Ambipolar Diffusion

For plasma simulations in which charge neutrality is assumed, the Ambipolar Diffusivity model allows you to account for migration in the species transport equation by defining an effective diffusivity.

The ambipolar assumption states that, when starting from a charge-neutral plasma state, if the net flux of all negatively charged species balances the net flux of all positively charged species, the plasma will remain charge neutral. This assumption—when expressed mathematically—ensures charge neutrality by providing an expression for the electric field that is induced by the charged species. From this electric field, referred to as the Ambipolar Electric Field, the effective ambipolar diffusivity for each of the charged species components can be computed.

The species transport equation is of the form:

Figure 1. EQUATION_DISPLAY

\frac{\partial n_{i}}{\partial t} + C = M + D + R

(4201)

where $n_{i}$ is the number density of species $i$ . $C$ , $M$ , $D$ , and $R$ denote the convection, migration, diffusion, and reaction terms, respectively.

Since all species convect with the same velocity, and the reaction sources ensure charge conservation, only the migrative and diffusive fluxes can cause charge imbalance.

The sum of the migrative and diffusive fluxes of species

i

can be represented by

Γ_{i}

where:

Figure 2. EQUATION_DISPLAY

Γ_{i} = μ_{i} n_{i} E - D_{i} \nabla n_{i}

(4202)

where $μ_{i}$ is the mobility (including the sign -/+ for charged species), $E$ is the electric field, and $D_{i}$ is the molecular diffusivity of species $i$ .

The ambipolar assumption states that the net flux of all of the negatively charged species (including the electron species) must balance the net flux of all of the positively charged species:

\sum_{i} Γ_{i}^{-} + Γ_{e} = \sum Γ_{i}^{+}

(4203)

where $Γ_{e}$ explicitly denotes the flux of electrons, and $Γ_{i}^{-}$ and $Γ_{i}^{+}$ denote the flux of other negatively and positively charged species, respectively.

The assumptions above lead to the following expression for the ambipolar electric field, $E_{A}$ :

Figure 3. EQUATION_DISPLAY

E_{A} = \frac{\sum_{i}^{+} D_{i} \nabla n_{i} - \sum_{i}^{-} D_{i} \nabla n_{i} - D_{e} \nabla n_{e}}{\sum_{i}^{+} μ_{i} n_{i} - \sum_{i}^{-} μ_{i} n_{i} - μ_{e} n_{e}}

(4204)

where the subscript $e$ denotes electrons, and the superscripts $+$ and $-$ denote positively and negatively charged ions, respectively.

Ambipolar Diffusivity for Two-Component Plasma

In the special case of only two charged components—one electron $e$ and one positively charged species $p$ —the expression for the ambipolar electric field $E_{A}$ simplifies to:

Figure 4. EQUATION_DISPLAY

E_{A} = \frac{D_{p} \nabla n_{p} - D_{e} \nabla n_{e}}{μ_{p} n_{p} - μ_{e} n_{e}}

(4205)

and since for charge neutrality

n_{e}

must equal

n_{p}

Figure 5. EQUATION_DISPLAY

E_{A} = (\frac{D_{p} - D_{e}}{μ_{p} - μ_{e}}) \frac{\nabla n_{e}}{n_{e}}

(4206)

This expression for the ambipolar electric field leads to the expression for ambipolar diffusivity:

Figure 6. EQUATION_DISPLAY

D_{A} = \frac{μ_{p} D_{e} - μ_{e} D_{p}}{μ_{p} - μ_{e}}

(4207)

The ambipolar diffusivity

D_{A}

is the same for electrons and positively charged species.

The flux for electrons or charged species is then given by:

Figure 7. EQUATION_DISPLAY

Γ_{e} = Γ_{p} = - D_{A} \nabla n_{e} = - D_{A} \nabla n_{p}

(4208)

Simplified Two-Component Ambipolar Diffusivity for Multi-Component Plasma

A simplified two-component diffusivity expression for multi-component plasma is derived (based on findings by Chau et al [842]) by defining averaged material properties as follows.

For all of the positively charged species, an averaged diffusivity ${\bar{D}}_{p}$ and an averaged mobility ${\bar{μ}}_{p}$ are defined:

Figure 8. EQUATION_DISPLAY

{\bar{D}}_{p} = \frac{\sum_{i}^{+} X_{i}}{\sum_{i}^{+} X_{i} / D_{i}}

(4209)

Figure 9. EQUATION_DISPLAY

{\bar{μ}}_{p} = \frac{\sum_{i}^{+} X_{i}}{\sum_{i}^{+} X_{i} / μ_{i}}

(4210)

If electrons and negatively charged ions are both present, then the averaged diffusivity and averaged mobility for negatively charged ions are taken to be those of the electrons:

Figure 10. EQUATION_DISPLAY

{\bar{D}}_{n} = D_{e}

(4211)

Figure 11. EQUATION_DISPLAY

{\bar{μ}}_{n} = μ_{e}

(4212)

otherwise (when negatively charged ions are present and electrons are not) the averaged properties are defined using the negatively charged ions:

Figure 12. EQUATION_DISPLAY

{\bar{D}}_{n} = \frac{\sum_{i}^{-} X_{i}}{\sum_{i}^{-} X_{i} / D_{i}}

(4213)

Figure 13. EQUATION_DISPLAY

{\bar{μ}}_{n} = \frac{\sum_{i}^{-} X_{i}}{\sum_{i}^{-} X_{i} / μ_{i}}

(4214)

Given the averaged properties for positively charged ions and negatively charged ions, the ambipolar diffusivity is defined using the two-component expression as follows:

Figure 14. EQUATION_DISPLAY

D_{A} = \frac{{\bar{μ}}_{p} {\bar{D}}_{n} - {\bar{μ}}_{n} {\bar{D}}_{p}}{{\bar{μ}}_{p} - {\bar{μ}}_{n}}

(4215)

The ambipolar electric field is then also calculated using the simplified two-component expression:

Figure 15. EQUATION_DISPLAY

E_{A} = (\frac{{\bar{D}}_{p} - {\bar{D}}_{n}}{μ_{p} - μ_{n}}) \frac{\nabla n_{e}}{n_{e}}

(4216)

Ohmic Heating Source Term

When computing the Ohmic heating source term with the ambipolar electric field, the electron flux

Γ_{e}

is calculated as:

Figure 16. EQUATION_DISPLAY

Γ_{e} = - D_{A} \nabla n_{e}

(4217)

Using this electron flux

Γ_{e}

, the Ohmic source term is then calculated as:

Figure 17. EQUATION_DISPLAY

s_{Ω} = - q_{e} Γ_{e} \cdot E_{A}

(4218)

where

q_{e}

is the electron charge, and

E_{A}

is the ambipolar electric field.