values at the left edge (anodic side) of the gap are ${\mathsf{e}}_{\mathrm{FDD}}^{-}=1.44\times {10}^{17}{\mathrm{m}}^{-3}$ and ${\mathsf{e}}_{\mathrm{BDD}}^{-}=1.41\times {10}^{17}{\mathrm{m}}^{-3}$. This agreement is important because the number densities at the two edges of the gap are several orders of magnitude larger than in the bulk for both considered species. This means that, given the dependence of the reaction rates on the number density of the reactants, the largest physical contributions from kinetic processes will likely be generated in these regions. In addition, the charged species fluxes directed towards the walls, responsible for the surface charge accumulation process, are computed using the number density in the CVs shared between the dielectric layers and the gap. Therefore, the discussed agreement between the computed number densities at the edges of the domain is consistent with the compatibility shown by the trends in surface charge over time in Figure 4. In order to have similar incident wall fluxes, a similar electric field at the gap edges must also be present. The right axis in Figure 5 shows the electric potential obtained using the two methodologies. The value yielded by the BDD approach (which depends on the reference electric potential ${\varphi}_{0}$) has been shifted by a constant value of 410 $\mathrm{V}$ to allow comparison to ${\phi}_{\mathrm{FDD}}$. The two obtained electric potentials are very close throughout the whole gap, meaning that the two electric fields will also be quite similar to each other.

#### Computational Performance

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Numerical Validation of the Semi-Implicit Source Term Integrator

Process | Reactants | Product(s) | Source | |
---|---|---|---|---|

Ionization | N_{2} + e^{−} | → | N_{2}^{+} + 2e^{−} | [54] |

O_{2} + e^{−} | → | O_{2}^{+} + 2e^{−} | [54] | |

Recombination | N_{2}^{+} + e^{−} | → | N_{2} | [54] |

O_{2}^{+} + e^{−} | → | O_{2} | [54] | |

N_{2}^{+} + O_{2}^{−} | → | N_{2} + O_{2} | [54] | |

O_{2}^{+} + O_{2}^{−} | → | 2O_{2} | [54] | |

N_{2} + N_{2}^{+} + O_{2}^{−} | → | 2N_{2} + O_{2} | [54] | |

N_{2} + O_{2}^{+} + O_{2}^{−} | → | N_{2} + 2O_{2} | [54] | |

O_{2} + N_{2}^{+} + O_{2}^{−} | → | N_{2} + O_{2} + O_{2} e | [54] | |

O_{2} + O_{2}+ + O_{2}^{−} | → | O_{2} + O_{2} + O_{2} | [54] | |

Attachment | N_{2} + O_{2} + e^{−} | → | N_{2} + O_{2}^{−} | [54] |

O_{2} + O_{2} + e^{−} | → | O_{2} + O_{2}^{−} | [54] | |

O_{2} + O + e^{−} | → | O_{2} + O^{−} | [56] | |

O_{3} + e^{−} | → | O_{2} + O^{−} | [56] | |

O_{3} + e^{−} | → | O_{2}^{−} + O | [56] | |

Detachment | O_{2} + O_{2}^{−} | → | O_{2} + O_{2} + e^{−} | [54] |

O_{2} + O^{−} | → | O_{3} + e^{−} | [56] | |

Dissociation | O_{2} + e^{−} | → | O + O + e^{−} | [56] |

O_{3} + e^{−} | → | O_{2} + O + e^{−} | [56] | |

O_{3} formation | O + O_{2} + N_{2} | → | O_{3} + N_{2} | [56] |

O + O_{2} + O_{2} | → | O_{3} + O_{2} | [56] |

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**Figure 1.**Solution of the non-linear Poisson problem (12) over a 5 $\mathrm{m}$$\mathrm{m}$ domain, with a uniform ions number density ${\phi}_{0}=0\mathrm{V}$ and boundary conditions ${\phi}_{L}=5\mathrm{V}$ and ${\phi}_{R}=-5\mathrm{V}$.

**Figure 2.**Total electric charge dependence from the employed reference electric potential; the target value of ${Q}_{t}$ meeting the required charge neutrality condition, highlighted.

**Figure 3.**Simulation of the volumetric dielectric barrier discharge reactor with two different numerical methodologies; comparison between the gap voltage obtained with the full drift diffusion (FDD) and Boltzmann drift diffusion (BDD) approaches.

**Figure 4.**Surface charge density deposited onto dielectric layers I and II over two cycles of the externally applied voltage; comparison between the full drift diffusion (FDD, black line) and Boltzmann Drift diffusion (BDD, red line) approaches.

**Figure 5.**Spatial distribution of N

_{2}

^{+}and e

^{−}(left) and electric potential (right) yielded by the full drift diffusion (FDD, solid lines) and Boltzmann drift diffusion (BDD, dashed lines) approaches at $\tau =57\mu \mathrm{s}$.

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## Share and Cite

**MDPI and ACS Style**

Popoli, A.; Ragazzi, F.; Pierotti, G.; Neretti, G.; Cristofolini, A.
A Boltzmann Electron Drift Diffusion Model for Atmospheric Pressure Non-Thermal Plasma Simulations. *Plasma* **2023**, *6*, 393-407.
https://doi.org/10.3390/plasma6030027

**AMA Style**

Popoli A, Ragazzi F, Pierotti G, Neretti G, Cristofolini A.
A Boltzmann Electron Drift Diffusion Model for Atmospheric Pressure Non-Thermal Plasma Simulations. *Plasma*. 2023; 6(3):393-407.
https://doi.org/10.3390/plasma6030027

**Chicago/Turabian Style**

Popoli, Arturo, Fabio Ragazzi, Giacomo Pierotti, Gabriele Neretti, and Andrea Cristofolini.
2023. "A Boltzmann Electron Drift Diffusion Model for Atmospheric Pressure Non-Thermal Plasma Simulations" *Plasma* 6, no. 3: 393-407.
https://doi.org/10.3390/plasma6030027