COMSOL 6.4 - Negative Streamer in Nitrogen

Streamers are transient filamentary electric discharges that can develop in a nonconducting background in the presence of an intense electric field. These discharges can attain high electron number density and, consequently, a high concentration of chemical active species that are relevant for numerous applications. Industrial applications include: ozone production, pollution control, and surface processing.

The propagation of streamers is driven by very nonlinear dynamics that involve very steep density gradients and high space-charge density distributed in very thin layers. The charge separation at the front (or head) of the streamer generates intense electric fields that are responsible for sharp ionization fronts propagating into the neutral medium.

In negative (anode directed) streamers, ionizing electrons are accelerated outward by the space-charge (the streamer extends toward the anode). These high energy electrons may have been transported by drift or diffusion, or created by another mechanism that provides preionization ahead of the streamer such as photoionization or ionization from runway electrons. In positive (or cathode directed) streamers the space-charge field in the streamer head accelerates the electrons inward. Consequently, the ionizing electrons must be produced by a preionization mechanism. The preionization of the streamer is a complex subject that is believed to be critical for both negative and positive streamers propagation and is still under intense investigation. This document is an introduction to streamer modeling with focus on basic concepts of streamer propagation. With that in mind it is followed a simplified approach where all preionization is neglected and only negative streamers are discussed.

This example presents a study of a negative streamer in atmospheric pressure nitrogen. The streamer propagates with a constant electric field of 100 kV/cm in front of the streamer. The model here presented is similar to the fluid model used in Ref. 1 and gives similar results. Furthermore, in Ref. 1, results from a fluid and particle-in-cell model are compared to agree very well for fields below 50 kV/cm (at atmospheric pressure).

Model Definition

The model is one dimensional and describes the transient behavior of an initial electron seed in the presence of a strong electric field using fluid-type equations.

The simulation is design with emphasis in the streamer propagation. The streamer develops from an initial small electron density placed near the cathode. Without preionization only a negative streamer develops toward the anode. The initial number density and exact location of the seed are chosen to form a streamer long before it reaches the anode. The distance between electrodes is 1.15 mm. The cathode (placed on the left) is grounded and at the anode is given a constant electric field of −100 kV/cm. A constant electric field ahead of the streamer allows for an equilibrium in the electron density growth and a constant propagation velocity.

The model solves the electron and ion continuity and momentum equations, in the drift-diffusion approximation, self-consistently coupled with Poisson’s equation. The local field approximation is used, which means that transport and source coefficients are assumed to be well parameterized through the reduced electric field (E/N). In the local field approximation the fluid equation for the mean electron energy is not solved, which reduces significantly the complexity of the numerical problem.

The local field approximation is valid in a situation where the rate of electron energy gain from the electric field is locally balanced by the energy loss rate. When this condition is met the electrons are said to be in local equilibrium with the electric field and the electron mean properties can be expressed as a function the reduced electric field.

Domain Equations

Electron transport is modeled by solving the continuity equation and the momentum equation under the drift-diffusion approximation (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.

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

The present simulation is arranged in a way that the charged particle interaction with the wall is irrelevant to what happen to the streamer propagation. Nevertheless boundary conditions must be given. Electrons are lost to the wall due to random motion within a few mean free paths of the wall resulting in the boundary condition for the electron flux

(1)

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:

(2)

The streamer propagates (from left to right) with a constant electric field ahead of -100 kV/cm. The cathode on the left is grounded,

Plasma Chemistry

The chemistry of a plasma sustained in nitrogen can be very complex and a detailed study of the main excited states could easily have hundreds of reactions. The main goal of this model is to study the charge particle density profiles in the presence of strong electric fields. With that in mind it is used one single ionization reaction as presented in Table 1 that describes correctly the creation of charged species in a background of nitrogen. In this work it is used the Townsend coefficient as a function of the mean electron energy provided in Ref. 1.

Table 1: Ionization reaction.
Reaction	Formula	Type	Δε (eV)	kf (m3/s)
1	e+N=>2e+N+	Ionization	15.5	-

In addition to the volumetric reactions, the following surface reaction is implemented:

Table 2: Table of surface reaction.
Reaction	Formula	Sticking coefficient
1	N+=>N	1

When an ions reach the wall, they are assumed to change back to neutral atoms.

Results and Discussion

The results in this section are for a streamer propagating in a background gas kept at a constant density as obtained by the ideal gas law at atmospheric pressure and at a temperature of 293.15 K. All transport and source coefficients used in this simulation are form Ref. 1. There, the electron mobility and diffusion coefficients, the Townsend coefficient, and the mean electron energy as a function of the reduced electric field are obtain using particle swarm simulations.

Figure 1 presents the spatial distribution of the electron and ion density for several instants during the streamer simulation. The streamer is initiated by a localized number of electrons near the cathode (left in all figures below). In a first phase there is electron drift and growth in an unperturbed electric field. The first instant in Figure 1 marks the end of this period. A streamer is formed if the amplification of the electron density is enough to generate intense space-charge electric fields and electric shielding before the streamer arrives to the anode. The last three instants of Figure 1 correspond to the streamer phase.

In the streamer phase the maximum electron and ion density reach a constant value and propagate at constant velocity (because the electric field ahead of the streamer is kept constant). The streamer morphology is composed of (i) a region of strong charge separation and strong density gradients (the streamer head), and (ii) a quasi-neutral region with flat profiles (the streamer body) that increases its length with the streamer propagation.

Figure 2, Figure 3, and Figure 4 show the spatial distribution of the electric field, space-charge density, and mean electron energy, respectively, for several instants during the streamer simulation. Note how the quasi-neutral streamer body shields the electric field to very small values causing the electrons to cool down. This makes the electron creation in the body negligible.

On the left there is also a charge separation region. This region is created because the electrons are pulled toward the anode leaving the created ions behind. The ions, being much less mobile than electrons, do not have time to drift in this 0.9 ns simulation. Note also that this charge separation region does not move in time, which is achieved by setting low levels of preionization.

The ionization front moves with a velocity larger than the electron drift velocity

(3)