## 2 Fluid Model, Plasma Chemistry, and Reactor Configuration
### Fluid Model Formulae
#### 2.1.1 Electrons Equation  
The equations of electron density and energy are given as follows:  
\[\begin{split}\frac{\partial n_{e}}{\partial t}+\
abla\cdot \mathbf{\Gamma}_{e}=R_{e},\\ \frac{\partial n_{e}}{\partial t}+\
abla\cdot\mathbf{\Gamma}_{e} +\mathbf{E}\cdot\mathbf{\Gamma}_{e}=R_{e}+P_{ohm}.\end{split} \tag{1}\]  
The vectoral fluxes of electron density and energy at the assumption of drift and diffusion are described in Equation (2).  
\[\begin{split}\mathbf{\Gamma}_{e}&=-\mu_{e}n_{e} \mathbf{E}-D_{e}\
abla n_{e},\\ \mathbf{\Gamma}_{e}&=-\mu_{e}n_{e}\mathbf{E}-D_{e} \
abla n_{e}.\end{split} \tag{2}\]  
Herein, \(n_{e}\) and \(n_{e}\) are the number density and energy density of electrons, respectively. Herein, \(n_{e}=\frac{3}{2}n_{e}k_{B}T_{e}\). \(k_{B}\) is the [PERSON]'s constant and \(T_{e}\) is the electron temperature in a unit of Kelvin. \(\mu_{e}\) and \(\mu_{e}\) are the electron mobility and electron energy mobility, respectively. \(D_{e}\) and \(D_{e}\) are electron diffusivity and electron energy diffusivity, respectively. The relations among the above mass and energy transport coefficients are \(D_{e}=\mu_{e}T_{e}\), \(D_{e}=\mu_{e}T_{e}\), and \(\mu_{e}=\frac{5}{3}\mu_{e}\). \(\mathbf{E}\) is the vectoral and electrostatic field intensity caused by the ambi-polar diffusion and sheath.  
\(R_{e}\) and \(R_{e}\) in Equation (1) are the respective source terms of number density and energy density of electrons. Their expressions are stated in Equation (3).  
\[\begin{split} R_{e}&=\sum\limits_{j=1}^{M}l_{j}k_{ j}\prod\limits_{m=1}^{p}{n_{m}}^{v_{jm}},\\ R_{e}&=\sum\limits_{j=1}^{M}\varepsilon_{j}l_{j}k_{ j}\prod\limits_{m=1}^{p}{n_{m}}^{v_{jm}}.\end{split} \tag{3}\]  
Herein, \(l_{j}\) is the number of electrons created (lost) per electron-impact reaction that generates (depletes) electrons. \(M\) is the number of these reactions. \(k_{j}\) is the rate coefficient of reaction \(j\), which is expressed in the Sections 2.2 and 2.3 for the Ar/SF\({}_{6}\) and Ar/O\({}_{2}\) plasmas, respectively. \(n_{m}\) is the number density of reactant m of reaction \(j\). \(\
u\) is the stoichiometric coefficient of the reaction and \(P\) is the number of reactants. \(\varepsilon_{j}\) is the electron energy loss per electron-impact reaction.  
\(P_{ohm}\) is the deposited power density via the [PERSON]'s heating scheme, illustrated in Equation (4).  
\[P_{ohm}=\frac{1}{2}\text{Re}(\sigma|E_{\theta}|^{2}). \tag{4}\]  
Herein, \(\sigma\) is the electron conductivity. \(E_{\theta}\) is the radio frequency azimuthal field, which is calculated from [PERSON]'s equation; see next the Section 2.1.3. Without the secondary electron emissions, the boundary conditions for the above equations are set in Equation (5).  
\[\begin{array}{l}\mathbf{n}\cdot\mathbf{\Gamma}_{e}=\frac{1-\
u_{e}}{1+\
u_{e}}( \frac{1}{2}\
u_{e,th}n_{e}),\\ \mathbf{n}\cdot\mathbf{\Gamma}_{\varepsilon}=\frac{1-\
u_{e}}{1+\
u_{e}}( \frac{5}{6}\
u_{e,th}n_{e}).\end{array} \tag{5}\]  
Herein, \(v_{e,th}\) is the thermal velocity of electrons. \(r_{e}\) is the reflection coefficient of electrons from the reactor wall, which is set to 0.2 in this model.