Plasma Physics

1
Plasma Physics Engineering

• Lecture 15

2
STEADY-STATE REGIMES OF NON-EQUILIBRIUM ELECTRIC
DISCHARGES
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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
• ?DCharacteristic diffusion length, calculated
  different shapes of the discharge chambers

(4.159)
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• 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 gt 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)
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Discharge Regime Controlled by Electron-Ion
Recombination

• Some plasma, destruction of negative ions by
  associative electron detachment gt 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)
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• 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
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• 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)
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• important peculiarity of the recombination-control
  led 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).
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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)
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• 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.
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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)
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• Table 4.5.
• The numerical parameters of the Engel-Steenbeck
  relation.

Gas Gas
N2 2104 410-2 Ar 2104 410-2
He 2102 410-3 Ne 4.5102 610-3
H2 1.25103 10-2
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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
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• 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

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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 gtgtneeded to
  sustain
• Thermal plasma propagation -- related heat
  transfer processes
• non-thermal plasma propagation -- provided just
  by electron diffusion in front of the discharge

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• 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 gt 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)
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• If CO-concentration gt critical value, the
  recombination-controlled balance gives the
  relatively high electron density
• Conversely if CO concentration lt 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.

(4.177)
(4.178)
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• 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
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• 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.

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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
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(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 a shows 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
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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 (g0) 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
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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)
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• 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.

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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

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• 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
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• 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)
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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.