COMSOL 6.4 - DC Glow Discharge, 1D

DC glow discharges in the low pressure regime have long been used for gas lasers and fluorescent lamps. DC discharges are good to study because the solution is time independent. This model shows how to use the 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):

Figure 1: Schematic of the DC discharge. The voltage applied across the electrodes leads to formation of a plasma.

domain equations

Electron transport is modeled by solving the continuity equation, the momentum equation under the drift-diffusion approximation, and the mean electron energy equation (for detailed information on electron transport, see Theory for the Drift Diffusion Interface in the Plasma Module User’s Guide)

The source coefficients in the above equations are determined by the plasma chemistry. The electron rate expression is defined as

where νe,j is the stoichiometric coefficient, and the reaction rate is defined as

where kjf is the forward rate constant and kjr is the reversed rate constant. Both the Electron Impact Reaction feature and Reaction feature can contribute to the electron rate expression. However, when using the Reaction feature it is important to note that the associated electron energy gain or loss is not included in the source term of the electron mean energy equation.

The rate constants can be computed from electron impact cross-section data

where γ = (2q/me)1/2 (SI unit: C1/2/kg1/2), me is the electron mass (SI unit: kg), ε is the electron energy (SI unit: V), σ is the electron impact collision cross section (SI unit: m2), and f is the electron energy distribution function.

When Townsend coefficients are used, the reaction rate is defined as

where αj/Nn is the reduced Townsend coefficient for reaction j (SI unit: m2) and Γe is the electron flux as defined above (SI unit: 1/(m2·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 total electron energy loss or gained is calculated by summing the collisional energy changes from all reactions defined with the Electron Impact Reaction feature as

where Δεj is the energy loss from reaction j (SI unit: V) and F is the Faraday constant (SI unit: C/mol). For excitation and ionization collisions Δεj corresponds to the energy of the excited state being excited/deexcited or ionized, for attachment Δεj is set to zero, and for elastic collisions

where me and mk are the electron and heavy species mass in kg, Te is the electron temperature in eV, and Tgas is the gas temperature in K.

For heavy species, the following equation is solved for the mass fraction of each species (for detailed information on the transport of the nonelectron species, see Theory for the Heavy Species Transport Interface in the Plasma Module User’s Guide):

The electrostatic field is computed using the following equation:

The space charge density ρ is automatically computed based on the plasma chemistry specified in the model using the formula

For detailed information about electrostatics see Theory for the Electrostatics Interface in the Plasma Module User’s Guide.

boundary conditions

Unlike RF discharges, the mechanism for sustaining the discharge is emission of secondary electrons from the cathode. An electron is emitted from the cathode surface with a specified probability when struck by an ion. These electrons are then accelerated by the strong electric field close to the cathode where they acquire enough energy to initiate ionization. The net result is a rapid increase in the electron density close to the cathode in a region often known as the cathode fall or Crookes dark space.

Electrons are lost to the wall due to random motion within a few mean free paths of the wall and gained due to secondary emission effects, resulting in the electron flux boundary condition

(1)

and the electron energy flux boundary condition

(2)

The second term on the right-hand side of Equation 1 is the gain of electrons due to secondary emission effects, γp being the secondary emission coefficient. The second term in Equation 2 is the secondary emission energy flux, εp being the mean energy of the secondary electrons. For the heavy species, ions are lost to the wall due to surface reactions and the fact that the electric field is directed toward the wall:

plasma chemistry

Argon is one of the simplest mechanisms to implement at low pressures. The electronically excited states can be lumped into a single species which results in a chemical mechanism consisting of only 3 species and 7 reactions (electrons impact cross sections are obtained from Ref. 2):

Table 1: table of collisions and reactions modeled.
Reaction	Formula	Type
1	e+Ar=>e+Ar	Elastic	0
2	e+Ar=>e+Ars	Excitation	11.5
3	e+Ars=>e+Ar	Superelastic	-11.5
4	e+Ar=>2e+Ar+	Ionization	15.8
5	e+Ars=>2e+Ar+	Ionization	4.24
6	Ars+Ars=>e+Ar+Ar+	Penning ionization	-
7	Ars+Ar=>Ar+Ar	Metastable quenching	-

In this discharge, the electron density and the density of the excited species are relatively low, so stepwise ionization is not as important as in high density discharges. In addition to volumetric reactions, the following surface reactions are implemented:

Table 2: table of surface reactions.
Reaction	Formula	Sticking coefficient
1	Ars=>Ar	1
2	Ar+=>Ar	1

When a metastable argon atom makes contact with the wall, it reverts to the ground state argon atom with some probability (the sticking coefficient).

Results and Discussion

The electric potential, electron density, and mean electron energy are all quantities of interest. Most of the variation in each of these quantities occurs along the axial length of the column. Figure 2 plots the electron density in the column. The electron density peaks in the region between the cathode fall and positive column. This region is sometimes referred to as Faraday dark space. The electron density obtained in this 1D model is different than that obtained in the 2D model because the diffusive loss of electrons to the outer walls and the accumulation of surface charge on the walls are not modeled.

Figure 2: Electron density along the axial length of the positive column.

In Figure 3 the electric potential is plotted along the axial length of the column. Notice that the potential profile is significantly different from the linear drop in potential which results in the absence of the plasma. The strong electric field in the cathode region can lead to high energy ion bombardment of the cathode. Heating of the cathode surface occurs which may in turn lead to thermal electron emission where additional electrons are emitted from the cathode surface.

Figure 3: Electron temperature along the axial length of the positive column.

Figure 4: Potential along the axial length of the positive column.

Figure 5: Mass fraction of excited argon atoms along the axial length of the positive column.

Figure 6: Number density of argon ions along the axial length of the positive column.

The plasma current due to electrons, ions and their sum is plotted in Figure 7. As expected, the ion current is highest at the cathode and increases sharply in the cathode fall region. The ion bombardment of the cathode results in an electron current released from the electrode. The electron current increases sharply in the cathode fall region because the high electron temperature results in production of new electrons which then contribute to the total electron current. Once the electrons pass the cathode fall region, the electron current density further increases due to production of new electrons through electron impact ionization with the background gas.