COMSOL 6.4 - Double-Headed 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.

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 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 double-headed streamer. An initial seed of electrons is placed between two electrodes spaced by 1 cm to which a voltage of 52 kV is applied (providing a constant initial electric field of 52 kV/cm). A negative and positive streamers propagate toward the electrodes. The electron density, electric field, and the propagation speed of the streamer agrees well with simulation results from Ref. 1.

Model Definition

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

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 surfaces does not influence much the streamer propagation. Nevertheless boundary conditions must be given. For electrons a fixed electron density of 1014 cm-3 is set at the electrodes and the default zero flux boundary condition is set at the right boundary. For ions the default zero flux is applied to all boundaries.

For Poisson’s equation, an electric potential of 0 V and 52 kV is set at the bottom and top electrodes, respectively, and the default zero charge boundary condition is applied at the right boundary.

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 this in mind, the model includes a single ionization reaction, as presented in Table 1, that correctly describes the creation of charged species in a background of nitrogen. This work uses the Townsend coefficient as a function of reduced electric field provided in Ref. 1.

Table 1: Ionization reaction.
Reaction	Formula	Type	Δε (eV)
1	e+M=>2e+M+	Ionization	10

Results and Discussion

The results in this section are for a double-headed 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. There is a general good agreement with the results from the model here presented and the ones from Ref. 1.

Figure 1, Figure 2, and Figure 3 show 2d plots of the electron density, space charge density, and electric potential at 2.5 ns. Figure 4 and Figure 5 present the spatial distribution of the electron and ion densities, and the z component of the electric field for several instants during the streamer simulation.

Starting from the center, two streamers develop toward the electrodes. These streamers have different propagation mechanisms that result in different morphology and propagation speeds. The top streamer is anode directed and develops a negative space-charge density since the electric field pulls electrons ahead of the streamer. The bottom streamer is cathode directed and the electrons are drifting in the opposite direction of the streamer propagation. The propagation of the cathode-directed streamer is only possible because it is given a high enough background electron density. Note how the quasineutral streamer body shields the electric field to very small values causing the electrons to cool down.

Figure 3 and Figure 5 can be compared with figures 6 and 7 of Ref. 1. In general, the agreement is very good with the velocity of propagation and the spread being captured accurately.