Plasma Physics & Engineering

1. Plasma Physics & Engineering Lecture 15

2. STEADY-STATE REGIMES OF NON-EQUILIBRIUM ELECTRIC DISCHARGES

3. Steady-State Discharges Controlled by Volume and Surface Recombination Processes. • If the degree of ionization-- relatively high and diffusion considered as ambipolar, the frequency of charge losses due to diffusion to the walls--- • Da--coefficient of ambipolar diffusion & • ΛD—Characteristic diffusion length, calculated different shapes of the discharge chambers (4.159)

4. for cylindrical discharge chamber of radius R and length L: • for parallelepiped with side lengths L1, L2, L3: • for a spherical discharge chamber with radius R: • a criterion for the volume-process-related steady-state regime of sustaining the non- equilibrium discharges (4.163)

5. Criterion restricts pressure, ( and ) • When pressure > 10--30 Torr • diffusion -- relatively slow & • balance of charge particles due to volume processes. • kinetics of electrons (&positive & negative ions -- characterized by (4.164) (4.165) (4.166)

6. Discharge Regime Controlled by Electron-Ion Recombination • Some plasma, destruction of negative ions by associative electron detachment > ion-ion recombination: • in plasma processes of CO2 and H2O dissociation, and NO-synthesis in air • the associative electron detachment processes-- very fast; these require ~0.1 μsec at concentrations of the CO, NO and H2 molecules  (4.167) (4.167a)

7. electron attachment & detachment - • in dynamic quasi-equilibrium in the recombination regime during the time intervals sufficient for electron detachment • Then, concentration of negative ions→ in dynamic quasi-equilibrium with electron concentration: • Using the quasi-constant parameter reduce the set of Eqs.(4.164)-(4.166) to the kinetic equation for electron concentration. (4.168) 4.169

8. Parameter shows the detachment ability to compensate the electron losses due to attachment. • If , the attachment influence on electron balance negligible • kinetic Eq.(4.169) becomes equivalent to that for non-electronegative gases including only ionization and electron-ion recombination. • kinetic equation include • effective rate coefficient of ionization, • coefficient effective coefficient of recombination. • Eq (4.169) describes the electron concentration evolution to the steady-state ne magnitude of the recombination-controlled regime: (4.170)

9. important peculiarity of the recombination-controlled regime ---there is the steady state degree of ionization ( ) for each value of electron temperature Te • Note, criterion of recombination-controlled regime can be rewritten using only rate coefficients, taking into account the plasma quasi-neutrality & degree of ionization: • criterion means -- recombination-controlled regime takes place when the electron detachment rate coefficient kd is sufficiently large (4.171).

10. Discharge Regime Controlled by Electron Attachment • Balance of charged particles-- due to volume processes and the discharge parameters correspond to inequalities opposite to Eqs.(4.167) and (4.171). • Here negative ions produced by electron attachment go almost instantaneously into ion-ion recombination, and electron losses mostly due to the attachment process. • The steady-state solution - for the attachment-control regime (4.172)

11. In the attachment-controlled regime, • the electron attachment is usually faster than recombination and • Eq.(4.172) actually requires • The exponential functions & usually appear as shown on Fig. 4.31 • the only crossing point Tst.---- determines the steady-state electron temperature • steady-state non-equilibrium discharge can be controlled by electron attachment only at high electron temperatures when Fig. 4.31 --: Rate coefficients of ionization (1) and dissociative attachment (2) for CO2.

12. Discharge Regime Controlled by Charged Particles Diffusion to the Walls, the Engel-Steenbeck relation • The balance of direct ionization by electron impact and ambipolar diffusion to the walls of a long discharge chamber of radius R → relation between Te & P (or the similarity parameter pR): Engel-Steenbeck relation for the diffusion-controlled regime of non-equilibrium discharges • If T – fixed & parameters constant, rewritten as • constant C only depends on the type of gas. (4.173) (4.174)

13. Table 4.5. The numerical parameters of the Engel-Steenbeck relation.

14. The universal relation between and the similarity parameter cpR for the diffusion-controlled regime is usually presented as a graph Fig.4.32 : Universal relation between electron temperature, pressure and discharge tube radius

15. in contrast to steady-state regimes sustained by volume processes, • the diffusion-controlled regimes of non-equilibrium discharges sensitive to radial density distribution of charged particles. • Such radial distribution for a long cylindrical discharge tube can be described by Bessel functions:

16. Propagation of Electric Discharges • not just a continuous breakdown of newer portions of gas coming into a high electric field zone, • incorrect -- breakdown and steady-state discharge conditions are usually quite different • E fields needed initiate a discharge >>needed to sustain • Thermal plasma propagation -- related heat transfer processes • non-thermal plasma propagation -- provided just by electron diffusion in front of the discharge

17. Consider 1D non-thermal discharge propagating in CO2 in uniform E field, Te≈ 1eV • CO2 breakdown controlled by dissociative attachment, requires large E fields and Te > 2 eV • However, CO2 dissociation → produces CO to provide effective electron detachment and the recombination-controlled regime corresponding to the lower E fields under consideration • parameters of CO2 discharge propagating in fast gas flow are: • critical value of CO-concentration, separating the attachment and recombination-controlled regimes is: (4.176)

18. (4.177) • If CO-concentration > critical value, the recombination-controlled balance gives the relatively high electron density: • Conversely if CO concentration < critical limit , • the electron concentration is very low, controlled by the dissociative attachment and is proportional to the CO concentration • Thus propagation of the electron concentration and of the discharge -- related to the propagation of the CO-concentration. (4.178)

19. Most of CO production -- due to dissociation of vibrationally excited CO2 molecules and takes place in the main plasma zone III • CO diffusion from the zone III into zone II provides the sufficiently high CO concentration for sustaining the high electron concentration that subsequently provides the vibrational excitation and CO2 dissociation in zone III. • Further decrease of the CO concentration below the critical value in the zone I corresponds to a dramatic fall of the electron concentration, Fig. 4.33 : Electron and CO density distributions in the front of propagating discharge. I-low electron concentration zone; II-discharge zone where CO-diffusion provides effective detachment and sufficient electron density; III-effective CO2 dissociation zone

20. Thus the discharge propagation can be interpreted • as the propagation of a self-sustained ionization wave, • supported by CO production after the ionization front, • which diffuses ahead and facilitates the ionization conditions.

21. Propagation of the Non-Thermal Ionization Wave, Self-Sustained by Diffusion of Plasma Chemical Products • Electron concentration profile & velocity of the ionization wave evolution, described by linear 1D differential eq with only the variable • g is a model source of CO as a result of CO2 dissociation (4.179) (4.180) vibrational excitation time in the zone II maximum concentration of CO at the end of zone III

22. (4.180) where vibrational excitation time in the zone II maximum concentration of CO at the end of zone III the total chemical reaction time in zone III parameter αshows the exponential smallness of dissociation rate at end of zone III when process is actually completed. BC for Eq.(4.179) should be taken as:

23. source g() is not powerful at negative values of , & perturbation theory used to solve the non-linear equations (4.179), (4.180). The non-perturbed equation (g=0) gives the solution . Contribution of the source g() in the first order of the perturbation theory leads to the following linear equation (4.181) where (4.182) as solution of this equation is

24. In a similar manner, for first order perturbation theory gives (4.184) Eqs(4.183) and (4.184) → concentration profiles for both positive and negative magnitudes of the auto-model variable . To find entire solution Eq(4.183 &184)matched at the wave front e.g. at the magnitude of the velocity of the ionization wave: (4.185) Where The approximate solution of the transcendent equation (4.185) for the velocity of the ionization wave can be expressed as: (4.186)

25. This velocity of the ionization wave and non-thermal discharge propagation -- physically interpreted as the velocity of diffusion transfer of the detachment active heavy particles (CO) ahead of the discharge front on a distance necessary for effective vibrational excitation of CO2 molecules with their further dissociation. • For numerical calculations it is convenient to rewrite Eq.(4.186) in terms of speed of sound , Mach number M and the ionization degree in plasma : (4.187) • Velocity of the non-equilibrium ionization wave propagation depends • mostly on the degree of ionization in the main plasma zone • and also on the critical amount ( ) of the ionization active species (e.g CO), which should be transported in front of the discharge to facilitate ionization. • does not strongly depend on the details of propagation mechanism • This means, that the final relation for the ionization wave velocity can be used for other similar mechanisms of non-thermal discharge propagation related to diffusion of some active heavy plasma species in front of the discharge to facilitate further propagation of the ionization wave.

26. Non-Equilibrium Behavior of Electron Gas, Difference Between Electron and Neutral Gas Temperatures • Principal aspects of non-equilibrium behavior • temperature differences between electrons and heavy particles, • significant deviation of the degree of ionization from that predicted by the Saha equilibrium • Ionization in plasma -- provided by electron impact and the ionization process should be quite intensive to sustain the steady-state plasma. - -Te - on the level of the ionization potential (1eV ) • True for both thermal and non-equilibrium plasma • the gas temperature T0, determines the equilibrium or non-equilibrium plasma behavior

27. For thermal discharges T0 ≈ Te system close to equilibrium, • in non-thermal discharges T0 is low and the degree of non-equilibrium can be high, sometimes up to 100. • in low pressure discharges related to intensive heat losses to the discharge chamber walls. • The difference gas temperature in plasma T0 and room temperature T00 in such discharges can be estimated from the simple relation P is the discharge power per unit volume

28. moderate and high pressure NE discharges (usually more than 20-30 Torr ) • heat losses to the wall are low, • neutral gas overheating can be prevented either by high velocities and low residence times or by short time of discharge pulses • Estimates over-heating is then given (4.189)

29. Non-Equilibrium Behavior of Electron Gas, Deviations From the Saha – Degree of Ionization • The quasi-equilibrium electron concentration and degree of ionization easily found as the function of one temperature, based on the Saha formula • Although the ionization processes (both in thermal and non-thermal discharges) are provided by the electron gas, • for non-equilibrium discharges the Saha formula with electron temperature Te gives the ionization degree several orders of value higher than the real one. • Obviously, the Saha formula assuming the neutral gas temperature gives even much less electron concentrations and much worse agreement with reality. • This non-equilibrium effect is due to the presence of additional channels of charged particles losses in cold gas.