Atmospheric Pressure Corona Discharge

This model simulates a corona discharge occurring between two coaxially fashioned conductors. The negative electric potential is applied to the inner conductor and the exterior conductor is grounded. The discharge gas simulated is argon at atmospheric pressure.

Model Definition

Figure 1 shows a cross section of the model. By considering a long and uniform coaxial conductor configuration, the model can be viewed as axisymmetric and thus simplified to a 1D problem.

The model presented in the following section is used to simulate the ionization of the neutral gas (Ar) as well as the flux of charged particles (Ar+ and electrons) when the negative electric potential is applied at the inner conductor (cathode). The high electric field generated by the combination of high potential and small conductor curvature radius (inner conductor, ri) causes electron drift and ionization of the neutral gas surrounding the cathode. The resulting ions generate more electrons through secondary emission at the cathode surface. These electrons are accelerated through a small region away from the cathode, where they can acquire significant energy. This can lead to ionization which creates new electron-ion pairs. The secondary ions migrate toward the cathode where they eject more secondary electrons. This process is responsible for sustaining the discharge.

The model is based on the fluid equations for electrons and ions as well as on Poisson’s equation. Secondary electrons generated by ion bombardment of the cathode surface are taken into account. The model uses a Scharfetter–Gummel upwind scheme to eliminate numerical instabilities in the number density of the charged particles associated with the finite element method. This is needed, in particular close to the cathode, where the ion flux is particularly high.

Figure 1: Not-to-scale cross section of the coaxial configuration. The negative potential (−Vin) is applied at the inner conductor (cathode) and the outer electrode is grounded (anode). The shaded area represents the ionization region created by the positive space charge distribution generated in the vicinity of the cathode.

Domain Equations

The electron density and mean electron energy are computed by solving the drift-diffusion equations for the electron density and mean electron energy. Convection of electrons due to fluid motion is neglected. For more detailed information on electron transport see the section Theory for the Drift Diffusion Interface in the Plasma Module User’s Guide.

The electron source Re and the energy loss due to inelastic collisions Rε are defined later. The electron diffusivity, energy mobility, and energy diffusivity are computed from the electron mobility using the relations

The source coefficients in the above equations are determined by the plasma chemistry using rate coefficients. Suppose that there are M reactions that contribute to the growth or decay of electron density and P inelastic electron-neutral collisions. In general

. In the case of rate coefficients, the electron source term is given by

where xj is the mole fraction of the target species for reaction j, kj is the rate coefficient for reaction j (SI unit: m3/s), and Nn is the total neutral number density (SI unit: 1/m3). The electron energy loss is obtained by summing the collisional energy loss over all reactions:

Here Δεj is the energy loss from reaction j (SI unit: V). The rate coefficients can be computed from cross-section data as the integrals

where γ = (2q/me)1/2 (SI unit: C1/2/kg1/2), me is the electron mass (SI unit: kg), ε is the energy (SI unit: V), σk is the collision cross section (SI unit: m2), and f is the electron energy distribution function. In this case, a Maxwellian EEDF is assumed.

For non-electron species, the following equation is solved for the mass fraction of each species:

For detailed information on the transport of the non-electron species see the section Theory for the Heavy Species Transport Interface in the Plasma Module User’s Guide.

The electrostatic field is computed using the 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

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 boundary condition

(1)

for the electron flux and

(2)

for the electron energy flux. 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:

The discharge is driven by the electric potential applied to the inner conductor of the coaxial geometry (at coordinate r = ri).

where the function tanh(t/τ) is used to generate the voltage step function (−1000 V). The other boundary (at coordinate r = ro) is grounded.

Note that, during the simulation, the cathode is submitted to an intense flux of ions that generate an important amount of secondary electrons which, in turn, increase the cathode ion bombardment and so on. In order to prevent this avalanche from increasing indefinitely, a RC circuit has been added in series with the system. In order to model the circuit addition, the voltage at the inner conductor is modified using the differential equation

where Ip is defined as

and where n · Ji is the normal component of the total ion current density at the wall, n · Je is the normal component of the total electron current density at the wall, and n · D the normal electrical displacement at the surface.

Plasma Chemistry

Argon is an attractive gas to use in a benchmark problem because only a handful of reactions and a few species need to be considered. Table 1 lists the chemical reactions considered.

Table 1: Table of collisions and reactions modeled.
Reaction	Formula	Type	Δε (eV)	kf (m3/(s·mol))
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	Ar+Ars=>Ar+Ar	Reaction	-	1807
7	Ars+Ars=>Ars+Ar	Reaction	-	2.3·107

Initially, a small number of seed electrons are present. These electrons are necessary in order to initiate the discharge. In addition to the 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 the ion atoms reach the wall, they are assumed to change back to neutral argon atoms and donate their charge to the wall. Note that the secondary emission coefficient is set to 0.2 on the cathode boundary (at coordinate r = ri) and to 0 at the outer electrode (at coordinate r = ro). The mean electron energy of the secondary electron is set to 4 eV.

Results and Discussion

When solving the problem, you can choose to plot the electron and ion density as the solver runs. Doing so, you can observe the behavior of the charged species as the model is solving.

Looking at the evolution of the densities during the solving process shows that the gas is initially weekly ionized (electrons and ions having relatively low densities compared to neutral atoms). As the negative voltage is applied to the cathode, the highly mobile electrons are accelerated toward the anode leaving behind a positively charged gas in the cathode vicinity. With increasing negative potential, ion bombardment becomes more persistent on the cathode surface, which generates more secondary electrons, ionizing more neutral atoms, and engendering greater ion current. As the negative potential rises, the population of charged particles becomes larger as a consequence of this avalanche effect.

As the ion current becomes more significant, the RC circuit reduces the cathode negative potential such that an equilibrium is reached between the generation of the charged particles, preventing the transition of the plasma into an arcing regime.

Figure 2 shows the electron and ion densities at the end of the simulation (t = 0.1 s). Note that the figure is plotted on a log-log scale. In the figure, you can see that the ion density is approximately three orders of magnitude higher than the electron density in the vicinity of the cathode. The tiny positive space charge distribution generated by the ion density defines an ionization region that screens the cathode potential from the anode. This can be observed by displaying the electric potential along the radius of the coaxial assembly; see Figure 3.

Figure 2: Plot of electron (blue) and ion (green) number density at the end of the simulation (t = 0.1 s).

Figure 3: Electric potential along the radius of the coaxial assembly at the end of the simulation (t = 0.1 s).

The high voltage and important current density at the cathode enhance the electron temperature near the electrode, thus boosting ionization of the neutral atoms in the ionization zone. This can be seen in Figure 4, which displays the electron temperature at the end of the simulation.

Figure 4: Electron temperature along the radius of the assembly at the end of the simulation (t = 0.1 s).

To see the effect of the RC circuit on the system, plot the secondary electron flux as a function of time at the cathode surface; see Figure 5. A close look at the figure shows a rapid rise of the flux that tends to stabilize as the circuit sees higher currents. Figure 6 also shows the potential at both electrodes as a function of time. Comparing figure Figure 5 with Figure 6 reveals the effect of the circuit on both potential and secondary emission at the cathode surface. You can also observe a similar effect by plotting the electron current density at the electrodes, Figure 6. Doing so shows a direct relation between the cathode secondary emission (positive current flow), the anode recombination (negative flow), and the RC circuit. Figure 7 displays a surface plot of the logarithm of the electron density at the last time step.