# A Practical Method for Controlling the Asymmetric Mode of Atmospheric Dielectric Barrier Discharges

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{d}to a different controlling frequency f

_{c}for a certain period of time, right after the establishment of AP1 discharge. Once the discharge stabilizes under f

_{c}, the driving frequency will be changed back to the initial value (i.e., f

_{d}). The rest of this paper is organized as follows: Section 2 briefly introduces the one-dimensional fluid model and its qualitative validation; the numerical regulating example and its underneath mechanism are shown in Section 3; the key conclusions are drawn in Section 4; Appendix A presents the chemical scheme used in our simulation.

## 2. Model Description

_{am}of 2 kV, whereas the lower one is grounded. The width of the gas gap d

_{g}is fixed to 4.4 mm and the dielectric layer thickness d

_{b}is 1 mm. Note that the gap width is not very large (<5 mm) and much shorter compared with the radius of electrodes (≈56.4 mm) [20]; in this case, the radially homogeneous assumption should be reasonable and accepted. On this basis, a one-dimensional fluid model is appropriate to simulate the discharge process, which has also been successfully applied to investigate the nonlinear phenomena in [6,21,22].

_{e}, n

_{ε},

**Γ**

_{e}, and

**Γ**

_{ε}respectively denote the electron number density, electron energy, total electron flux, and electron energy flux;

**J**

_{k}is the flux vector for heavy species k, and its mass fraction and charge number are ω

_{k}and z

_{k}, respectively; S

_{e}, S

_{en}, and S

_{k}respectively represent the source terms describing the net changing rate of electron density, electron energy loss/gain, and the source term for heavy species k; the electron mobility μ

_{e}is solved by Bolsig

^{+}with the cross-section data from IST-Lisbon Database [28,29,30], then the electron diffusion coefficient D

_{e}can be obtained through Einstein’s relation [29]; the mobility and diffusion coefficient for heavy species k, i.e., μ

_{k}and D

_{k}, are referenced from [25];

**E**is the electric field intensity; the mixture properties ρ and M

_{n}stand for the density of mixture and its mean molar mass.

**n**is the unit normal vector towards the surface; both

**Γ**

_{k}and

**Γ**

_{i}stand for the boundary flux of heavy species but the latter only denotes the term for ions; v

_{e,th}is the thermal velocity; M

_{k}represents the molar weight; R

_{surf,k}is the surface reaction rate; γ

_{i}is the emission coefficient of the secondary electron cited from [25] and $\overline{{{\displaystyle \epsilon}}_{i}}$ is its mean energy; σ

_{s}represents the surface charge density on the wall, and

**J**

_{e}and

**J**

_{i}respectively denote the electron density and the ion density there;

**D**

_{1}and

**D**

_{2}are the electric displacement vectors on both sides of the interface; α

_{s}is a switching function as given below.

_{2}*, He

^{+}and He

_{2}

^{+}) and 27 reactions which are listed in Table A1 in the Appendix A. The initial densities of e, He

^{+}, and He

_{2}

^{+}were set to 1 × 10

^{13}m

^{−3}, 5 × 10

^{12}m

^{−3}, and 5 × 10

^{12}m

^{−3}, respectively, to ensure the initial neutral condition. The whole calculating domains were discretized through finite element method (log formulation, linear shape function), and a direct solver PARDISO of COMSOL software was employed to solve the above equations. Besides, we have taken a numerical test in terms of the initial densities and concluded that such values did not significantly influence the results in the steady state but affected the calculation time. Based on the above, a qualitative validation compared with the experimental results presented by Mangolini et al. [28] is further given, as shown in Figure 2.

_{g}) waveforms predicted by our model principally match well with those observed in the experiment. Note that such a discrepancy in the current waveform might be contributed to the Penning ionization induced by the nitrogen impurities; that is, the extra Penning ionization causes a higher seed electron level which boosts the breakdown though this causes extinguishment more quickly. Therefore, our model is qualified to study the discharge characteristics in the homogeneous He DBD under atmospheric pressure.

## 3. Examples and Mechanisms of the Discharge Mode Control

#### 3.1. SP1, AP1P, and AP1N

#### 3.2. Numerical Regulating Example and the Underlying Mechanism

_{d}of 14 kHz will finally stabilize in the AP1P mode after about 10 applied voltage cycles, as shown in Figure 4a. Starting from the 15th cycle, we change f

_{d}to a different frequency for 20 cycles, and then adjust the frequency back to 14 kHz at the 35th cycle. The results indicate that the final stabilized discharge mode depends on f

_{c}to some extent. Figure 4b shows that when f

_{c}is set to 8 kHz, the discharge mode at the frequency-altered stage is SP1. However, after this stage, the discharge mode evolves back to AP1P; when f

_{c}is set to 20 kHz, the discharge mode of the control section and the section after the 35th cycles are both AP1N, as shown in Figure 4c; when f

_{c}increases to 30 kHz, an AP1P discharge is observed at the frequency-altered stage, but the discharge mode after the 35

^{th}cycle evolves into SP1, as shown in Figure 4d. Note that, when f

_{c}rises beyond 30 kHz (the largest f

_{c}considered in our simulation is 100 kHz), the initial AP1P discharge under 14 kHz can always be adjusted to SP1.

_{c}is lower than f

_{d}(14 kHz). On this basis, when f

_{c}is set to a lower value like 8 kHz, the discharge in the controlling section develops from a relatively lower seed electron density and provides sufficient dissipative time for the residual electrons before the next breakdown, leading to an SP1 mode. The opposite leading to AP1 mode can be inferred by analogy.

^{13}m

^{−3}), then the discharge intensity of the subsequent discharge will be strong (>1.23 mA/cm

^{2}). Correspondingly, the residual electron density after the breakdown, as further depicted in Figure 7a, can not be dissipated completely and exceeds 1 × 10

^{17}m

^{−3}(Figure 6b). Such a high initial value of electron density can only ignite an unmatured discharge, forming an AP1 mode. Analogously, if the seed electron density reaches a high level at the beginning of discharge (>8 × 10

^{15}m

^{−3}), then the subsequent discharge cannot develop to a mature one and will form several weak current pulses with the maximum density being less than 0.2 mA/cm

^{2}, as shown in Figure 6a and Figure 7b.

^{13}to 8 × 10

^{15}m

^{−3}under our simulation conditions. Ignited by an appropriate seed electron level, the intensity of the subsequent discharge should be moderate so that the residual electrons can be dissipated completely, as shown in Figure 7c, leading to the SP1 mode. It should be pointed out that, being a control method of seed electron density, the higher driving frequency is able to limit the dissipative process of discharge. Therefore, the cooperation between seed electron level and dissipative time will lead to a less intense AP1 mode when f

_{c}is beyond 30 kHz, as can be seen from the current waveform and seed electron evolution illustrated by Figure 4 and Figure 5. We have carried out a parameterized sweep in terms of f

_{c}from 30 to 100 kHz, and the results show that, within this range of frequency, such a less intense AP1 discharge can impose a restriction on the drastic variation of the seed electron density and finally, the initial density of electrons before the breakdown varies within the abovementioned range (from 2 × 10

^{13}to 8 × 10

^{15}m

^{−3}). That is to say, increasing the driving frequency would be a practical strategy to manipulate the discharge symmetry and some more in-depth investigations would be carried out in our future study.

## 4. Conclusions

- (1)
- The practical control strategy proposed here first changes the original driving frequency to a relatively larger one until the discharge stabilizes again, and then turns the driving frequency back to the original one;
- (2)
- Three period-one discharge modes can be converted to each other by applying different control frequencies;
- (3)
- The effectiveness of the control strategy is determined by the seed electron level at the frequency-altered phase, and there is a critical range of the seed electron density. Under the original driving frequency of 14 kHz, the seed electron level approximately ranges from 2 × 10
^{13}m^{−3}to 8 × 10^{15}m^{−3}; - (4)
- The higher driving frequency in the controlling section can limit the dissipative process of discharge, and further induce a less intense AP1 mode through the cooperation between seed electron level and dissipative time. In our simulations, the discharges with an initial driving frequency of 14 kHz can always be converted to SP1 mode when the control frequency is beyond 30 kHz.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Index | Reaction | Rate Coefficient | Reference |
---|---|---|---|

R1 | $\mathrm{e}+\mathrm{He}=>\mathrm{e}+\mathrm{He}$ | $f\left({T}_{\mathrm{e}}\right)$ | [30] |

R2 | $\mathrm{e}+\mathrm{He}=>\mathrm{e}+{\mathrm{He}}^{*}$ | $f\left({T}_{\mathrm{e}}\right)$ | [30] |

R3 | $\mathrm{e}\text{}+\text{}\mathrm{He}=2\mathrm{e}+{\mathrm{He}}^{+}$ | $f\left({T}_{\mathrm{e}}\right)$ | [30] |

R4 | $\mathrm{e}+{\mathrm{He}}^{*}=>2\mathrm{e}+{\mathrm{He}}^{+}$ | $1.28\times {10}^{-7}\times {T}_{\mathrm{e}}^{0.6}\times \mathrm{exp}\left(-4.78/{T}_{\mathrm{e}}\right)$ | [31] |

R5 | $\mathrm{e}+{\mathrm{He}}^{*}=>\mathrm{e}+\mathrm{He}$ | $2.9\times {10}^{-9}$ | [31] |

R6 | $\mathrm{e}+{\mathrm{He}}_{2}^{*}=>\mathrm{e}+2\mathrm{He}$ | $3.8\times {10}^{-9}$ | [31] |

R7 | $2\mathrm{e}+{\mathrm{He}}^{+}=>\mathrm{e}+{\mathrm{He}}^{*}$ | $5.82\times {10}^{-20}\times {\left({T}_{\mathrm{e}}/0.026\right)}^{-4.4}$ | [31] |

R8 | $2\mathrm{e}+{\mathrm{He}}_{2}^{+}=>{\mathrm{He}}^{*}+\mathrm{He}+\mathrm{e}$ | $2.8\times {10}^{-20}$ | [31] |

R9 | $\mathrm{e}+\mathrm{He}+{\mathrm{He}}_{2}^{+}=>{\mathrm{He}}^{*}+2\mathrm{He}$ | $3.5\times {10}^{-27}$ | [31] |

R10 | $2\mathrm{e}+{\mathrm{He}}_{2}^{+}=>{\mathrm{He}}_{2}^{*}+\mathrm{e}$ | $1.2\times {10}^{-21}$ | [31] |

R11 | $\mathrm{e}+\mathrm{He}+{\mathrm{He}}_{2}^{+}=>{\mathrm{He}}_{2}^{*}+\mathrm{He}$ | $1.5\times {10}^{-27}$ | [31] |

R12 | $\mathrm{e}+{\mathrm{He}}^{+}=>{\mathrm{He}}^{*}$ | $6.76\times {10}^{-13}\times {T}_{\mathrm{e}}^{-0.5}$ | [32] |

R13 | $\mathrm{e}+\mathrm{He}+{\mathrm{He}}^{+}=>\mathrm{He}+{\mathrm{He}}^{*}$ | $1\times {10}^{-26}\times {\left({T}_{\mathrm{e}}/0.026\right)}^{-2}$ | [33] |

R14 | $\mathrm{e}+{\mathrm{He}}_{2}^{+}=>{\mathrm{He}}^{*}+\mathrm{He}$ | $8.82\times {10}^{-9}\times {\left({T}_{\mathrm{e}}/0.026\right)}^{-1.5}$ | [33] |

R15 | $\mathrm{e}+{\mathrm{He}}_{2}^{+}=>2\mathrm{He}$ | $1.0\times {10}^{-8}$ | [33] |

R16 | $\mathrm{e}+\mathrm{He}+{\mathrm{He}}_{2}^{+}=>3\mathrm{He}$ | $2.0\times {10}^{-27}$ | [31] |

R17 | ${\mathrm{e}\text{}+\text{}\mathrm{He}}_{2}^{*}=2\mathrm{e}+{\mathrm{He}}_{2}^{+}$ | $9.75\times {10}^{-16}\times {T}_{\mathrm{e}}^{0.71}\times \mathrm{exp}\left(-3.4/{T}_{\mathrm{e}}\right)$ | [32] |

R18 | ${\mathrm{He}}^{*}+2\mathrm{He}=>3\mathrm{He}$ | $2.0\times {10}^{-34}$ | [31] |

R19 | ${\mathrm{He}}^{*}+{\mathrm{He}}^{*}=>\mathrm{e}+{\mathrm{He}}_{2}^{+}$ | $2.03\times {10}^{-9}$ | [31] |

R20 | ${\mathrm{He}}^{*}+{\mathrm{He}}^{*}=>\mathrm{e}+\mathrm{He}+{\mathrm{He}}^{+}$ | $8.7\times {10}^{-10}$ | [15] |

R21 | ${\mathrm{He}}^{+}+2\mathrm{He}=>{\mathrm{He}}_{2}^{+}+\mathrm{He}$ | $1.4\times {10}^{-31}$ | [31] |

R22 | ${\mathrm{He}}^{*}+2\mathrm{He}=>{\mathrm{He}}_{2}^{*}+\mathrm{He}$ | $2.0\times {10}^{-34}$ | [34] |

R23 | ${\mathrm{He}}^{*}+{\mathrm{He}}_{2}^{*}=>{\mathrm{He}}^{+}+2\mathrm{He}+\mathrm{e}$ | $5.0\times {10}^{-10}$ | [15] |

R24 | ${\mathrm{He}}^{*}+{\mathrm{He}}_{2}^{*}=>{\mathrm{He}}_{2}^{+}+\mathrm{He}+\mathrm{e}$ | $2.0\times {10}^{-9}$ | [35] |

R25 | ${\mathrm{He}}_{2}^{*}+{\mathrm{He}}_{2}^{*}=>{\mathrm{He}}^{+}+3\mathrm{He}+\mathrm{e}$ | $3.0\times {10}^{-10}$ | [35] |

R26 | ${\mathrm{He}}_{2}^{*}+{\mathrm{He}}_{2}^{*}=>{\mathrm{He}}_{2}^{+}+2\mathrm{He}\text{}+\text{}\mathrm{e}$ | $1.2\times {10}^{-9}$ | [15] |

R27 | ${\mathrm{He}}_{2}^{*}+\mathrm{He}=>3\mathrm{He}$ | $1.5\times {10}^{-15}$ | [33] |

^{-3}. The unit of reaction rate coefficient for two body reactions and three body reactions are m

^{3}/s and m

^{6}/s, respectively. $f\left({T}_{\mathrm{e}}\right)$ indicates the rate coefficient as function of the electron mean temperature calculated by Bolsig+ [30], and the cross-section data from IST-Lisbon database [36] was used as the input parameter of the calculation.

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**Figure 1.**Schematic of the model geometry. ε denotes the relative permittivity of the dielectric layer; d

_{b}and d

_{g}represent the widths of the dielectric layer and gas gap, respectively.

**Figure 2.**Comparison between (

**a**) and (

**c**) previous experimental waveforms and (

**b**) and (

**d**) their corresponding simulation results obtained from our model under mostly the same conditions. J, V

_{a}, and V

_{g}respectively denote the total discharge current density, applied voltage, and gap voltage. The experiment was assumed to have 100 ppm nitrogen impurities, whereas our simulation considers the pure helium. The main external parameters in (

**a**) and (

**b**) are: V

_{am}= 1776 V, f = 10 kHz, d

_{g}= 5 mm, d

_{b}= 1 mm; in (

**c**) and (

**d**) are: V

_{am}= 1813 V, f = 10 kHz, d

_{g}= 5 mm, d

_{b}= 1.5 mm.

**Figure 3.**Waveforms of the total discharge current density (

**a**–

**c**) and the corresponding spatiotemporal distributions of electron density (

**d**–

**f**) under different driving frequencies. The voltage amplitude V

_{am}, gap width d

_{g}, and dielectric thickness d

_{b}are fixed at 2 kV, 4.4 mm, and 1 mm, respectively.

**Figure 4.**Temporal profiles of current density when the controlling frequency f

_{c}is set to be (

**a**) 14 kHz, (

**b**) 8 kHz, (

**c**) 20 kHz and (

**d**) 30 kHz respectively.

**Figure 5.**Evolution of the seed electron level as a function of the breakdown number, i.e., the average electron density right at the beginning moment of each breakdown. The f

_{c}in (

**a**–

**c**) are 8 kHz, 20 kHz, 30 kHz, respectively. The red circles indicate the transition points of the driving frequency.

**Figure 6.**The relationship between the average electron density and discharge current density. (

**a**) average electron density at the starting moment of the discharge vs. current density at the instant of current peak, (

**b**) current density at the instant of current peak vs. maximum average electron density.

**Figure 7.**Spatiotemporal distribution of log

_{10}(n

_{e}) during the 33, 34, 35, 36 and 37 applied voltage cycle when f

_{c}is set to be (

**a**) 8 kHz, (

**b**) 20 kHz, (

**c**)30 kHz. The black dashed lines divide off different applied voltage cycles.

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## Share and Cite

**MDPI and ACS Style**

Luo, L.; Wang, Q.; Dai, D.; Zhang, Y.; Li, L.
A Practical Method for Controlling the Asymmetric Mode of Atmospheric Dielectric Barrier Discharges. *Appl. Sci.* **2020**, *10*, 1341.
https://doi.org/10.3390/app10041341

**AMA Style**

Luo L, Wang Q, Dai D, Zhang Y, Li L.
A Practical Method for Controlling the Asymmetric Mode of Atmospheric Dielectric Barrier Discharges. *Applied Sciences*. 2020; 10(4):1341.
https://doi.org/10.3390/app10041341

**Chicago/Turabian Style**

Luo, Ling, Qiao Wang, Dong Dai, Yuhui Zhang, and Licheng Li.
2020. "A Practical Method for Controlling the Asymmetric Mode of Atmospheric Dielectric Barrier Discharges" *Applied Sciences* 10, no. 4: 1341.
https://doi.org/10.3390/app10041341