# 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] |

## References

- Neretti, G.; Popoli, A.; Scaltriti, S.G.; Cristofolini, A. Real Time Power Control in a High Voltage Power Supply for Dielectric Barrier Discharge Reactors: Implementation Strategy and Load Thermal Analysis. Electronics
**2022**, 11, 1536. [Google Scholar] [CrossRef] - Kogelschatz, U. Atmospheric-pressure plasma technology. Plasma Phys. Control. Fusion
**2004**, 46, B63–B75. [Google Scholar] [CrossRef] - Monrolin, N.; Plouraboué, F. Multi-scale two-domain numerical modeling of stationary positive DC corona discharge/drift-region coupling. J. Comput. Phys.
**2021**, 443, 110517. [Google Scholar] [CrossRef] - Belinger, A.; Dap, S.; Naudé, N. Influence of the dielectric thickness on the homogeneity of a diffuse dielectric barrier discharge in air. J. Phys. D Appl. Phys.
**2022**, 55, 465201. [Google Scholar] [CrossRef] - Shang, J.S.; Huang, P.G. Surface plasma actuators modeling for flow control. Prog. Aerosp. Sci.
**2014**, 67, 29–50. [Google Scholar] [CrossRef] - Adamiak, K. Quasi-stationary modeling of the DBD plasma flow control around airfoil. Phys. Fluids
**2020**, 32, 085108. [Google Scholar] [CrossRef] - Adamiak, K. Approximate Formulations in Mean Models of Dielectric Barrier Discharge Plasma Flow Actuator. AIAA J.
**2022**, 60, 4215–4226. [Google Scholar] [CrossRef] - Li, S.; Dang, X.; Yu, X.; Abbas, G.; Zhang, Q.; Cao, L. The application of dielectric barrier discharge non-thermal plasma in VOCs abatement: A review. Chem. Eng. J.
**2020**, 388, 124275. [Google Scholar] [CrossRef] - Nayak, G.; Simeni Simeni, M.; Rosato, J.; Sadeghi, N.; Bruggeman, P. Characterization of an RF-driven argon plasma at atmospheric pressure using broadband absorption and optical emission spectroscopy. J. Appl. Phys.
**2020**, 128, 243302. [Google Scholar] [CrossRef] - Jenns, K.; Sassi, H.; Zhou, R.; Cullen, P.; Carter, D.; Mai-Prochnow, A. Inactivation of foodborne viruses: Opportunities for cold atmospheric plasma. Trends Food Sci. Technol.
**2022**, 124, 323–333. [Google Scholar] [CrossRef] - Massima Mouele, E.S.; Tijani, J.O.; Badmus, K.O.; Pereao, O.; Babajide, O.; Zhang, C.; Shao, T.; Sosnin, E.; Tarasenko, V.; Fatoba, O.O.; et al. Removal of Pharmaceutical Residues from Water and Wastewater Using Dielectric Barrier Discharge Methods—A Review. Int. J. Environ. Res. Public Health
**2021**, 18, 1683. [Google Scholar] [CrossRef] - Guo, H.; Su, Y.; Yang, X.; Wang, Y.; Li, Z.; Wu, Y.; Ren, J. Dielectric Barrier Discharge Plasma Coupled with Catalysis for Organic Wastewater Treatment: A Review. Catalysts
**2023**, 13, 10. [Google Scholar] [CrossRef] - Feizollahi, E.; Misra, N.; Roopesh, M.S. Factors influencing the antimicrobial efficacy of Dielectric Barrier Discharge (DBD) Atmospheric Cold Plasma (ACP) in food processing applications. Crit. Rev. Food Sci. Nutr.
**2021**, 61, 666–689. [Google Scholar] [CrossRef] - Seri, P.; Nici, S.; Cappelletti, M.; Scaltriti, S.G.; Popoli, A.; Cristofolini, A.; Neretti, G. Validation of an indirect nonthermal plasma sterilization process for disposable medical devices packed in blisters and cartons. Plasma Process. Polym.
**2023**, e2300012. [Google Scholar] [CrossRef] - Colonna, G.; Pintassilgo, C.D.; Pegoraro, F.; Cristofolini, A.; Popoli, A.; Neretti, G.; Gicquel, A.; Duigou, O.; Bieber, T.; Hassouni, K.; et al. Theoretical and experimental aspects of non-equilibrium plasmas in different regimes: Fundamentals and selected applications. Eur. Phys. J. D
**2021**, 75, 183. [Google Scholar] [CrossRef] - Lu, X.; Bruggeman, P.; Reuter, S.; Naidis, G.; Bogaerts, A.; Laroussi, M.; Keidar, M.; Robert, E.; Pouvesle, J.M.; Liu, D.; et al. Grand challenges in low temperature plasmas. Front. Phys.
**2022**, 10, 28–36. [Google Scholar] [CrossRef] - Roth, M.; Schollmeier, M. Ion Acceleration—Target Normal Sheath Acceleration. CERN Yellow Rep.
**2016**, 1, 231–270. [Google Scholar] [CrossRef] - Golubovskii, Y.B.; Maiorov, V.A.; Behnke, J.; Behnke, J.F. Modelling of the homogeneous barrier discharge in helium at atmospheric pressure. J. Phys. D Appl. Phys.
**2002**, 36, 39. [Google Scholar] [CrossRef] - Tochikubo, F.T.F.; Chiba, T.C.T.; Watanabe, T.W.T. Structure of Low-Frequency Helium Glow Discharge at Atmospheric Pressure between Parallel Plate Dielectric Electrodes. Jpn. J. Appl. Phys.
**1999**, 38, 5244. [Google Scholar] [CrossRef] - Becker, K.H.; Kogelschatz, U.; Schoenbach, K.; Barker, R. Non-Equilibrium Air Plasmas at Atmospheric Pressure; CRC Press: Boca Raton, FL, USA, 2004. [Google Scholar]
- Shaygani, A.; Adamiak, K. Mean model of the dielectric barrier discharge plasma actuator including photoionization. J. Phys. Appl. Phys.
**2023**, 56, 055203. [Google Scholar] [CrossRef] - Sato, S.; Shiroto, T.; Takahashi, M.; Ohnishi, N. A fast solver of plasma fluid model in dielectric-barrier-discharge simulation. Plasma Sources Sci. Technol.
**2020**, 29, 075007. [Google Scholar] [CrossRef] - Nakai, K.; Komuro, A.; Nishida, H. Effect of chemical reactions on electrohydrodynamic force generation process in dielectric barrier discharge. Phys. Plasmas
**2020**, 27, 063518. [Google Scholar] [CrossRef] - Hua, W.; Fukagata, K. Near-surface electron transport and its influence on the discharge structure of nanosecond-pulsed dielectric-barrier-discharge under different electrode polarities. Phys. Plasmas
**2019**, 26, 013514. [Google Scholar] [CrossRef] - Sato, S.; Furukawa, H.; Komuro, A.; Takahashi, M.; Ohnishi, N. Successively accelerated ionic wind with integrated dielectric-barrier-discharge plasma actuator for low-voltage operation. Sci. Rep.
**2019**, 9, 5813. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Emmons, D.J.; Weeks, D.E. Steady-State Model of an Argon-Helium High-Pressure Radio Frequency Dielectric Barrier Discharge. IEEE Trans. Plasma Sci.
**2020**, 48, 2715–2722. [Google Scholar] [CrossRef] - Zhong, L.; Wu, B.; Wang, Y. Low-temperature plasma simulation based on physics-informed neural networks: Frameworks and preliminary applications. Phys. Fluids
**2022**, 34, 087116. [Google Scholar] [CrossRef] - Zhang, Y.T.; Gao, S.H.; Ai, F. Efficient numerical simulation of atmospheric pulsed discharges by introducing deep learning. Front. Phys.
**2023**, 11, 1125548. [Google Scholar] [CrossRef] - Boeuf, J.P.; Lagmich, Y.; Unfer, T.; Callegari, T.; Pitchford, L.C. Electrohydrodynamic force in dielectric barrier discharge plasma actuators. J. Phys. D Appl. Phys.
**2007**, 40, 652. [Google Scholar] [CrossRef] - Cristofolini, A.; Popoli, A. A multi-stage approach for DBD modelling. J. Phys. Conf. Ser.
**2019**, 1243, 012012. [Google Scholar] [CrossRef] - Boeuf, J.P.; Merad, A. Fluid and Hybrid Models of Non Equilibrium Discharges. In Plasma Processing of Semiconductors; Williams, P.F., Ed.; Springer: Dordrecht, The Netherlands, 1997; pp. 291–319. [Google Scholar] [CrossRef]
- Ventzek, P.L.G.; Sommerer, T.J.; Hoekstra, R.J.; Kushner, M.J. Two-dimensional hybrid model of inductively coupled plasma sources for etching. Appl. Phys. Lett.
**1993**, 63, 605–607. [Google Scholar] [CrossRef] [Green Version] - Ventzek, P.L.G.; Hoekstra, R.J.; Kushner, M.J. Two-dimensional modeling of high plasma density inductively coupled sources for materials processing. J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom.
**1994**, 12, 461–477. [Google Scholar] [CrossRef] - Punset, C.; Cany, S.; Boeuf, J.P. Addressing and sustaining in alternating current coplanar plasma display panels. J. Appl. Phys.
**1999**, 86, 124–133. [Google Scholar] [CrossRef] - Lin, K.M.; Hung, C.T.; Hwang, F.N.; Smith, M.; Yang, Y.W.; Wu, J.S. Development of a parallel semi-implicit two-dimensional plasma fluid modeling code using finite-volume method. Comput. Phys. Commun.
**2012**, 183, 1225–1236. [Google Scholar] [CrossRef] - Teunissen, J. Improvements for drift-diffusion plasma fluid models with explicit time integration. Plasma Sources Sci. Technol.
**2020**, 29, 015010. [Google Scholar] [CrossRef] [Green Version] - Tamura, H.; Sato, S.; Ohnishi, N. Numerical simulation of atmospheric-pressure surface dielectric barrier discharge on a curved dielectric with a curvilinear mesh. J. Phys. D Appl. Phys.
**2022**, 56, 045202. [Google Scholar] [CrossRef] - Kwok, D.T.K. A hybrid Boltzmann electrons and PIC ions model for simulating transient state of partially ionized plasma. J. Comput. Phys.
**2008**, 227, 5758–5777. [Google Scholar] [CrossRef] - Holgate, J.T.; Coppins, M. Numerical implementation of a cold-ion, Boltzmann-electron model for nonplanar plasma-surface interactions. Phys. Plasmas
**2018**, 25, 043514. [Google Scholar] [CrossRef] [Green Version] - Cartwright, K.L.; Verboncoeur, J.P.; Birdsall, C.K. Nonlinear hybrid Boltzmann—Particle-in-cell acceleration algorithm. Phys. Plasmas
**2000**, 7, 3252–3264. [Google Scholar] [CrossRef] - Kang, S.H. PIC-DSMC Simulation of a Hall Thruster Plume with Charge Exchange Effects Using pdFOAM. Aerospace
**2023**, 10, 44. [Google Scholar] [CrossRef] - Brieda, L. Plasma Simulations by Example, 1st ed.; CRC Press: Boca Raton, FL, USA, 2019. [Google Scholar]
- Cristofolini, A.; Popoli, A.; Neretti, G. A multi-stage model for dielectric barrier discharge in atmospheric pressure air. Int. J. Appl. Electromagn. Mech.
**2020**, 63, S21–S29. [Google Scholar] [CrossRef] - Lieberman, M.A.; Lichtenberg, A.J. Principles of Plasma Discharges and Materials Processing, 2nd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2005. [Google Scholar]
- LeVeque, R.J. Finite Volume Methods for Hyperbolic Problems; Cambridge University Press: Cambridge, UK, 2002; Volume 31. [Google Scholar]
- Hirsch, C. Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics; John Wiles & Sons: Chichester, UK, 1988; Volume 1. [Google Scholar]
- Scharfetter, D.; Gummel, H. Large-signal analysis of a silicon Read diode oscillator. IEEE Trans. Electron Devices
**1969**, 16, 64–77. [Google Scholar] [CrossRef] - Kulikovsky, A.A. A More Accurate Scharfetter-Gummel Algorithm of Electron Transport for Semiconductor and Gas Discharge Simulation. J. Comput. Phys.
**1995**, 119, 149–155. [Google Scholar] [CrossRef] - Liu, L.; van Dijk, J.; ten Thije Boonkkamp, J.H.M.; Mihailova, D.B.; van der Mullen, J.J.A.M. The complete flux scheme—Error analysis and application to plasma simulation. J. Comput. Appl. Math.
**2013**, 250, 229–243. [Google Scholar] [CrossRef] - Nguyen, T.D.; Besse, C.; Rogier, F. High-order Scharfetter-Gummel-based schemes and applications to gas discharge modeling. J. Comput. Phys.
**2022**, 461, 111196. [Google Scholar] [CrossRef] - Chen, F.F. Introduction to Plasma Physics and Controlled Fusion, 3rd ed.; Springer: Cham, Switzerland, 2016. [Google Scholar]
- Metcalf, M.; Reid, J.; Cohen, M. Modern Fortran Explained: Incorporating Fortran 2018; Oxford University Press: Oxford, UK, 2018. [Google Scholar]
- Sharma, G.; Martin, J. MATLAB
^{®}: A Language for Parallel Computing. Int. J. Parallel Program.**2009**, 37, 3–36. [Google Scholar] [CrossRef] [Green Version] - Parent, B.; Macheret, S.O.; Shneider, M.N. Electron and ion transport equations in computational weakly-ionized plasmadynamics. J. Comput. Phys.
**2014**, 259, 51–69. [Google Scholar] [CrossRef] - Hosea, M.; Shampine, L. Analysis and implementation of TR-BDF2. Appl. Numer. Math.
**1996**, 20, 21–37. [Google Scholar] [CrossRef] - Kossyi, I.A.; Kostinsky, A.Y.; Matveyev, A.A.; Silakov, V.P. Kinetic scheme of the non-equilibrium discharge in nitrogen-oxygen mixtures. Plasma Sources Sci. Technol.
**1992**, 1, 207–220. [Google Scholar] [CrossRef]

**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