## 4 Simulation
### Governing Equations
#### 4.3.2 Plasma Model  
Normally, it is known that the gas flow and the temperature highly affect the spatial spreading of neutral gases [43]. However, in this model, the neutral fluid velocity was set to 0 and the neutral gas temperature was set to 300 K to simplify the model in the 3-D domain. Hence, the equations used in this simulation and the terms related to the neutral fluid velocity were omitted. \[\frac{\partial}{\partial t}(n_{e})+\
abla\cdot\Gamma_{e}=R_{e} \tag{7}\]  
\[\Gamma_{e}=-\big{(}\mu_{e}\cdot\overline{E}\big{)}n_{e}-(D_{e}\cdot\
abla n_{e}) \tag{8}\]  
\[\mu_{e}=\frac{q}{m_{e}v_{en}},\ D_{e}=\frac{k_{B}T_{e}}{m_{e}v_{en}} \tag{9}\]  
The continuity equation for the electron density is given as above. Here, \(\overline{E}\) denotes the electric field(DC field), \(n_{e}\) is the electron density and \(R_{e}\) is electron production rate. In this simulation, the mobility of electrons was calculated as a function of the electron energy by using BOLSIG+ [50], and the diffusivity was determined from [PERSON]'s relation. In this model, the electron mobility used to calculate the electron transport properties was taken from the electron impact reactions set to \(v_{en}\). Here, \(\mu_{e}\) is the electron mobility, and \(D_{e}\) is the electron diffusivity [49,50,51,52,53]. \[\frac{\partial}{\partial t}(n_{e})+\
abla\cdot\Gamma_{e}+\overline{E}\cdot \Gamma_{e}=S_{en}+Q_{h}/q \tag{10}\]  
\[\Gamma_{e}=-\big{(}\mu_{e}\cdot\overline{E}\big{)}n_{e}-\
abla(D_{e}n_{e}) \tag{11}\]  
\[\mu_{e}=(\frac{5}{3})\mu_{e},\ D_{e}=\mu_{e}T_{e} \tag{12}\]The conservation equation for the electron energy density is given by the aforementioned equations, where, \(n_{e}\) is the electron energy density, \(S_{en}\) is the energy loss/gain from the collisions, and \(Q_{k}\) is the heat source for plasma model absorbed from the electromagnetic wave power. In addition, \(T_{e}\) is the electron temperature, \(\mu_{e}\) is the electron energy mobility, and \(D_{e}\) is the electron energy diffusivity [51; 52; 53; 54]. Both \(\mu_{e}\) and \(D_{e}\) are calculated from the electron mobility \(\mu_{e}\), using the Einstein relation. \[\rho\frac{\partial}{\partial t}(w_{k})=\
abla\cdot\mathrm{j}_{k}+R_{k} \tag{13}\]  
\[\mathrm{j}_{k}=\rho w_{k}\mathrm{v}_{k} \tag{14}\]  
\[\mathrm{v}_{k}=D_{k}\frac{\
abla\omega_{k}}{\omega_{k}}-z_{k}\mu_{k}\overline {E} \tag{15}\]  
As aforementioned, for non-electron species (neutral, excited species, and ions), the equation is solved for the mass fraction of each species. This is the simplified form of the Maxwell-Stefan equation, which makes it computationally less expensive in solving the transport equations. Here, \(w_{k}\) is the mass fraction of the kth species, \(\mathrm{j}_{k}\) is the diffusive flux vector, and \(R_{k}\) is the rate coefficient for species \(k\). The rate coefficients of electron-heavy particle reactions are calculated from the cross-section of the corresponding reaction \(\sigma\) as a mean value [34]. In addition, \(\rho\) is the density of the mixture, \(\mathrm{v}_{k}\) is the diffusion velocity for species \(k\), \(D_{k}\) is the diffusion coefficient (diffusion coefficient of Ar: \(D_{Ar}\) is 0.01 m\({}^{2}\)/s, which is same for Ar\({}^{*}\) and Ar\({}^{+}\)), \(z_{k}\) is the charge number of species \(k\), and \(\mu_{k}\) is the mobility for species \(k\)[55; 56; 57; 58; 59]. \[\overline{E}=-\
abla V,\ \
abla\left(\epsilon_{0}\epsilon_{r}\overline{E} \right)=\rho \tag{16}\]  
\[\Gamma_{e}=-\big{(}\mu_{e}\cdot\overline{E}\big{)}n_{e}-(D_{e}\cdot\
abla n_{e}) \tag{8}\]0  
As aforementioned, the electrostatic field can be calculated using [PERSON]'s equation. The space charge density, \(\rho\), is computed based on the plasma chemistry (electron density and ion density) [35]. \(V\) is the electric potential and \(n_{k}\) is the number density of species \(k\). The boundary conditions of the waveguiding structure power input ports were used to supply electromagnetic waves (input power) in the active ports.