Streamer Modeling
The two-dimensional plasma modeling code PLASMATOR™ marketed by Kinema Research & Software is a direct descendent of the computer program used by Vitello, et al. [1, 2] for modeling streamer development. PLASMATOR has evolved primarily toward modeling applications in ICP and CCP plasma etching and deposition [3, 4] but the underlying physical models remain appropriate for modeling streamers and dielectric barrier discharges (DBDs).
We show here a test calculation modeling a streamer in a DBD device using a complex chemistry. The model of the DBD device with a non-uniform mesh, as used by PLASMATOR in the calculations, is shown in the following figure:
The calculation is initiated using a very localized electron density just above the cathode, which is the lower electrode in the reactor.
Streamers typically take several nanoseconds to develop and traverse the gap. The next figure shows the calculated electron temperature distribution Te (r, z) at 0.7 ns. The electron energy distribution is Maxwell-Boltzmann where f(u) is proportional to exp[-u/kTe] with u being the electron energy.
The next figure shows the electron density distribution Ne (r, z) in the gap at 0.7 ns.
The electron density distribution also showing the mesh is graphed in the next figure:
The final product distribution for the gas mixture of 79% N2, 7% O2, 7% CO2, and 7% H2O, 300 ppm CO, 300 ppm NO, and 600 ppm CH4 is shown next:
The atomic and molecular species shown here are only the original gases and the products formed by electron collisions with those molecules. The full list including species formed by ion-molecule, ion-ion, and neutral reactions from these products of electron collisons is shown in the following table:
These calculations were performed using a Maxwell-Boltzmann model for the electron energy distribution function. In molecular gases at atmospheric pressure this is a bad approximation. Fortunately solving Boltzmann's equation for the correct f(u) at high gas pressure is straight forward because, unlike the situation with low pressure ICPs and CCPs, the local approximation is valid. The true electron energy distributions look like those in the next figure:
These calculations were performed using our ELENDIF™ code [5]. The reduced mean energies, i.e. 2<u>/3, for the three distribution functions at values of E/N of 50, 100, and 200 Td (1 Td = 1 Townsend = 10^-17 V-cm^2) are 0.76, 1.68, and 3.47 eV.
PLASMATOR is able to use ELENDIF as a subroutine for calculating correct electron energy distributions.
DBD Modeling
Dielectric barrier discharges (DBD), also known as "silent discharges", were invented by W. Siemens in 1857 for the purpose of "ozonizing" air. They are stable high pressure gas discharges capable of producing large densities of radical atomic and molecular species. The proceedings cited as Reference [6] contains a number of papers discussing the use of DBDs for producing processing plasmas.
A typical DBD configuration is shown in the next figure [7]
Due to the presence of the dielectric, DBDs cannot be operated as DC discharges. The discharge is homogeneous in neither space nor time. It consists of a number of independent transient filaments or microdischarges or streamers randomly distributed across the electrode surfaces. The next figure [7] is a photograph showing many such streamers.
There are typically 10 such microdischarges per square centimeter at any given time. They have the following properties:
- Duration: 1-10 ns
- Filament radius: approximately 0.1 mm or less
- Peak current: 0.1 A
- Current density: 100-1000 A/cm^2
- Total charge: 0.1-1 nano-Coulombs
- Electron density: 10^14-10^15/CC
- Electron energy: 1-10 eV
- Gas temperature: close to ambient
The filaments ignite when the breakdown field is reached and are extinguished due to electron attachment and recombination when the field falls slightly, due to space charge buildup, and the conductivity is reduced. Charge buildup at the location of the filament on the dielectric terminates the microdischarge a few nanoseconds after breakdown. The next figure [8] shows the relationship between the formation time of microdischarges and the phase of an applied sinusoidal voltage.
A recent and extensive review [9] of DBDs by Kogelschatz discusses their physics and applications and cites 310 references.
The difficulty in modeling or simulating a DBD stems from the inhomogeniety of the discharge. The ionization and other electron collision rates are very non-linear functions of electric field strength or E/N. The results that we would obtain for the fields, production rates of ions and radicals, and the subsequent plasma chemistry can be expected to be very different within a streamer than outside of the streamer. With ordinary computing resources modeling the whole discharge, using a code such as PLASMATOR where the device dimensions are measured in mm or cm and the time scales are 10-100 ms or longer but the streamer dimensions are tenths or hundredths of a mm and the time scales are 1-10 ns, is pretty much computationally intractable. For this reason, most of the modeling work that has been performed has focused on the physics of the streamers themselves.
We have seen above that we can model the individual streamers on microdischarges. Because the streamers are very small in radial dimension compared to the DBD device and because they occur on very short time scales compared to the AC cycle, we might treat the streamers as a delta function source driving the larger scale plasma chemistry of the DBD. That is, the source of electrons, ions and radical species densities Nj can be written in cylindrical coordinates as
where the density Nj(z) comes from detailed streamer modeling. The streamer appears at random time tk, which is distributed in the AC cycle as shown in the diagram above. It appears at random location ri = (xi, yi) uniformly distributed as shown in the photograph above.
Since a single streamer represents a certain amount of energy dissipation in the plasma, the number of streamers per unit area in (x, y) is directly proportional to the power being dissipated by the DBD.
If P is the power being dissipated in the DBD as a whole and dE is the average energy dissipated per microdischarge, the number of streamers or microdischarges per area per time (/cm^2/sec) on the dielectric surface of area A is Ns = (P/dE)(1/A).
A volumetric source term representing the rate of production of species (/cc/sec) in the reactor volume V would be
dNi/dt = (ni/V)(P/dE)
where ni is the total number of species (/cc/sec) produced by an average streamer.
It's clear from the diagram above of the AC cycles that the microdischarges are not uniformly distributed in time. If fAC is the AC frequency then the frequency of microdischarges is 2 fAC or the typical time between such microdischarges is 1/(2 fAC). In modeling then we should consider the fundamental interval to be the half AC cycle so that the number of streamers per area in that interval is
(P/dE)(1/A)/(2 fAC)
Another way of looking at this involves using the measured current I. If Q is the amount of charge transferred per microdischarge then the number of streamers per area per time is
Ns = I/(AQ)
or
I/(2AQ fAC)
per half AC cycle.
A zeroth approximation for finding the spatial and temporal distribution of the density of a species in two dimensions (x, y) might be the following. The diffusion equation for the density Nj(r, t) of a particular molecular species
where Dj is the diffusion coefficient for the given molecules in the background gas and q(r,t) is the stochastic source function representing the source of species of interest in a streamer at random location ri and random time ti. kij is the loss rate coefficient.
A more sophisticated approach would be to use the coupled drift-diffusion, Poisson's, and rate equations. Fluid flow could be incorporated in an approximate manner using models such as the well stirred reactor model on the plug-flow model.
The most sophisticated approach would be to use the results of streamer modeling as source functions in computational fluid dynamics code such as FLUENT.
Kinema Research is actively engaged in research on modeling and simulation of dielectric barrier and corona discharges.
References
1. Vitello, Penetrante, and Bardsley, in Non-Thermal Plasma Techniques for Pollution Control, eds. B. M. Penetrante and S. E. Schulthesis (Springer-Verlag, Berlin, 1993).
2. Vitello, Penetrante, and Bardsley, Phys. Rev. E 49, 5574 (1994).
3. Bukowski, Graves, and Vitello, J. Appl. Phys. 80, 2614 (1996).
4. Font, Morgan, and Mennenga, J. Appl. Phys. 91, 3530 (2002).
5. "ELENDIF: A Time Dependent Boltzmann Solver for Partially Ionized Plasmas", W. L. Morgan and B. Penetrante, Computer Physics Communications 58, 127 (1990).
6. Non-Thermal Plasma Techniques for Pollution Control, eds. B. M. Penetrante and S. E. Schulthesis (Springer-Verlag, Berlin, 1993).
7. U. Kogelschatz, et al., ICPIG XXIII, Toulose, France (1997).
8. B. Eliasson and U. Kogelschatz, IEEE Trans. Plasma Sci 19, 1063 (1991).
9. U. Kogelschatz, Plasma Chemistry and Plasma Processing 23, 1 (2003).