# A Practical Method for Controlling the Asymmetric Mode of Atmospheric Dielectric Barrier Discharges
## 2 Model Description
\[\mathbf{n}\cdot\mathbf{\Gamma}_{\rm e}=\frac{1}{2}\
u_{\rm e,th}n_{\rm e}-\alpha_ {\rm s}n_{\rm e}\mu_{\rm e}\mathbf{E}\cdot\mathbf{n}-\sum_{i}\gamma_{i}(\mathbf{ \Gamma}_{i}\cdot\mathbf{n}) \tag{7}\]  
\[\mathbf{n}\cdot\mathbf{\Gamma}_{\rm e}=\frac{5}{6}\
u_{\rm e,th}n_{\rm e}-\alpha_ {\rm s}n_{\rm e}\mu_{\rm e}\mathbf{E}\cdot\mathbf{n}-\sum_{i}\gamma_{i}\overline {\varepsilon_{i}}(\mathbf{\Gamma}_{i}\cdot\mathbf{n}) \tag{8}\]  
\[\mathbf{n}\cdot\mathbf{\Gamma}_{k}=M_{k}R_{\rm surf,}k+\alpha_{\rm s}M_{\rm c}k_{ \rm i}\mu_{\rm e,m}z_{k}\mathbf{E}\cdot\mathbf{n} \tag{9}\]  
\[\frac{\mathrm{d}\sigma_{\rm s}}{\mathrm{d}t}=\mathbf{J}_{\rm e}\cdot\mathbf{n}+\bm {J}_{i}\cdot\mathbf{n} \tag{10}\]  
\[\frac{\partial n_{\varepsilon}}{\partial t}+\
abla\cdot\mathbf{\Gamma}_{\mathrm{ c}}+\mathbf{EI}_{\mathrm{e}}=S_{\mathrm{en}} \tag{2}\]0  
Here, \(\mathbf{n}\) is the unit normal vector towards the surface; both \(\mathbf{\Gamma}_{k}\) and \(\mathbf{\Gamma}_{i}\) stand for the boundary flux of heavy species but the latter only denotes the term for ions; \(\
u_{\rm e,th}\) is the thermal velocity; \(M_{k}\) represents the molar weight; \(R_{\rm surf,}k\) is the surface reaction rate; \(\gamma_{i}\) is the emission coefficient of the secondary electron cited from [25] and \(\overline{\varepsilon_{i}}\) is its mean energy; \(\sigma_{\rm s}\) represents the surface charge density on the wall, and \(\mathbf{J}_{\rm e}\) and \(\mathbf{J}_{\rm i}\) respectively denote the electron density and the ion density there; \(\mathbf{D}_{1}\) and \(\mathbf{D}_{2}\) are the electric displacement vectors on both sides of the interface; \(\alpha_{\rm s}\) is a switching function as given below. \[\alpha_{\rm s}=\left\{ \begin{array}{c}1,\ \mathrm{sgn}(q)\mathbf{E}\cdot \mathbf{n}>0\\ 0,\ \mathrm{sgn}(q)\mathbf{E}\cdot\mathbf{n}\leq 0\end{array} \right. \tag{12}\]  
Figure 1: Schematic of the model geometry. \(\varepsilon\) denotes the relative permittivity of the dielectric layer; \(d_{\rm b}\) and \(d_{\rm g}\) represent the widths of the dielectric layer and gas gap, respectively. Here, \(q\) represents the signed charge. For chemical kinetics, we reused ones from our previous work [19]. To be specific, the model sustains itself under the pressure of 760 Torr and temperature of 300 K, and it considers six particles (i.e., e (electron), He, He\({}^{*}\), He\({}_{2}\)\({}^{*}\), He\({}^{+}\) and He\({}_{2}\)\({}^{+}\)) and 27 reactions which are listed in Table 1 in the Appendix A. The initial densities of e, He\({}^{+}\), and He\({}_{2}\)\({}^{+}\) were set to \(1\times 10^{13}\) m\({}^{-3}\), \(5\times 10^{12}\) m\({}^{-3}\), and \(5\times 10^{12}\) 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 [PERSON] et al. [28] is further given, as shown in Figure 2.  
Obviously, although the discharge current phase in the experiment was slightly ahead of that in our model, and the current pulse magnitude shows a difference between the experiment and simulation, the sketches of the total discharge current density (\(J\)) and gap voltage (\(V_{\text{g}}\)) 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.