Cold, Non-Thermal Plasma

Cold, non-thermal plasma consists of weakly ionized particles which are generated by heating electrons through inductive or capacitive coupling.

Transport Equations

The Coupled Plasma Electron model solves two transport equations using a coupled approach—one for the electron number density $n_{e}$ , and another for the electron energy density $\in_{e}$ . The Charge Accumulation model solves for the ion flux to the electrodes and walls, and the associated accumulated charge.

Number Density: The number density of the electron is evolved using the drift-diffusion-convection-reaction equation:
Figure 1. EQUATION_DISPLAY

$\frac{\partial n_{e}}{\partial t} + \nabla \cdot Γ_{e} = S_{e}$

(4184)
Electron Flux: The electron flux is given by:
Figure 2. EQUATION_DISPLAY

$Γ_{e} = - μ_{e} E n_{e} - D_{e} \nabla n_{e} + υ n_{e}$

(4185 4191)
where $E$ is the electric field, $υ_{}$ is the gas velocity, ${D_{e}}_{}$ is the electron molecular diffusivity, and $μ_{e}$ is the electron mobility. The energy density of the electrons is evolved similarly:
Figure 3. EQUATION_DISPLAY

$\frac{\partial \in_{e}}{\partial t} + \nabla \cdot Γ_{\in} = S_{\in}$

(4186); For interfaces on which charge accumulation is simulated, the electron flux normal that is directed towards the electrodes or walls (assuming no reflection or secondary transmission) is given by [843]:
Figure 4. EQUATION_DISPLAY

$Γ_{e} = \frac{1}{4} n_{e} υ_{e, t h}$

(4187)
where $υ_{e, t h}$ is the electron thermal velocity, given by Eqn. (4197).
Electron Energy Flux: The electron energy flux is given by [843]:
Figure 5. EQUATION_DISPLAY

$Γ_{\in} = \frac{5}{3} (- μ_{e} E \in_{e} - D_{e} \nabla \in_{e} + υ \in_{e})$

(4188); where a Maxwellian electron energy distribution function (EEDF) is assumed and the electron temperature $T_{e}$ is given by:
Figure 6. EQUATION_DISPLAY

$T_{e} = \frac{2}{3} \frac{\in_{e}}{k_{B} n_{e}}$

(4189); For interfaces on which charge accumulation is simulated, the electron energy flux to the electrodes and walls is given by:
Figure 7. EQUATION_DISPLAY

$Γ_{\in} = \frac{1}{4} n_{e} υ_{e, t h} (2 k_{B} T_{e})$

(4190)
Ion Flux: When simulating charge accumulation, and the ion drift velocity is directed to the electrodes and walls, the ion flux is given by [843]:
Figure 8. EQUATION_DISPLAY

$Γ_{p, n} = n_{p} μ_{p} E_{n} - D \nabla c_{p}$

(4185 4191)
otherwise, the ion flux is zero.; $D$ is the effective diffusivity and $\nabla c_{p}$ is the gradient of species $c_{p}$ .

Source Terms

Source term Contributions for Electron Number Density.

The electron number density source term is given by:

Figure 9. EQUATION_DISPLAY

S_{e} = S_{e, u s r}

(4192)

where

S_{e, u s r}

is the electron production / consumption user source term.

Since the electrons are added without energy, the electrons cool down—unless an electron energy source is added.

The collective electron energy density source term is given by:

Figure 10. EQUATION_DISPLAY

S_{\in} = S_{Ω} + S_{e l} + S_{\in, u s r}

(4193)

where:

$S_{Ω}$ is the Ohmic heating source term and is given by:
Figure 11. EQUATION_DISPLAY

$S_{Ω} = - q_{e} Γ_{e} \cdot E$

(4194)

This source term accounts for the increase in energy when electrons are running against an electric field and the loss of energy when the electrons are running in the direction of an electric field.
$S_{e l}$ is the elastic collision source term which accounts for the elastic collisions of the electrons with other main gas species. This source term arises from the collisions of the electrons with the background gas components and is given by:
Figure 12. EQUATION_DISPLAY

$S_{e l} = \sum_{ζ} - 3 \frac{M_{e}}{M_{ζ}} k_{B} {(T}_{e} {- T) K}_{ζ e}$

(4195)
where $ζ$ is the sum over the background gas components with a molar mass of $M_{ζ}$ . The electron momentum transfer collisional rate $K_{ζ e}$ is given by:
Figure 13. EQUATION_DISPLAY

$K_{ζ e} = σ_{ζ e} n_{ζ} n_{e} υ_{t h, e}$

(4196)
where
- $σ_{ζ e}$ is the user-specified effective elastic collision momentum transfer cross-section which is specific to the type of background gas components.
- $n_{ζ}$ is the number density of the main gas.
- $υ_{t h, e}$ is the thermal velocity of the electrons:
  Figure 14. EQUATION_DISPLAY
  
  $υ_{t h, e} = \sqrt{\frac{8}{π} \frac{k_{B} T_{e}}{m_{e}}}$
  
  (4197)

$S_{ϵ, u s r}$ is the electron energy density user source term.