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## Numerical simulation study on microdischarge via a unified fluid model

Wang Zhen, Zhao Zhi-Hang, Fu Yang-Yang
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• #### 摘要

数值仿真技术已发展成为气体放电领域的重要研究手段, 常用于研究揭示某一具体放电形式的微观物理过程. 本文介绍了气体放电的统一流体模型, 包括粒子的连续性方程、能量守恒方程及泊松(Poisson)方程, 考虑阴极电子发射(二次电子、热电子发射)、反应焓变与气体加热、阴极热传导等基本过程, 可模拟得到包含盖革-米勒(Geiger-Müller)放电、汤森(Townsend)放电、辉光放电、电弧放电等各区域的完整伏安特性曲线. 基于该模型, 仿真得到的气体放电伏安特性曲线与已有文献结果一致, 验证了该模型的正确性. 在此基础上, 对间距为400 µm、气压分别为50和500 Torr (1 Torr ≈ 133.322 Pa)的放电过程进行了具体研究, 对比分析了不同气压条件下放电典型参量的分布特性. 该模型实现了广域参数范围条件下的气体放电数值仿真, 拓展了气体放电流体模型的应用范围, 促进了对放电参数特性的系统性分析.

#### 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.

#### 作者及机构信息

###### 通信作者: 付洋洋, fuyangyang@tsinghua.edu.cn
• 基金项目: 国家自然科学基金(批准号: 52277154)资助的课题.

#### Authors and contacts

###### Corresponding author: Fu Yang-Yang, fuyangyang@tsinghua.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 52277154).

#### 施引文献

• 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} = $$20 \;{\text{mm}} Fig. 1. Schematic diagram of the Ar DC microdischarge. The gap distance between the anode and cathode is L_\text{gap} = 400\;{\text{μm}}. The radius of electrodes is R_\text{el} = 2\; {\text{mm}}. The cathode is tungsten and it’s length is L_\text{cath} = 20 \;{\text{mm}}. 图 2 阴极材料(钨)的比热容C_\text{p,M}、导热率 \kappa_\text{M} 、发射率 \epsilon_\text{M}随温度的变化 Fig. 2. Specific heat capacity (C_\text{p, M}), thermal conductivity ( \kappa_\text{M} ), and emissivity ( \epsilon_\text{M}) of tungsten cathode scaling with temperature. 图 3 不同参数条件下得到的放电CVC曲线 (a) p_0 =$$ 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

Fig. 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 $p_\text{2} = 500 \;\text{Torr}$.

图 4  气压在50—3000 Torr范围内得到的CVC曲线

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

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

Fig. 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.

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

Fig. 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.

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

Fig. 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.

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

Fig. 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.

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

Fig. 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.

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

Fig. 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.

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$T_\text{g,max} = 11850 \;\text{K}$

Fig. 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 $500 \;\text{Torr}$.

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

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

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

Fig. 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.