## 2 Methodology
### Numerical Method
#### 2.2.1 One-Dimensional Plasma Fluid Model (1D PFM)  
This section briefs the 1D PFM developed previously [24] with the framework of discharge chemistry including helium and nitrogen related species. Figure 2a shows the numerical domain of the 1D PFM. The discharge gap is 5.0 mm and the thickness of each glass plate is 0.7 mm as used in experimental measurements. The non-uniform mesh is employed with the mesh size covering from 30 \(\upmu\)m to 100 \(\upmu\)m for capturing discharge dynamics appropriately in the sheath and bulk regions. The mesh independence test has been conducted to ensure the mesh quality. The continuity equation of each species considered is solved as:  
\[\frac{\partial n_{e,i,nc}}{\partial t}+\frac{\partial\Gamma_{e,i,nc}}{\partial x }=S_{e,i,nc} \tag{1}\]  
where \(n\) is the species density, \(\Gamma\) is the species flux, and the subscripts \(e\), \(i\), and \(uc\) represent electron, ion, and neutral species, respectively. The source term \(S\) of each continuity equation is evaluated from the discharge chemistry introduced next to account for the generation and destruction reactions of each species. The species flux terms calculated with the drift-diffusion approximation are written as:  
Figure 1: The schematic diagrams of (**a**) the experimental configuration; (**b**) the measurement of the emission spectrum with an optical emission spectrometer (OES). \[\Gamma_{e,i}=sign(q_{e,i})\mu_{e,i}n_{e,i}E-D_{e,i}\frac{\partial n_{e,i}}{\partial x} \tag{2}\]  
\[\Gamma_{uc}=-D_{uc}\frac{\partial n_{uc}}{\partial x} \tag{3}\]  
where \(q\) is the species charge, \(\mu\) is the species mobility, \(E\) (\(=-\frac{\partial\varphi}{\partial x}\), \(\varphi\) is the potential) is the electric field, and \(D\) is the species diffusivity. The Poisson equation is solved to calculate the electric field built inside the discharge gap as:  
\[\
abla\cdot(\varepsilon\
abla\varphi)=-\sum_{c=1}^{N_{c}}q_{c}n_{c} \tag{4}\]  
where \(\varepsilon\) (\(=\varepsilon_{0}\varepsilon_{r}\), \(\varepsilon_{0}\) is the permittivity of free space and \(\varepsilon_{r}\) is the relative permittivity of each region) is the permittivity of each region and \(N_{c}\) represents the number of charged species. The electron energy density equation is solved to evaluate the electron temperature for determining rate constants of electron-related reactions and electron transport properties (i.e., \(D_{e}\) and \(\mu_{e}\)) as:  
\[\frac{\partial n_{e}}{\partial t}+\frac{\partial\Gamma_{e}}{\partial x}=- \varepsilon\Gamma_{e}\cdot E-\sum_{k=1}^{N_{B}}\left[3\frac{m_{e}}{M_{k}}n_{ e}k_{B}v_{m,k}\Big{(}T_{e}-T_{g}\Big{)}\right]-n_{e}\sum_{j=1}^{N_{e}} \varepsilon_{j}k_{j}n_{j} \tag{5}\]  
where \(n_{e}\Big{(}\)\(=\frac{3}{2}n_{e}k_{B}T_{e}\Big{)}\) is the electron energy density, \(k_{B}\) is the Boltzmann constant, \(T_{e}\) is the electron temperature, \(m_{e}\) is the electron mass, \(M\) is the mass of the background gas (i.e., He or N\({}_{2}\)), \(v_{m}\) is the frequency of elastic collision, \(N_{B}\) is the number of background gases (i.e., \(N_{B}\) = 2 for He and N\({}_{2}\)), \(T_{g}\) is  
Figure 2: The numerical domains: (**a**) 1D PFM; (**b**) three-dimensional gas flow model (3D GFM). the gas temperature, \(\varepsilon_{j}\) is the threshold energy of the \(j\)th electron-impact reaction, \(k\) is the rate constant, and \(N_{e}\) is the number of electron-impact reactions considered in the chemical model. On the right-hand side of Equation (5), the first term is the electron Joule heating evaluating the energy acquired by electrons from the externally applied voltage. The second term calculates the energy loss due to elastic collisions among electrons and neutral background gases (i.e., He and N\({}_{2}\)). The last term calculates the energy consumption due to electron-impact reactions considered in the discharge chemistry. It is assumed that charged species accumulate and neutral species quench at the glass surface.