Numerical simulation study on microdischarge via a unified fluid model

Wang Zhen; Zhao Zhi-Hang; Fu Yang-Yang

doi:10.7498/aps.73.20240392

Abstract

Numerical simulation has become an indispensable tool in the study of gas discharge. However, it is typically used to reveal microscopic properties in a discharge under specific conditions. In this work, a unified fluid model for discharge simulation is introduced in detail. The model includes the continuity equation, the energy conservation equation of the species (electrons and heavy particles), and Poisson’s equation. The model takes into account some processes such as cathode electron emission (secondary electron emission and thermionic emission), reaction enthalpy change, gas heating, and cathode heat conduction. The full current-voltage characteristic (CVC) curve covers a range of discharge regimes, such as the Geiger-Müller discharge regime, Townsend discharge regime, subnormal glow discharge regime, normal glow discharge regime, abnormal glow discharge regime, and arc discharge regime. The obtained CVC curve is consistent with the results in the literature, confirming the validity of the unified fluid model. On this basis, the CVC curves are obtained in a wide pressure range of 50–3000 Torr. Simulation studies are carried out focusing on the discharge characteristics for microgap of 400 µm at pressures of 50 Torr and 500 Torr, respectively. The distributions of typical discharge parameters under different pressure conditions are analyzed by comparison. The results indicate that the electric field in the discharge gap is uniform, and that the space charge effect can be ignored in Townsend discharge regime. The cathode fall region and the quasi-neutral region both appear in glow discharge regime, and the space charge effect is significant. In particular, the electric field reversal occurs in abnormal discharge regime due to the heightened particle density gradient. The electron density reaches about 10²² m^–3 in arc discharge regime dominated by thermionic emission and thermal ionization, with the current density increasing. The gas temperature peak is 11850 K when the pressure is 500 Torr, and the cathode surface is heated to nearly 4000 K due to heat conduction. The present model can be used to simulate gas discharge across a wide range of condition parameters, promoting and expanding fluid model applications, and assisting in a more comprehensive investigation of discharge parameter properties.

Keywords:

Authors and contacts

Department of Electrical Engineering, Tsinghua University, Beijing 100084, China

Corresponding author: Fu Yang-Yang, fuyangyang@tsinghua.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 52277154).

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图 1 氩气直流微放电结构示意图. 其中, 电极间距$L_\text{gap} = 400\;{\text{μm}}$, 电极半径$R_\text{el} = 2\; {\text{mm}}$, 阴极导体材料为钨, 长度$L_\text{cath} = $$ 20 \;{\text{mm}}$

Figure 1. Schematic diagram of the Ar DC microdischarge. The gap distance between the anode and cathode is $L_\text{gap} = 400\;{\text{μm}}$. The radius of electrodes is $R_\text{el} = 2\; {\text{mm}}$. The cathode is tungsten and it’s length is $L_\text{cath} = 20 \;{\text{mm}}$.

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图 2 阴极材料(钨)的比热容$C_\text{p,M}$、导热率$ \kappa_\text{M} $、发射率$ \epsilon_\text{M}$随温度的变化

Figure 2. Specific heat capacity ($C_\text{p, M}$), thermal conductivity ($ \kappa_\text{M} $), and emissivity ($ \epsilon_\text{M}$) of tungsten cathode scaling with temperature.

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图 3 不同参数条件下得到的放电CVC曲线　(a) $p_0 = $$ 760 \;\text{Torr} \;(1 \;\text{atm})$条件下, 本文模拟结果与文献[40]结果对比; (b) $p_0 = 760 \;\text{Torr}$条件下, 电阻扫描与电压扫描所得CVC曲线结果一致, 具体放电模式取决于外电路$V{\text{-}}I$曲线与CVC曲线的交点; (c)气压分别为$p_1 = 50 \;\text{Torr}$, $p_2 = $$ 500 \;\text{Torr}$时, 仿真得到的CVC曲线

Figure 3. CVC curves of the microdischarges under different conditions: (a) Benchmark between the calculation results of the unified fluid model in this article with the Ref. [40] at $p_0$ = 760 Torr; (b) overlapping CVC curves obtained by ballast resistor sweeping and voltage source sweeping at $p_\text{0}$ = 760 Torr, the discharge regime depends on the intersection of external circuit $V{\text{-}}I$ curves and the CVC curve; (c) CVC curves at $p_\text{1} = 50 \;\text{Torr}$ and $p_\text{2} = 500 \;\text{Torr}$.

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图 4 气压在50—3000 Torr范围内得到的CVC曲线

Figure 4. The CVC curves obtained with the gas pressure ranging from 50 Torr to 3000 Torr.

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图 5 汤森放电区的参数特性($x = 0$的位置为阴极, $x = 400\; \text{μm}$的位置为阳极)　(a) p₁ = 50 Torr 与 (b) p₂ = 500 Torr条件下电子密度($n_\text{e}$)、离子密度($n_\text{i}=n_{\text{Ar}^+} + n_{\text{Ar2}^+}$)、电子电流密度($J_\text{e}$)、离子电流密度($J_\text{i}$)的空间分布; (c) p₁ = 50 Torr 与 (d) p₂ = 500 Torr条件下电势(ϕ)和电场(E)的空间分布

Figure 5. Discharge characteristics in Townsend discharge regime. The position of $x = 0$ is the cathode and $x = 400\;{\text{μm}}$ is the anode. Spatial distributions of the electron density ($n_\text{e}$), ion density ($n_\text{i}=n_{\text{Ar}^+} + n_{\text{Ar2}^+}$), electron current density ($J_\text{e}$), and ion current density ($J_\text{i}$) at (a) 50 Torr and (b) 500 Torr. The corresponding spatial distributions of the electric potential (ϕ) and the electric field (E) at (c) 50 Torr and (d) 500 Torr.

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图 6 亚正常辉光放电区的参数特性　(a) p₁ = 50 Torr 与 (b) p₂ = 500 Torr条件下电子密度($n_\text{e}$)、离子密度($n_\text{i}$)、电子电流密度($J_\text{e}$)、离子电流密度($J_\text{i}$)的空间分布; (c) p₁ = 50 Torr 与 (d) p₂ = 500 Torr条件下电势(ϕ)和电场(E)的空间分布

Figure 6. Discharge characteristics in subnormal glow discharge regime. Spatial distributions of the electron density ($n_\text{e}$), ion density ($n_\text{i}$), electron current density ($J_\text{e}$), and ion current density ($J_\text{i}$) at (a) 50 Torr and (b) 500 Torr. The corresponding spatial distributions of the electric potential (ϕ) and the electric field (E) at (c) 50 Torr and (d) 500 Torr.

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图 7 正常辉光放电区的参数特性　(a) p₁ = 50 Torr 与 (b) p₂ = 500 Torr条件下电子密度($n_\text{e}$)、离子密度($n_\text{i}$)、电子电流密度($J_\text{e}$)、离子电流密度($J_\text{i}$)的空间分布; (c) p₁ = 50 Torr 与 (d) p₂ = 500 Torr条件下电势(ϕ)和电场(E)的空间分布

Figure 7. Discharge characteristics in normal glow discharge regime. Spatial distributions of the electron density ($n_\text{e}$), ion density ($n_\text{i}$), electron current density ($J_\text{e}$), and ion current density ($J_\text{i}$) at (a) 50 Torr and (b) 500 Torr. The corresponding spatial distributions of the electric potential (ϕ) and the electric field (E) at (c) 50 Torr and (d) 500 Torr.

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图 8 反常辉光放电区的参数特性　(a) p₁ = 50 Torr 与 (b) p₂ = 500 Torr条件下电子密度($n_\text{e}$)、离子密度($n_\text{i}$)、电子电流密度($J_\text{e}$)、离子电流密度($J_\text{i}$)的空间分布; (c) p₁ = 50 Torr 与 (d) p₂ = 500 Torr条件下电势(ϕ)和电场(E)的空间分布

Figure 8. Discharge characteristics in abnormal glow discharge regime. Spatial distributions of the electron density ($n_\text{e}$), ion density ($n_\text{i}$), electron current density ($J_\text{e}$), and ion current density ($J_\text{i}$) at (a) 50 Torr and (b) 500 Torr. The corresponding spatial distributions of the electric potential (ϕ) and the electric field (E) at (c) 50 Torr and (d) 500 Torr.

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图 9 反常辉光放电区, (a) 50 Torr和(b) 500 Torr条件下电子电流密度($J_\text{e}$)、电子扩散电流密度($J_\text{e, dif}$)、电子漂移电流密度($J_\text{e, dr}$)的空间分布, 其中虚线为电场反转临界位置$x_\text{r}$; (c), (d)两个气压条件下电场反转临界位置附近的总电流密度($J_\text{total}$)与其他电流密度分量的空间分布; (e), (f)对应气压下电场反转临界位置的总电流密度与电子电流密度的空间分布

Figure 9. Spatial distributions of the electron current density ($J_\text{e}$), the diffusion ($J_\text{e, dif}$), and the drift ($J_\text{e, dr}$) component of the electron current density in the abnormal glow regime at (a) 50 Torr and (b) 500 Torr. The dotted line is the critical position of the electric field reversal ($x_\text{r}$). Spatial distributions of total current density ($J_\text{total}$) and other current density components near the $x_\text{r}$ at (c) 50 Torr and (d) 500 Torr. Spatial distributions of the $J_\text{total}$ and $J_\text{e}$ near the $x_\text{r}$ at (e) 50 Torr and (f) 500 Torr.

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图 10 电弧放电区的参数特性　(a) p₁ = 50 Torr 与 (b) p₂ = 500 Torr条件下电子密度($n_\text{e}$)、离子密度($n_\text{i}$)、电子电流密度($J_\text{e}$)、离子电流密度($J_\text{i}$)的空间分布; (c) p₁ = 50 Torr 与 (d) p₂ = 500 Torr条件下电势(ϕ)和电场(E)的空间分布

Figure 10. Discharge characteristics in arc discharge regime. Spatial distributions of the electron density ($n_\text{e}$), ion density ($n_\text{i}$), electron current density ($J_\text{e}$), and ion current density ($J_\text{i}$) at (a) 50 Torr and (b) 500 Torr. The corresponding spatial distributions of the electric potential (ϕ) and the electric field (E) at (c) 50 Torr and (d) 500 Torr.

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图 11 (a) 50 Torr和(b) 500 Torr条件下气体温度空间分布随放电电流密度的关系. 放电初始气体温度均为300 K, 其中, 500 Torr时, 气体温度最大值$T_\text{g,max} = 11850 \;\text{K}$

Figure 11. Spatial distributions of discharge gap gas temperature scaling with current density at $p=$ (a) $50 \;\text{Torr}$ and (b) $500 \;\text{Torr}$. The initial temperature is 300 K. The maximum gas temperature ( $ T_\text{g,max} $ ) is 11850 K at $500 \;\text{Torr}$.

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图 12 p = 50, 500 Torr气压条件下, 不同放电模式气体温度的空间分布

Figure 12. Spatial distributions of gas temperature in different discharge regimes at $p=50,500 \;\text{Torr}$.

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图 13 (a) $p_1 = 50 \;\text{Torr}$与(b) $p_2 = 500 \;\text{Torr}$时不同电流密度下阴极表面温度随时间的演化. 初始温度为300 K. 500 Torr时, 阴极表面温度最大值为$T_\text{c,max} = 3961 \;\text{K}$. 电流密度为$1 \times 10^7 \;\text{A}/\text{m}^2$时(c) $p_1 = 50 \;\text{Torr}$与(d) $p_2 = 500 \;\text{Torr}$条件下阴极表面温度随时间的变化

Figure 13. Evolution of cathode surface temperature scaling with current density at (a) 50 Torr and (b) 500 Torr. The initial temperature is 300 K. The maxmium cathode surface temperature ($ T_\text{c,max}$) is 3961 K at 500 Torr. The temperature of cathode surface scaling with time when the current density is $J_\text{total} = 1 \times 10^7 \;\text{A}/\text{m}^2$ at (c) 50 Torr and (d) 500 Torr.

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表 1 该模型中计算的等离子体化学反应

Table 1. Reactions involved in the model

序号	反应过程	反应系数	参考文献	$\Delta E$/eV^[41]
R1	e + Ar $\rightarrow $ e + Ar	BOLSIG+	[47]	0
R2	e + Ar $\rightarrow $ e + $\text{Ar}^*$	BOLSIG+	[47]	11.5
R3	e + $\text{Ar}^*$ $\rightarrow $ e + Ar	BOLSIG+	[47]	–11.5
R4	e + Ar $\rightarrow $ 2e + $\text{Ar}^+$	BOLSIG+	[47]	15.8
R5	e + $\text{Ar}^*$ $\rightarrow $ 2e + $\text{Ar}^+$	BOLSIG+	[47]	4.3
R6	e + $\text{Ar}_2^*$ $\rightarrow $ 2e + $\text{Ar}_2^+$	BOLSIG+	[47]	3.66
R7	e + $\text{Ar}_2^*$ $\rightarrow $ e + 2Ar	BOLSIG+	[47]	–11.27
R8	e + $\text{Ar}_2^+$ $\rightarrow $ $\text{Ar}^*$ + Ar	$ 1.04\times {{10}^{-12}} {(0.026/{T_\text{e})}^{0.67}}\dfrac{1-\exp (-418/{T_\text{g}})}{1-0.31\exp (-418/{T_\text{g}})} \; [\text{m}^3/\text{s}] $	[48]	–3.03
R9	e + $\text{Ar}_2^+$ $\rightarrow $ e + $\text{Ar}^+$ + Ar	$ 1.11\times {{10}^{-12}}{{T}_\text{e}^{-1}}{\exp \{-[2.94+3({T_\text{g}}/11600-0.026)]\}}\; [\text{m}^3/\text{s}] $	[49,50]	4.53
R10	2e + $\text{Ar}^+$ $\rightarrow $ e + Ar	$ \left\{ \begin{array}{l}8.75 \times 10^{-39} T_\text{e}^{-4.5}\;[\text{m}^6/\text{s}], \; T_\text{e} \leqslant 0.276 \;\text{eV}\\1.29 \times 10^{-44}\left({11.659}/{T_{\mathrm{e}}}+2\right) \exp \left({4.11}/{T_\text{e}}\right)\;[\text{m}^6/\text{s}], \; T_\text{e} > 0.276 \;\text{eV}\end{array}\right. $	[50]	–15.8
R11	$\text{Ar}^$ + $\text{Ar}^$ $\rightarrow $ e + $\text{Ar}^+$ + Ar	$ 1.62\times {{10}^{-16}}{{T_\text{g}}^{0.5}} \; [\text{m}^3/\text{s}] $	[51]	–13.26
R12	$\text{Ar}^*$ + Ar $\rightarrow $ Ar + Ar	$ 3\times {{10}^{-21}}\; [\text{m}^3/\text{s}] $	[52,53]	–11.5
R13	2$\text{Ar}_2^*$ $\rightarrow $ e + 2Ar + $\text{Ar}_2^+$	$ 1.6248\times {{10}^{-16}}{{T_\text{g}}^{0.5}}\; [\text{m}^3/\text{s}] $	[54]	–8.01
R14	2Ar + $\text{Ar}^+$ $\rightarrow $ Ar + $\text{Ar}_2^+$	$ 7.5\times {{10}^{-41}}/T_\text{g} \; [\text{m}^6/\text{s}] $	[54]	–1.27
R15	2Ar + $\text{Ar}^$ $\rightarrow $ Ar + $\text{Ar}_2^$	$ 3.3\times {{10}^{-44}}\; [\text{m}^6/\text{s}] $	[39]	–0.23
R16	Ar + $\text{Ar}_2^+$ $\rightarrow $ 2Ar + $\text{Ar}^+$	$ 6.06\times {{10}^{-12}}{T_\text{g}^{-1}}{\exp (-1.258\times {{10}^{5}}/{{R}_{\text{g}}}/{T_\text{g}})}\; [\text{m}^3/\text{s}] $	[49,50]	1.27
R17	$\text{Ar}^*$ $\rightarrow $ Ar + $h\nu$	$ 3.145\times {{10}^{5}}\; [1/\text{s}] $	[55]	–11.5
R18	$\text{Ar}_2^*$ $\rightarrow $ 2Ar + $h\nu$	$ 6.00\times {{10}^{7}} \; [1/\text{s}] $	[39]	–11.27

DownLoad: CSV

表 2 p = 50, 500 Torr时反常辉光放电区电场反转、离子密度最大值、电势最大值、电子扩散电流密度与总电子电流密度相等、离子电流密度为零值的位置

Table 2. Position where the electric field is reversed ($x_{\text{r}}$), the ion density is maximum ($x_{\text{i}_\text{max}}$), the electric potential is maximum ($x_{\phi _ \text{max}}$), the electron diffusion current density equals the total electron current density ($x_{J_\text{e,dif}=J_\text{e}}$), and the ion current density equals zero ($x_{J_\text{i} = 0}$) at 50 Torr and 500 Torr in abnormal glow regime.

p/Torr	$x_{\text{r}}/\text{μm}$	$x_{\text{i}_\text{max}}/\text{μm}$	$x_{\phi _ \text{max}}/\text{μm}$	$x_{J_\text{e,dif}=J_\text{e}}/\text{μm}$	$x_{J_\text{i} = 0}/\text{μm}$
50	305.7	305.1	305.7	305.8	305.7
500	183.1	179.6	183.1	181.4	183.1

DownLoad: CSV

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