## 2 Description of Models  
In this paper, two parallel-plate metal electrodes are used to generate the gas-mixture discharge. The discharge model is shown in Figure 1. The left-plate electrode is driven by a dual-frequency source. The dual-radio-frequency source can be expressed as follows: \(V=V_{Hf}sin(2\pi f_{Hf}t)+V_{L}\)\(sin(2\pi f_{L}t)\), where HF represents the high-frequency source and LF represents the low-frequency source. The voltage of the high-frequency source is 200 V, the high frequency is the typical radio frequency 13.56 Mhz. The voltage of the low-frequency source is 100 V and the low frequency is 2 Mhz. The right-plate electrode is grounded. In Figure 1, the letter L on the \(x\)-axis represents the distance between the two plates. In this study, we chose a typical distance of 0.2 mm for the atmospheric pressure discharge. Generally, electrodes are separated by millimeters in atmospheric pressure discharges. The fluid model includes the particles' continuity equations, the  
In fact, the real reactions in the Ar/O\({}_{2}\) atmospheric pressure discharge are more complex than those are listed in Table 1. For an argon discharge or oxygen discharge, the main particles listed in Table 1. are sufficient for the simple discharge model [22], and the main dominant collision processes are included. Thus, the model can predict the Ar/O\({}_{2}\) atmospheric pressure discharge characteristics. The basic assumptions are the same as those in Refs. [24; 25]. According to these assumptions, the equations can be written as follows:  
\[\frac{\partial n_{e,i,n,m}}{\partial t}+\frac{\partial J_{e,i,n,m}}{\partial x} =S_{e,i,n,m} \tag{1}\]  
where \(e\), \(i\), \(n\), and \(m\) denote electrons, positive ions, negative ions, and metastable atoms, respectively. In addition, \(n\), \(J\), and \(S\) denote density, flux, and the source term, respectively. \[J_{e,i,n,m}=\pm\mu_{e,i,n}En_{e,i,n}-D_{e,i,n,m}\frac{\partial n_{e,i,n,m}}{ \partial x} \tag{2}\]  
Here, \(\mu\) denotes the mobility of the particles, and \(D\) is the diffusion coefficient of the particles. In this study, we obtained these values from Refs. [22; 23; 26]. \[\frac{\partial}{\partial t}\left(\frac{3}{2}kn_{e}T_{e}\right)=-\frac{ \partial}{\partial x}q_{e}-eJ_{e}\cdot E-\sum_{j}H_{j}Nk_{j}n_{e} \tag{3}\]  
\[q_{e}=-\frac{3}{2}kD_{e}n_{e}\frac{\partial T_{e}}{\partial x}+\frac{5}{2}kT_ {e}J_{e} \tag{4}\]  
Here, \(q_{e}\) is the electron energy flux; \(T_{e}\) is the electron temperature in units of eV; e denotes the elementary charge; \(k\) denotes the Boltzmann coefficient; \(E\) is taken to denote the electric field; \(N\) is the background gas number density; \(H_{j}\) is the coefficient of energy loss between electrons and particles \(j\); \(k_{j}\) is the reaction coefficient between electrons and particles \(j\). The Poisson equation is as follows:  
\[\frac{\partial^{2}V}{\partial x^{2}}=\frac{e}{\varepsilon_{0}}(n_{e}+n_{n}-n_ {i}) \tag{5}\]  
The relationship between the electric field and the electric potential is as follows:  
\[E=-\frac{\partial V}{\partial x} \tag{6}\]  
Here, \(\varepsilon_{0}\) is the permittivity of free space, and V is the electric potential. The gas temperature is fixed at 300 K. The boundary conditions are listed in Table 2[22; 27; 28]. The boundary conditions are very important and complex in the numerical calculations. In this study, the complex boundary conditions are not considered. The electron flux at the wall is caused by the secondary emission and recombination with the wall. The fluxes of ions are determined by the electric drift. For the neutral species, these are unaffected by the electric drift and are due to the fact that the mass of the particles is much higher than that of electrons; we can assume that these are zero at the wall. The electron temperature was set to 0.5 eV according to Ref. [22]. The electron temperature of 1 eV at the walls was used to test the model.