43rd AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, January 10-13, 2005
AIAA 2005-0202
Ionization in Strong Electric Fields and Dynamics of Nanosecond-Pulse Plasmas Sergey O. Macheret,* Mikhail N. Shneider,† Robert C. Murray,‡ and Richard B. Miles§ Princeton University, Princeton, NJ 08544
[email protected] The paper describes experimental and computational studies of air plasmas sustained by high repetition rate high-voltage nanosecond pulses. Current and voltage measurements, together with earlier microwave diagnostics, allowed us to determine the efficiency of ionization. The energy cost per newly produced electron in these diffuse volumetric plasmas was found to be on the order of 100 eV, two orders of magnitude lower than in diffuse quasineutral DC and RF plasmas, and comparable with or even lower than in the cathode sheaths of glow discharges. A plasma kinetic model was developed and tested against the experimental Paschen breakdown curve in argon. The kinetic model was found to adequately describe the Paschen curve, and the important role of ionization by fast ions and atoms near the cathode, as well as the increase in secondary emission coefficient in strong fields described in the literature, was confirmed. Modeling of plasma dynamics in highvoltage nanosecond pulses yielded the energy cost of ionization, which was found to agree well with the experimental values. Both experiments and modeling revealed that the ionization cost per electron in these plasmas is relatively insensitive to the gas density. Detailed investigations of the plasma dynamics revealed a critical role of the cathode sheath that was found to take up most of the peak voltage applied to the electrodes. The extremely high E/N, much higher than the Stoletov’s field at the Paschen minimum point, results in a very high ionization cost in the sheath. In contrast, the E/N in the quasineutral plasma is closer to that associated with the Stoletov’s point, resulting in a near-optimal electron generation. This behavior (the reversal of ionization efficiencies in the sheath and in the plasma) is opposite to that in conventional glow discharges. The positive space charge in the sheath and its relatively slow relaxation due to the low ion mobility was also found to result in reversal of electric field direction in the plasma at the tail of the high-voltage pulse.
I. Introduction For practical aerospace applications of nonequilibrium weakly ionized plasmas, the power budget of sustaining the plasmas is critical.1-3 In our earlier work, we have shown that electron beams are characterized by the minimum power cost of sustaining a plasma with a prescribed electron density.4 An alternative to electron beams is to sustain the plasma by repeated application of short (~nanosecond) pulses of very high voltage.5 In the pulses, the electrons that survived the recombination after the previous pulse are accelerated to very high energies (hundreds or thousands of electron volts) and efficiently generate new electrons; the pulse repetition rate should approximately match the rate of recombination. We have theoretically shown that the power budget of plasmas sustained by such pulses can be orders of magnitude lower than that in conventional discharges and can be within a factor of 3-5 of the power budget of e-beam sustained plasmas with the same average electron density.5 However, our kinetic modeling has revealed that the actual dynamics and kinetics of nanosecond-pulse plasmas is quite complex.5 A major factor is that, due to the pre-existing plasma, the high voltage and electric field applied during the pulse are rapidly displaced *
Senior Research Scientist, Associate Fellow AIAA Research Scientist, Senior Member AIAA ‡ Graduate Student § Professor, Fellow AIAA †
Copyright © 2005 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission 1 American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, January 10-13, 2005
AIAA 2005-0202
into a cathode sheath, and there is only a relatively weak field in the rest of the plasma volume. This sheath formation complicates the plasma dynamics and affects the overall efficiency of ionization. In the present paper, we address in detail the sheath formation and ionization efficiency in high-voltage nanosecond pulses. First, we present a preliminary analysis of ionization cost. We then report the results of experimental studies of air plasmas sustained by 2-ns, 10-100 kHz repetition rate, ~10 kV pulses in air at 1-20 Torr. Prior to modeling the dynamics of plasmas in nanosecond pulses, we test the mechanisms of ionization in strong electric fields by computing the left branch of the well-known Paschen breakdown curve, where significant new ionization processes, such as ionization by collisions of molecules with high-energy ions and an increase in the secondary emission from the cathode play a role. The plasma model, supplemented by equations describing the thin dielectric layer covering the electrodes, is used to model the dynamics of nanosecond pulses under experimental conditions, and a comparison is made between the computations and the experiments. To further clarify the pulse dynamics and the ionization efficiency, parametric studies are performed for pulses of varying duration and amplitude.
II. Ionization Cost in DC and Pulsed Discharges The power budget of sustaining a nonequilibrium plasma by a constant or oscillating electric field is determined by two factors: the ionization rate that should match the rate of electron losses in recombination, attachment, and diffusion, and the ionization efficiency that can be expressed as energy cost, in electron volts, per newly produced electron.1,4,5 While recombination and attachment losses can certainly vary somewhat depending on the plasma conditions, it is the energy cost of electrons that is dramatically, by many orders of magnitude, different in various types of nonequilibrum plasmas.1.4.5 Therefore, in this work, we focus on the electron energy cost. If a plasma is sustained by an electric field, the energy cost of ionization obviously is equal to the ratio of the work done by electric field on an electron per unit length of its drift in the field to the number of new electrons produced in this unit-length path. That number of newly produced electrons per unit length is called the Townsend ionization coefficient, α. Note that α N = f ( E / N ) ,6 where E is the electric field strength, and N is the number density of gas. Thus, the ionization cost in eV is:6 Yi = E α
(1)
This quantity is plotted in Fig. 1 as a function of E/N. Fig. 1 illustrates the potential advantage of generating electrons with high E/N, showing that the energy cost close to the Stoletov’s constant of 66 eV, several orders of magnitude lower than that in electrodeless RF and microwave plasmas, can be achieved at E/N≈10-14 V⋅cm2.6 This value of E/N at the Stoletov’s point corresponds to the minimum of Paschen breakdown curve6 (see below). Note also that the energy cost increases again at very high E/N. 4
10
3
10
2
Yi, eV
10
e-beam (E/N)c
10
1
DC or RF plasma
10
-15
10 .
-14
E/N, V cm
10
-13
2
Figure 1. Ionization cost, in eV per newly produced electron, in air as a function of E/N. The value (E/N)c corresponds to the Stoletov’s point. The energy cost of ionization by e-beams and the typical E/N in DC glow and RF discharges are also indicated by arrows.
2 American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit Reno, Nevada, January 10-13, 2005
AIAA 2005-0202
Although Eq. (1) and Fig. 1 can provide a general idea of ionization efficiency, the actual energy cost in discharges with electrodes is affected by the strong non-uniformity of electric field (i.e., the cathode sheath). Consider a discharge between the cathode and anode, with the total voltage being the sum of the cathode voltage xa
fall, Vc, and the voltage on the positive column, V p = ∫ Edx . Here, E=E(x) is the field in the plasma column, and xs xs
and xa are the coordinates of the cathode sheath edge and the anode, respectively. Assuming that electrons are born at the cathode as a result of secondary emission with the coefficient γ , and introducing the local Townsend ionization coefficient α = α ( E / N , N , x ) , the following formula can be easily derived for the energy cost, in eV, per newly produced electron in the discharge: Yi =
(V
c
+ Vp )
(2)
x
a 1 + ∫ α dx 1 + γ xs
For example, in a normal laboratory-size glow discharge at low gas density: γ
