2024年5月11日发(作者：)

Solved with COMSOL Multiphysics 5.2

DC Glow Discharge

Introduction

DC glow discharges in the low pressure regime have long been used for gas lasers and

fluorescent lamps. DC discharges are attractive to study because the solution is time

independent. This model shows how to use the DC Discharge interface to set up an

analysis of a positive column. The discharge is sustained by emission of secondary

electrons at the cathode.

Model Definition

The DC discharge consists of two electrodes, one powered (the anode) and one

grounded (the cathode). The positive column is coupled to an external circuit:

1000

Ω

Cathode

Plasma

Anode

1 pF

V

Figure 1: Schematic of the DC discharge and external circuit.

DOMAIN EQUATIONS

The electron density and mean electron energy are computed by solving a pair of

drift-diffusion equations for the electron density and mean electron energy.

Convection of electrons due to fluid motion is neglected. For detailed information on

electron transport see Theory for the Drift Diffusion Interface in the Plasma Module

User’s Guide.

∂

n

()+∇⋅[

–

n

e

(

μ

e

•E)

–

D

e

•∇n

e

]

=

R

e

∂t

e

∂

n

()+∇⋅[

–

n

ε

(

μ

ε

•E)

–

D

ε

•∇n

ε

]+E⋅Γ

e

=

R

ε

∂t

ε

where:

1 |

DC GLOW DISCHARGE

Solved with COMSOL Multiphysics 5.1

Γ

e

=

–(

μ

e

•E)n

e

–

D

e

•∇n

e

The electron source R

e

and the energy loss due to inelastic collisions R

ε

are defined

later. The electron diffusivity, energy mobility and energy diffusivity are computed

from the electron mobility using:

5

D

e

=

μ

e

T

e

,

μ

ε

=

--

μ

e

,

D

ε

=

μ

ε

T

e

3

The source coefficients in the above equations are determined by the plasma chemistry

using rate coefficients. Suppose that there are M reactions which contribute to the

growth or decay of electron density and P inelastic electron-neutral collisions. In

general P >> M. In the case of rate coefficients, the electron source term is given by:

M

R

e

=

x

j

k

j

N

n

n

e

j=1

where x

j

is the mole fraction of the target species for reaction j, k

j

is the rate coefficient

for reaction j (m

3

/s), and N

n

is the total neutral number density (1/m

3

). For DC

discharges it is better practice to use Townsend coefficients instead of rate coefficients

to define reaction rates. Townsend coefficients provide a better description of what

happens in the cathode fall region Ref 1. When Townsend coefficients are used, the

electron source term is given by:

M

R

e

=

x

j

α

j

N

n

Γ

e

j=1

where α

j

is the Townsend coefficient for reaction j (m

2

) and Γ

e

is the electron flux as

defined above (1/(m

2

·s)). Townsend coefficients can increase the stability of the

numerical scheme when the electron flux is field driven as is the case with DC

discharges. The electron energy loss is obtained by summing the collisional energy loss

over all reactions:

P

R

ε

=

x

j

k

j

N

n

n

e

Δε

j

j=1

where Δε

j

is the energy loss from reaction j (V). The rate coefficients may be computed

from cross section data by the following integral:

2 |

DC GLOW DISCHARGE