# Investigation of Plasma Propagation in Packed-Bed Dielectric Barrier Discharge Based on a Customized Particle-in-Cell/Monte Carlo Collision Model

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## Abstract

**:**

## 1. Introduction

_{2}O

_{3}[28]. Similarly, Butterworth and Allen experimentally observed two main types of discharge in a single-catalyst-pellet DBD reactor using nitrogen: point-to-point local microdischarges and surface streamers; these were found to be influenced by the material’s dielectric constant [29]. Wang et al., furthermore, examined streamer propagation and discharge characteristics in a packed-bed DBD reactor using ICCD [30]. They concluded that, in addition to the discharge mode transition, a higher dielectric constant constrains the discharge to the contact points of the beads, resulting in enhanced intensity. The production of reactive species is most prominent in the positive restrikes, surface discharges, and local microdischarges occurring between the beads. Electrons emitted by the electric field serve as a pre-ionization source and additional electron sources, which subsequently amplify the partial discharge.

## 2. Computational Model

_{2}O

_{3}(${\epsilon}_{\mathrm{r}}$ = 1.5–5), CeO

_{2}(${\epsilon}_{\mathrm{r}}$ ≈ 24)), semiconductor materials (e.g., WO

_{3}(${\epsilon}_{\mathrm{r}}$ = 10–20), and TiO

_{2}(${\epsilon}_{\mathrm{r}}$ = 40–100)), and ferroelectric materials (e.g., CaTiO

_{3}(${\epsilon}_{\mathrm{r}}$ ≈ 200), SrTiO

_{3}(${\epsilon}_{\mathrm{r}}$ ≈ 300), and BaTiO

_{3}(${\epsilon}_{\mathrm{r}}$ ≈ 10,000)). The simulated discharge geometry is divided into square cells, with each cell having dimensions of 11 μm × 11.2 μm, resulting in a grid consisting of 1000 × 500 cells. For the majority of the simulated geometry, the plasma density is relatively low and the spatial step length is smaller than the Debye length. However, in the bulk region of the plasma streamer, although the electron density is high, both the spatial charge separation and the electric field are weak. As a result, even if the spatial step length is greater than the Debye length in the bulk streamer region, it does not have a significant impact on the numerical accuracy (especially in such a short discharge time ~ns).

^{−13}s. In this setup, the upper electrode is subjected to a DC voltage of 16 kV, while the lower electrode is maintained at ground potential.

**E**represents the electric field.

**D**represents the electric displacement vector, $\epsilon $ is the permittivity ($\epsilon ={\epsilon}_{0}{\epsilon}_{\mathrm{r}}$), and ${\epsilon}_{0}$ is the dielectric constant of vacuum. Given the inclusion of dielectric material in the model, the relative permittivity ${\epsilon}_{\mathrm{r}}$ is a spatially varying function, denoted as ${\epsilon}_{\mathrm{r}}$(r), and cannot be moved outside the divergence operator. This process implements the “Poisson solver” in Figure 1b.

_{2}) and nitrogen (N

_{2}), with a consistent gas temperature of 300 K and an oxygen ratio of 50%. Throughout the simulation, free electrons, as well as N

_{2}

^{+}, O

_{2}

^{+}, and O

_{2}

^{−}ions, are meticulously tracked and represented as superparticles. The electron impact reaction mechanisms considered in the simulation include elastic collisions, as well as collisions involving attachment, excitation, and ionization with N

_{2}and O

_{2}gas molecules. This process corresponds to the “MC Collisions” in Figure 1b. These mechanisms are further elaborated in references [34,39]. The cross-sections and threshold energies utilized in the simulation are derived from the LXCat database and other relevant literature sources [40,41,42,43,44]. Particles such as metastable and free radicals are not directly considered. However, the MCC model indirectly accounts for their influence through electron impact reactions, which determines power deposition and electron energy.

^{15}m

^{−3}is introduced into the unoccupied space within the simulated geometry. In reality, seed electrons exist naturally as a result of cosmic radiation and environmental photo-ionization. These processes contribute to the generation of background electrons and the sustained presence of charges from previous plasma discharges. Following a similar approach to that in [39,45], during each time-step, a single new electron superparticle is randomly introduced into the simulation domain at various positions. This process corresponds to the “new electron addition” in Figure 1b, which is crucial in simulating positive streamers because initially arranged electrons move in the opposite direction due to the applied electric field and induced electric field in the streamer head. These electrons are quickly depleted during the simulation, causing the simulated streamer to stop. However, in reality, there exists a continuous source of electrons in front of the streamer head, which sustains the evolution of the positive streamer. The ‘new electron addition’ allows for the consideration of random events, such as cosmic radiation, photo-ionization, and various secondary electron emission (SEE) processes, detailed in [19,21,46]. Noting this, the ‘particle merger’ and ‘new electron addition’ steps are newly introduced mechanisms in the traditional PIC/MCC flow, and are of significance to simulate high-density discharge in restricted geometry.

## 3. Results and Discussion

#### 3.1. Plasma Streamer Evolution in Packed-Bed DBD

#### 3.2. Effect of Pellet Material on Plasma Streamer Propagation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic diagram depicting the 2D simulation geometry. The electrode gap is 4 mm, the electric layer thickness is 0.4 mm, the electrode length is 11 mm, the dielectric pellet radius is 6 mm, and the pellet height is 2.4 mm. (

**b**) Customized PIC/MCC algorithm flow chart. The weight of each superparticle is adjusted adaptively at ‘particle merger’, conserving both momentum and energy. Electrons produced via cosmic radiation, photoionization, and various secondary electron emission will be included in the “New electron addition” section.

**Figure 2.**Electron density profiles, electric field distributions, and charge density distributions at three different moments, with a dielectric constant of the pellet of 4, illustrating the evolution of a plasma streamer. (

**a**–

**c**) Electron density distribution; (

**d**–

**f**) absolute values of the electric field; (

**g**–

**i**) charge density distributions; (

**a**,

**d**,

**g**) 0.6 ns; (

**b**,

**e**,

**h**) 1.44 ns; (

**c**,

**f**,

**i**) 2.28 ns.

**Figure 3.**Electron density profiles, electric field distributions and charge density distributions at three different moments, with a dielectric constant of the pellet of 40, illustrating the evolution of a plasma streamer. (

**a**–

**c**) Electron density distribution; (

**d**–

**f**) absolute values of the electric field; (

**g**–

**i**) charge density distributions; (

**a**,

**d**,

**g**) 0.6 ns; (

**b**,

**e**,

**h**) 1.44 ns; (

**c**,

**f**,

**i**) 2.28 ns.

**Figure 4.**Electron density profiles, electric field distributions, and charge density distributions at three different moments, with a dielectric constant of the pellet of 400, illustrating the evolution of a plasma streamer. (

**a**–

**c**) Electron density distribution; (

**d**–

**f**) absolute values of the electric field; (

**g**–

**i**) charge density distributions; (

**a**,

**d**,

**g**) 0.6 ns; (

**b**,

**e**,

**h**) 1.44 ns; (

**c**,

**f**,

**i**) 2.28 ns.

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**MDPI and ACS Style**

Li, X.; Zhang, L.; Shahzad, A.; Attri, P.; Zhang, Q.
Investigation of Plasma Propagation in Packed-Bed Dielectric Barrier Discharge Based on a Customized Particle-in-Cell/Monte Carlo Collision Model. *Plasma* **2023**, *6*, 637-648.
https://doi.org/10.3390/plasma6040044

**AMA Style**

Li X, Zhang L, Shahzad A, Attri P, Zhang Q.
Investigation of Plasma Propagation in Packed-Bed Dielectric Barrier Discharge Based on a Customized Particle-in-Cell/Monte Carlo Collision Model. *Plasma*. 2023; 6(4):637-648.
https://doi.org/10.3390/plasma6040044

**Chicago/Turabian Style**

Li, Xufeng, Leiyu Zhang, Aamir Shahzad, Pankaj Attri, and Quanzhi Zhang.
2023. "Investigation of Plasma Propagation in Packed-Bed Dielectric Barrier Discharge Based on a Customized Particle-in-Cell/Monte Carlo Collision Model" *Plasma* 6, no. 4: 637-648.
https://doi.org/10.3390/plasma6040044