Download Energy coupling to the plasma in repetitive nanosecond pulse

PHYSICS OF PLASMAS 16, 113505 共2009兲
Energy coupling to the plasma in repetitive nanosecond pulse discharges
Igor V. Adamovich, Munetake Nishihara, Inchul Choi,
Mruthunjaya Uddi, and Walter R. Lempert
Department of Mechanical Engineering, Michael A. Chaszeyka Nonequilibrium Thermodynamics
Laboratories, The Ohio State University, Columbus, Ohio 43210, USA
共Received 8 September 2009; accepted 23 October 2009; published online 20 November 2009兲
A new analytic quasi-one-dimensional model of energy coupling to nanosecond pulse discharge
plasmas in plane-to-plane geometry has been developed. The use of a one-dimensional approach is
based on images of repetitively pulsed nanosecond discharge plasmas in dry air demonstrating that
the plasma remains diffuse and uniform on a nanosecond time scale over a wide range of pressures.
The model provides analytic expressions for the time-dependent electric field and electron density
in the plasma, electric field in the sheath, sheath boundary location, and coupled pulse energy. The
analytic model predictions are in very good agreement with numerical calculations. The model
demonstrates that 共i兲 the energy coupled to the plasma during an individual nanosecond discharge
pulse is controlled primarily by the capacitance of the dielectric layers and by the breakdown
voltage and 共ii兲 the pulse energy coupled to the plasma during a burst of nanosecond pulses
decreases as a function of the pulse number in the burst. This occurs primarily because of plasma
temperature rise and resultant reduction in breakdown voltage, such that the coupled pulse energy
varies approximately proportionally to the number density. Analytic expression for coupled pulse
energy scaling has been incorporated into the air plasma chemistry model, validated previously by
comparing with atomic oxygen number density measurements in nanosecond pulse discharges. The
results of kinetic modeling using the modified air plasma chemistry model are compared with
time-resolved temperature measurements in a repetitively pulsed nanosecond discharge in air, by
emission spectroscopy, and purely rotational coherent anti-Stokes Raman spectroscopy showing
good agreement. © 2009 American Institute of Physics. 关doi:10.1063/1.3264740兴
I. INTRODUCTION
Insight into the kinetics of breakdown development and
energy coupling to the plasma in nanosecond pulse duration
discharges is critical for their use for efficient ionization and
radical species generation, such as O atoms in air, H atoms
and OH radicals in hydrogen-air and hydrocarbon-air mixtures. Prediction and/or measurement of energy coupled to
the plasma during each high-voltage pulse are also critical
for rapidly emerging engineering applications, such as
plasma assisted combustion and high-speed flow control. Direct measurements of coupled pulse energy from current and
voltage waveforms may result in a significant uncertainty, in
particular, at the conditions when a significant fraction of
incident pulse energy is reflected from the load. Theoretical
prediction of energy coupling kinetics in nanosecond pulse
discharges may also be difficult since these discharges may
become nonuniform and develop complex unsteady structures on nanosecond time scale.1 Prediction and analysis of
such structure formation require the use of three-dimensional
time-dependent discharge models.
Our recent experiments demonstrated that repetitively
pulsed, nanosecond pulse duration plane-to-plane geometry
discharges in dry air remain spatially uniform in a wide
range of pressures 共P = 40– 100 torr兲 without developing
well-defined structures and filaments. In these experiments,
the discharge was sustained in air slowly flowing through a
rectangular cross section quartz channel 共220 mm long
⫻ 22 mm span⫻ 10 mm height兲 with two rectangular
1070-664X/2009/16共11兲/113505/13/$25.00
copper plate electrodes, 14 mm wide and 65 mm long, attached to the outside surface of the channel, as shown in Fig.
1. Thus, both electrodes are separated from the plasma by the
quartz channel walls of 1.75 mm thick. The electrode plates
are rounded at the corners to reduce the electric field nonuniformity. At room temperature, flow velocity in the channel is
approximately 1 m/s, and flow residence time in the discharge region is about 60 ms. High voltage, nanosecond duration pulses are generated in the flow by a Chemical Physics
Technologies plasma generator producing 20–25 kV peak
voltage pulses with individual pulse duration of 25–50 ns
and maximum pulse repetition rate of ␯ = 50 kHz. Figure 2
shows a typical single pulse voltage waveform in air at
P = 60 torr and ␯ = 40 kHz. To produce breakdown in the
discharge section on the very first pulse, the test cell was
irradiated by a deuterium UV lamp 共Resonance Ltd.兲 through
the side wall, as shown in Fig. 1, providing preionization of
the discharge volume.
Figure 3 shows a collage of single pulse plasma images
taken during a nanosecond pulse burst in air at a pressure of
P = 60 torr and pulse repetition rate of ␯ = 40 kHz 共time between consecutive pulses of 25 ␮s, 1600 pulses in the
burst兲. The images were taken using a gated PI-MAX intensified charge coupled device 共ICCD兲 camera with a UV lens
共UV-Nikkor 105 mm f/4.5, Nikon兲. The camera was triggered by a Stanford Research System DG535 delay/pulse
generator which used a low-voltage transistor-transistor logic
pulse output generated by the plasma generator as synchronization input. Broadband emission from the air plasma at
16, 113505-1
© 2009 American Institute of Physics
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-2
Adamovich et al.
Phys. Plasmas 16, 113505 共2009兲
FIG. 3. ICCD camera images of a repetitive nanosecond pulse plasma in air
flow. P = 60 torr, ␯ = 40 kHz 共time between consecutive pulses of 25 ␮s兲.
Camera gate of 1 ␮s.
FIG. 1. 共Color online兲 Schematic of the discharge section.
these conditions is dominated by nitrogen second positive
system bands with emission decay time of about 10 ns.2 In
Fig. 3, the camera gate is 1 ␮s. From Fig. 3, it can be seen
that although the first discharge pulse exhibits some nonuniform filamentary structure, the nonuniformity during the second discharge pulse is substantially reduced. All air plasma
images taken during subsequent pulses appear diffuse and
uniform with plasma filling the entire volume between the
electrode plates 共see Fig. 3兲. Note that individual images in
Fig. 3 were taken during different bursts and do not illustrate
discharge development during a single burst. When the nanosecond pulse discharge was operated at a much lower pulse
repetition rate, 10 Hz, all individual pulse images showed
filamentary structure such as pulse 1, shown in Fig. 3, although even in this case an image averaged over 100 pulses
appeared uniform,2 demonstrating random character of filaments formed during individual pulses.
FIG. 2. 共Color online兲 Nanosecond pulse waveform in dry air at
P = 60 torr and ␯ = 40 kHz shown with a Gaussian fit 共pulse 800,
Vpeak = 24.3 kV, ␶ = 14 ns, full width at half maximum of 23.5 ns兲.
To determine whether spatially nonuniform structures
might be present in the air plasma on a shorter time scale,
additional images were taken with shorter gate width. Figure
4 shows ICCD camera images of pulse 100 at the conditions
of Fig. 3, taken with gate width ranging from 1 ␮s to 10 ns.
From Fig. 4, it can be seen that the emission intensity distribution remains uniform in the entire range of gate widths
used. Further gate width reduction to 4 ns did not change this
result. This provides additional evidence that at these conditions, air plasmas generated by repetitive nanosecond pulses
between two plane electrodes covered with dielectric plates
at a high pulse repetition rate are diffuse and uniform on a
nanosecond time scale. Similar results were obtained in nitrogen and dry air at pressures ranging from 40 to 100 torr.
On the other hand, adding small amounts of nitric oxide
共0.7%兲 or ethylene 共0.07%–0.7%兲 to air at P = 60– 90 torr
resulted in well-pronounced filament development in the
plasma at the same discharge and flow parameters 共see Fig.
5兲. This most likely occurs due to ionization/heating instability development, accelerated in presence of species with
lower ionization energy,3 such as NO 共9.3 eV兲 or ethylene
共10.5 eV兲, compared with oxygen or nitrogen 共12.2 and
15.6 eV, respectively兲. The instability development may be
further accelerated by a more rapid energy thermalization
rate predicted in ethylene and NO containing air plasmas.4
The images shown in Figs. 3 and 4 demonstrate that
operating a nanosecond pulse discharge with dielectriccovered plane electrodes in air at a high pulse repetition rate
creates a uniform plasma. This is most likely due to residual
FIG. 4. ICCD camera images of a repetitive nanosecond pulse plasma in air
flow 共pulse 100 in a burst兲 at the conditions of Fig. 3 for different gate
widths from 1 ␮s to 10 ns.
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-3
Energy coupling to the plasma…
Phys. Plasmas 16, 113505 共2009兲
FIG. 5. ICCD camera images of a repetitive nanosecond pulse plasma in air,
ethylene-air 共equivalence ratio ⌽ = 0.1 and 0.01, ethylene mole fractions
of 0.7% and 0.07%兲, and NO-air 共NO mole fraction of 0.7%兲 flows.
P = 40 torr, ␯ = 40 kHz, pulse 100, camera gate of 1 ␮s.
ionization remaining in the volume from a previous pulse,
and possibly due to residual surface charges accumulated on
dielectric 共quartz兲 walls covering the electrode plates during
a previous pulse. This result leads to a key conclusion, that
repetitive nanosecond discharge plasmas in nitrogen and dry
air can be analyzed using a simple quasi-one-dimensional
kinetic model, taking into account charged species and electric field variation in time and across the distance between
the plane electrodes. Development of such a model would
result in approximate analytic solutions for time-dependent
electron density and electric field in the plasma and provide
insight into kinetics of energy coupling to the plasma by a
nanosecond duration pulse.
The general quasi-one-dimensional approach to spatially
uniform nanosecond pulse discharge plasma analysis used in
the present work is similar to the method employed recently
by Nikandrov et al.5 and by Tsendin and Nikandrov.6 In Ref.
5, the authors sought to obtain an analytic solution for a
general case of an arbitrary pulse voltage waveform and have
reduced the original system of partial differential equations
for electron and ion number densities coupled with the Poisson equation to a single integral equation for the electric field
in the plasma. Although good agreement between their
model and the numerical calculations has been shown, no
explicit closed-form solutions for the electric field and the
electron density in the plasma, as well as for the energy
coupled to the plasma have been obtained, even for a simple
test case of a linear voltage rise. In Ref. 6, the authors studied the effect of initial and boundary conditions for the electron density on the sheath dynamics, considering electron
trajectories in the Lagrangian description. Similar to Ref. 5,
it appears that obtaining a tractable analytic solution had a
somewhat lower priority, compared with making it more
general and incorporating several possible scenarios spanning a broad range of initial and boundary conditions. Also,
in Refs. 5 and 6 no quantitative comparison has been made
between the predicted time-dependent sheath thickness and
the results of numerical calculations, making it rather difficult to gage the accuracy of their model. In the present work,
we use a similar approach to analyze nanosecond pulse
breakdown kinetics, sheath development, and energy cou-
FIG. 6. 共Color online兲 Schematic of the discharge geometry.
pling to the plasma with the main emphasis on obtaining a
closed-form analytic solution and comparing it with the results of numerical modeling and with the experimental data.
In addition to predicting energy coupled to the plasma by
electron impact during the pulse, predictive kinetic modeling
of repetitively pulsed discharges in air as well as in air-fuel
mixtures requires insight into initial energy partition among
different internal energy levels of molecules and atoms, dissociation, and ionization, as well as into the rates of subsequent relaxation and plasma chemical processes. Since detailed air and air-fuel plasma chemistry models, such as that
developed in our previous work,2,7,8 incorporate a large number of species and processes, they typically assume plasma
quasineutrality and therefore cannot predict time-dependent
sheath formation and its effect on energy coupling to the
plasma. On the other hand, an overwhelming majority of
relaxation and plasma chemical processes occur on time
scales much longer than the ionizing pulse duration. Therefore kinetics of energy coupling during nanosecond discharge pulses and of subsequent plasma chemical processes
共occurring on microsecond to millisecond time scale兲, both
of which affect the rate of temperature rise in the plasma,
require separate analysis.
The objectives of the present work are as follows: 共i兲
theoretically analyze the energy coupling process in a nanosecond pulse discharge with dielectric covered electrodes in
a plane-to-plane geometry; obtain close-form analytic solutions for time-dependent electric field and electron density in
the plasma, as well as energy coupled to the plasma as a
function of temperature, pressure, and pulse waveform; 共ii兲
use the results of this analysis for kinetic modeling of air
plasma chemistry and temperature rise in repetitively pulsed
nanosecond discharges; and 共iii兲 compare the plasma chemistry model predictions with time-resolved temperature and
species concentration measurements in repetitive nanosecond
pulse discharges.
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-4
Phys. Plasmas 16, 113505 共2009兲
Adamovich et al.
II. NANOSECOND PULSE DISCHARGE MODEL
Eapp共t兲 =
The schematic of the discharge geometry used in the
present model is shown in Fig. 6. Using a drift-diffusion
approximation in a nonelectronegative gas,9 one has
⳵ n+ ⳵ ⌫+
+
= ␯ in e − ␤ n +n e,
⳵t
⳵x
⌫+ = − D+
⳵ ne ⳵ ⌫e
+
= ␯ in e − ␤ n +n e,
⳵t
⳵x
⌫e = − De
⳵ 2␸
e
共n+ − ne兲,
2 =−
⳵x
␧0
⳵␸
.
⳵x
E=−
⳵ n+
+ ␮+n+E,
⳵x
⳵ ne
− ␮eneE,
⳵x
共1兲
共2兲
共3兲
In Eqs. 共1兲–共3兲, n+ and ne are positive ion and electron number densities, ␸ and E = −⳵␸ / ⳵x are the potential and the
electric field, ␯i = ␣␮e兩E兩 is the ionization frequency, ␣ is the
Townsend ionization coefficient, ␤ is the electron-ion recombination rate coefficient, and ⌫+, ⌫e are the drift-diffusion
fluxes. In the expressions for the fluxes, ␮+, ␮e are the mobilities and D+ = ␮+kBTi / e, De = ␮ekBTe / e are the diffusion
coefficients.
Voltage across the discharge gap, Vgap共t兲, is calculated as
follows:9
冉 冊
⫻
冕
L
关⌫+共x,t兲 − ⌫e共x,t兲兴dx,
共4兲
where Vapp共t兲 is the applied voltage approximated as a
Gaussian pulse,
冦
冋 册
冋 册
␾共0兲 = 0,
x = 0:
E
⬍ 100 V/cm torr
pⴱ
342
12 exp −
cm−1 ,
E/pⴱ
E
100 ⬍ ⴱ ⬍ 800 V/cm torr,
p
The electron and ion mobilities are10
冉 冊
␮e = 400
760
cm2/V/s,
pⴱ
共11兲
冉 冊
␮i = 2.75
760
cm2/V/s,
pⴱ
共12兲
and the electron and the ion temperatures are assumed to be
Te = 1 eV and Ti = 300 K, respectively. Note that at the
冊册
2
2
共5兲
,
␾共L兲 = Vgap共t兲,
冉 冊
⳵ n+
⳵
⳵␸
n+
,
= ␮+
⳵t
⳵x
⳵x
␾共x,t = 0兲 =
冉 冊
共6兲
ne = new ,
共7兲
冉 冊
共8兲
⳵
⳵␸
⳵ ne
= − ␮e
ne
,
⳵x
⳵x
⳵t
ne = n+ = n0 ,
Vapp共t = 0兲
x,
2l
L+
␧
共9兲
where new is the electron density on the dielectric surface,
new Ⰶ n0. Approximate boundary conditions on the cathode
and on the anode, Eqs. 共7兲 and 共8兲, have been obtained in
Ref. 10.
Electron impact ionization and electron-ion recombination rate data in nitrogen are10
315
900
cm−1 ,
ⴱ exp −
E/p
E/pⴱ
␤ = 2 ⫻ 10−7共300/Te兲1/2cm3/s.
t − tpeak
␶
冊册
␧ is the dielectric constant, and L and l are the discharge gap
and the dielectric plate thickness. Basically, Eq. 共4兲 describes
gap voltage reduction during the pulse due to charge accumulation on surfaces of the dielectric plates, i.e., charging
of two capacitors with the combined capacitance of
Cd = ␧0␧A / 2l, where A is the dielectric plate surface area. In
the present calculations, peak voltage and pulse duration parameter are taken to be Vpeak = 20 kV and ␶ = 15 ns, respectively. Discharge geometry is assumed to be the same as in
the present experiments 共see Sec. I兲, L = 1 cm, l = 1.75 mm,
and A = 10 cm2, with the dielectric constant of quartz
␧ = 4.3. The system of Eqs. 共1兲–共4兲 is solved with the following initial and boundary conditions:
t = 0:
0
␣
=
pⴱ
冋冉
= Epeak exp −
x = L:n+ = 0,
2l dVgap共t兲 2le 1
dVapp共t兲
= 1+
+
dt
␧L
dt
␧ 0␧ L
冋冉
t − tpeak
Vapp共t兲 Vpeak
=
exp −
2l
2l
␶
L+
L+
␧
␧
冧
; pⴱ = p共torr兲
300
T
共10兲
present conditions, electron-ion recombination and ion drift
have almost no effect on electron and ion density distributions because of short voltage pulse duration, a few tens of
nanoseconds. Basically, recombination and ion drift occur
on a much longer time scale compared with the pulse duration. The two dominant processes at the present conditions
are electron impact ionization and charging the dielectric
surfaces.
Neglecting recombination and considering the quasineu-
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-5
Phys. Plasmas 16, 113505 共2009兲
Energy coupling to the plasma…
tral plasma region 共where ne ⬇ n+ ⬇ n, E = E p兲 and the sheath
region 共where ne ⬇ 0, E = Es兲 separately, for a suddenly applied voltage Vapp = const, the system of Eqs. 共1兲–共4兲 can be
simplified as follows:
dn
= ␯i共E p兲n,
dt
共13兲
冉 冊
冉 冊
2l
1 dls
n
dE p
,
1+
= − 共␯c + ␯s兲E p − 共Es − E p兲
n0
⳵t
␧L
L dt
共14兲
冉 冊
1 dls
dE p
ls
dEs ls 2l
,
+
=−
1−
− 共Es − E p兲
⳵ t L ␧L
⳵t
L
L dt
Vapp共t = 0兲
,
E p共t = 0兲 = Es共t = 0兲 =
2l
L+
␧
where the ionization frequency in the
= ␣␮e兩E p兩共s−1兲, is approximated as follows:
冋 册
␯i共E p兲 = A p pⴱ exp −
B p pⴱ
,
Ep
共16兲
plasma,
ls⬁ =
E p共t兲 =
冦
,
t⬘ ⱕ tⴱ
冉 冊
Eⴱ
␰2
1+
Eapp共t兲
E0
␰1
exp关共␰1 +
␰2兲␯0i 共t⬘
␰2
− t 兲兴 +
␰1
ⴱ
Es共t兲 = Eapp共t兲 + 关Eapp共t兲 − E p共t兲兴
, t⬘ ⬎ tⴱ ,
and
共18兲
冉 冊
共19兲
␯is共Es兲 = AsEs exp −
Bs pⴱ
,
Es
As = 3.6 ⫻ 106,
共20兲
冧
,
Bs = 342,
E
100 ⬍ ⴱ ⬍ 800 V/cm torr.
p
If the applied voltage, given by Eq. 共5兲, is varying in
time much slower than the frequency of ionization, such that
1 / ␶ Ⰶ ␯0i , then the solution for the electric field in the plasma,
electric field in the sheath, and the electron density in the
plasma is as follows:
E⬁p = 0
共21兲
冉
L − ls⬁
,
2l
⬁
ls +
␧
n共t兲 = n0 1 + ␥
冊
a − aEp共t兲/E0
a − aEp共t兲/E0
,
⬇ n⬁
a ln a
a−1
共23兲
共22兲
2l
L+
␧
Es⬁ =
Eapp共t兲,
2l
⬁
ls +
␧
冉 冊
␮Es⬁
1
n⬁
n⬁
ln
=
,
⬁ ln
␣s
n0
n0
␯is共Es 兲
A p = 3.7 ⫻ 108,
where ␯c + ␯s is the characteristic frequency of charging the
capacitors formed by the two dielectric layers and by the
sheath,
1
exp共␯0i t⬘兲 + 1
␥
2lse␮en0
,
␧ 0L
冋 册
␯i
Ep
⬍ 200 V/cm torr,
pⴱ
Eapp共t兲
冉 冊
␯s =
such that dls / dt ⬇ 0 in Eqs. 共14兲 and 共15兲. In Eq. 共19兲, n⬁
is the steady-state electron density in the plasma, Es⬁ is
the quasi-steady-state electric field in the sheath, and
␯is = ␣␮e兩Es兩共s−1兲 is the ionization frequency in the sheath,
共17兲
B p = 318,
2le␮en0
,
2l
␧0␧L 1 +
␧L
and ls is the location of the sheath boundary. Closing the
system of Eqs. 共13兲–共15兲 requires an additional equation for
ls共t兲. However, a simple analytic solution of this system can
be obtained assuming that after the sheath is formed, it rapidly moves to its quasistationary position near the cathode,
共15兲
with the initial conditions
n共t = 0兲 = n0,
␯c =
冉
n⬁ = n0 1 + ␥
冊
a−1
a−1
.
⬇ n 0␥
a ln a
a ln a
In Eqs. 共21兲–共23兲,
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-6
␥=
␯0i = ␯i共E0兲,
␰1 =
Phys. Plasmas 16, 113505 共2009兲
Adamovich et al.
Eⴱ/E0
a−a
,
a ln a
冉 冊
␯0i
,
␯c + ␯s
␰2 =
ⴱ
a = exp
3B p pⴱ
,
2E0
共24兲
Eⴱ/E0
E a
,
E0 3a
In Eq. 共21兲, t⬘ is time shifted by the difference between
breakdown delay time for the constant applied voltage,
V0 = E0共L + 2l / ␧兲, and that for a Gaussian voltage waveform
given by Eq. 共5兲,
const
Gauss
+ tbr
,
t⬘ = t − tbr
where tⴱ is the time when the power density coupled to the
plasma, w共t兲 = e␮en共t兲E2p共t兲, peaks
Eⴱ
=
xⴱ =
E0
ln
2a
2 + ln a
,
ln a
冉
ⴱ
yⴱ =
nⴱ a − ax
,
=
n⬁ a − 1
共25兲
冊
1 − xⴱ
tⴱ = 0 ln ␥ ⴱ ,
x
␯i
1
and E0 is the effective breakdown field 共i.e., applied field
when field in the plasma falls to one half of its value兲
determined from the following coupled equations:
E0 = Eapp共t0兲,
ls共t兲 =
冦
␯0i ␶
t0 − tpeak
E0
=
.
␶
4 ln共␥/2兲 B p pⴱ
␮e
冕
共26兲
冉
Gauss
t ⱕ tbr
␯is共Es兲
E s␧ 0
Gauss
共t − tbr
兲,
Gauss
ⴱ
n⬁ en⬁
Bs p
t ⬎ tbr
1+
ln
Es
n0
冊冉 冊
ls⬁,
if ls共t兲 ⬍
ls⬁
Integrating the power density given by Eqs. 共21兲 and
共23兲 and multiplying it by the plasma volume ⬇AL gives the
pulse energy coupled to the plasma during breakdown,
Qbreak =
where
冋
冉 冊
Qcap,break ⴱ 2 ⴱ a − 1
冑␲共1 + 冑2兲 + 2
共x 兲 y
2l
a ln a
1+
␧L
Qcap,break
⬇
,
2l
1+
␧L
册
gⴱ⬇1
冉 冊
共29兲
冉 冊
1 ␧ 0A 2
1
2l
E L+
Qcap,break = CtotV20 =
2
2 2l ⬁ 0
␧
+ ls
␧
2
共30兲
is the energy stored in the capacitor formed by the two dielectric layers and the sheath 共see Fig. 5兲 at the moment
when breakdown occurs. As expected, gⴱ ⬇ 1 in Eq. 共29兲, i.e.,
␯0i
,
共27兲
In Eq. 共27兲, breakdown delay time is defined as the time
when the field in the plasma deviates from the applied field
by 1%.
Note that if 2l / ␧ Ⰶ L, ls⬁ Ⰶ L, then Es⬁ ⬇ 共␧L / 2l兲Eapp共t兲,
n⬁ / n0 = 共␯0i / ␯c兲共a − 1兲 / ln a, and the quasi-steady-state location of the sheath boundary used in Eq. 共22兲 can be found
explicitly from Eq. 共19兲. From Eq. 共1兲, one can also obtain
that after the voltage is applied 共a兲 the sheath boundary
moves away from the cathode with the speed dls / dt
= ␮Eapp共t兲 until breakdown occurs and the field in the plasma
starts deviating from the applied field, 共b兲 after breakdown
the ionization front 共i.e., the sheath boundary兲 rapidly moves
toward the cathode, until 共c兲 it reaches the quasi-steady-state
position near the cathode, ls⬁, given by Eq. 共19兲,
0
Gauss
ls共tbr
兲−
ln共0.01␥兲
Gauss
Gauss
␯0i ␶
− tpeak
兲
Eapp共tbr
tbr
=
.
␶
4 ln共0.01␥/2兲 B p pⴱ
t
Eapp共t兲dt,
const
tbr
=
冧
.
共28兲
the energy dissipated in the plasma is close to the energy
stored in the capacitive load by the voltage pulse. Basically,
as is well known, the energy dissipated in a resistor and the
energy stored in a capacitor in an RC circuit after a voltage is
suddenly applied are the same. It can be seen that for the
fixed load geometry and dielectric material, the breakdown
pulse energy coupled to the plasma is controlled entirely by
the breakdown field, given by Eq. 共26兲. As can be seen from
Eq. 共26兲, the breakdown field is determined by the pulse
waveform 共i.e., by peak applied electric field Epeak and pulse
duration parameter ␶兲 given by Eq. 共5兲, as well as the reduced pressure pⴱ 共i.e., number density兲 given by Eq. 共10兲.
Further applied voltage increase or reduction after breakdown results in additional energy coupling to the plasma,
although if the voltage is varying in time slowly, such that
1 / ␶ Ⰶ ␯0i , this energy is significantly lower than the energy
coupled during breakdown, given by Eq. 共29兲. After breakdown, the load behaves as a constant resistance RC circuit
with the time constant
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-7
Phys. Plasmas 16, 113505 共2009兲
Energy coupling to the plasma…
FIG. 7. 共Color online兲 Time-dependent electron density and electric field in the plasma predicted by the numerical discharge model. Nitrogen, P = 60 torr,
L = 1 cm.
␯RC =
1
1
a−1
= ␯0i
Ⰷ ,
RCtot
a ln a ␶
共31兲
⬇
and the field in the quasi-steady-state plasma is found by
equating the displacement current in the external circuit and
the conduction current in the plasma, Eq共t兲 = Ėapp共t兲 / ␯RC.
Then the field in the plasma during both breakdown and
post-breakdown stages can be found as
冋
E共t兲 ⬇ max关E p共t兲,Eq共t兲兴 = max E p共t兲,
册
Ėapp共t兲
,
␯RC
共32兲
where E p共t兲 is given by Eq. 共21兲. Taking this effect into
account, additional energy coupled to the plasma by the
Gaussian pulse of Eq. 共5兲 after breakdown is
冑2␲
Qcap,peak
Qafter =
2l ␯RC␶
1+
␧L
冉 冊
⬇
冋 冉
Qcap,peak 冑2␲
,
2l ␯RC␶
1+
␧L
冉 冊
Qcoupled = Qbreak + Qafter
1 + erf
tpeak − t0
␶冑2
冊册
2
共33兲
2
/ 2 is the peak energy stored in the
where Qcap,peak = CtotVpeak
capacitor formed by the two dielectric layers and the sheath.
The error function approximation used in Eq. 共33兲 is accurate
at high overvoltages, when breakdown voltage V0 is
significantly lower than pulse peak voltage Vpeak, i.e.,
when tpeak − t0 Ⰷ ␶. At low overvoltages, when V0 ⬇ Vpeak
and tpeak − t0 Ⰶ ␶, energy coupled to the plasma after
breakdown decreases by approximately a factor of 2,
Qafter ⬇ Qcap,peak冑␲ / 2 / 关共1 + 2l / ␧L兲␯RC␶兴.
Total
energy
coupled to the plasma by the entire waveform of Eq. 共5兲,
assuming high overvoltage, is given by the sum of Eqs. 共29兲
and 共33兲,
冋
冉 冊
冉 冊
2
1 CtotVpeak
2
2l
1+
␧L
V0
Vpeak
冑2␲
2
+
␯RC␶
册
共34兲
.
The efficiency of pulse energy coupling to the plasma can be
defined as the ratio of the coupled and incident energies,
Qcoupled
␩=
⬇
Qcoupled + Qcap,peak
冋冉 冊 冑 册
冋冉 冊 冑 册
V0
Vpeak
1+
2
V0
Vpeak
+
2
2␲
␯RC␶
2␲
+
␯RC␶
,
共35兲
with the rest reflected to the high voltage pulse generator.
From Eq. 共35兲 it can be seen that at high overvoltages and
共V0 Ⰶ Vpeak兲 and relatively long pulse durations 共␯RC␶ Ⰷ 1兲,
most of the incident pulse energy is reflected.
The present approach can be generalized to more complex waveforms, such as dual polarity pulses 共e.g., see Fig.
2兲 or multiple pulse reflections from the load and from the
pulse generator, which can be approximated as superposition
of several Gaussian pulses, with coupled energy for each one
calculated using Eq. 共33兲. However, as discussed above, if
␯RC␶ Ⰷ 1, relative contribution of the postbreakdown waveform into the total coupled energy is relatively minor.
The present analytic model remains accurate if the initial
electron density 共i.e., residual electron density remaining
from the previous discharge pulse兲 is much less compared
with peak electron density in the plasma generated during the
pulse, i.e., if n0 Ⰶ n⬁. If this condition is satisfied, the coupled
pulse energy only weakly depends on the initial electron density, since the breakdown field E0 has a weak 共logarithmic兲
dependence on the initial electron density given by Eq. 共26兲.
To test the accuracy of the analytic model of the discharge shown schematically in Fig. 6, it was compared with
numerical solution of Eqs. 共1兲–共4兲. The calculations have
been done for nitrogen at P = 60 torr, T = 300 K, L = 1 cm,
A = 10 cm2, l = 1.75 mm, ␧ = 4.3, pulse peak voltage
Vpeak = 20 kV, tpeak = 100 ns, ␶ = 15 ns 共same as in our previ-
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-8
Adamovich et al.
Phys. Plasmas 16, 113505 共2009兲
FIG. 8. 共Color online兲 Comparison of time-dependent electric field 共left兲 and electron density 共right兲 in the plasma predicted by numerical and analytic
discharge models. Electric field is given by Eqs. 共32兲 and 共21兲, electron density by Eq. 共23兲. Applied voltage waveform, given by Eq. 共5兲, is also shown in the
left figure.
ous experiments2,7,8兲, and initial electron density of
n0 = 107 cm−3. Figure 7 plots time-dependent electron density and electric field distributions in the discharge predicted
by the numerical model. The results illustrate rapid electron
density rise and electric field reduction in the plasma during
breakdown, as well as sheath development and its gradual
reduction until it reaches a quasi-steady-state position near
the cathode on the left, x / L ⬇ 0.03 at t ⬇ 90 ns. From Fig. 7,
it can be seen that after the quasi-steady-state sheath is
formed, both the electron density and the electric field distributions over most of the plasma region outside the sheath 共at
x / L ⬎ 0.2兲 are uniform. The presence of a nonuniform region
between the sheath and the plasma at 0.03⬍ x / L ⬍ 0.2 illustrates a simplifying assumption used by the analytic model
共compare Figs. 6 and 7兲 and somewhat limits its accuracy.
Figures 8–10 compare time-dependent electric field,
electron density, and power density coupled to the plasma,
w共t兲 = e␮en共t兲E2共t兲, as well as electric field in the sheath and
the sheath boundary location predicted by the numerical
model 关Eqs. 共1兲–共4兲兴 with the analytic solution given by Eqs.
共21兲–共23兲, 共30兲, and 共32兲. The plasma and sheath parameters
predicted by the numerical model are taken at the discharge
centerline at x / L = 0.5 and near the cathode at x / L → 0 共see
Fig. 7兲. In the numerical calculations, the sheath boundary
was identified as the location where the space charge density
gradient peaks. It can be seen that the analytic solution accurately predicts these plasma and sheath parameters, as well
as breakdown time and breakdown voltage. The “kink” in the
electric field predicted by the numerical model 共see Fig. 8,
left兲 occurs during the rapid sheath motion to its quasisteady-state location ls⬁, which is not incorporated in the analytic solution for the field and the electron density given by
Eqs. 共21兲 and 共23兲. It can also be seen that the sheath boundary location is predicted by Eqs. 共28兲 and 共19兲 fairly accurately. From Fig. 10 共right兲, it can be seen that the quasisteady-state sheath thickness is small, such as ls⬁ Ⰶ 2l / ␧.
Basically, at these conditions the effect of the sheath on the
total capacitance of the load Ctot 关see Eq. 共30兲兴 and therefore
on the coupled pulse energy Qcoupled 关see Eq. 共34兲兴 is fairly
weak.
Note that since the applied voltage is slowly varying in
time, such that 1 / ␶ Ⰶ ␯0i , energy coupling to the plasma during further voltage increase after breakdown is fairly insignificant 共see Fig. 9兲. In this case the load acts as an RC
circuit with the time constant ␯RC Ⰷ 1 / ␶ 关see Eq. 共31兲兴 and
the ratio of the energy dissipated in the resistor 共i.e., in the
plasma兲 to the energy stored in the capacitor is of the order
of 1 / ␯RC␶ Ⰶ 1 关see Eq. 共34兲兴. On the other hand, voltage
increase after breakdown results in significant additional en-
FIG. 9. 共Color online兲 Comparison of time-dependent power density
coupled to the plasma during breakdown and postbreakdown predicted by
numerical and analytic discharge models. Applied voltage waveform, given
by Eq. 共5兲, is also shown.
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-9
Energy coupling to the plasma…
Phys. Plasmas 16, 113505 共2009兲
FIG. 10. 共Color online兲 Comparison of time-dependent electric field in the sheath 共left兲 and of the sheath boundary location 共right兲 predicted by numerical and
analytic discharge models. Sheath electric field is given by Eq. 共22兲, sheath boundary location by Eq. 共28兲.
ergy storage in the capacitors formed by the charged dielectric layers 共see Fig. 5兲, which is reflected back when the
applied voltage is reduced to zero. At the conditions of
Figs. 8–10, the energy coupling efficiency, defined as the
ratio of the coupled pulse energy to the incident 共i.e., coupled
plus stored兲 energy, given by Eq. 共35兲, is approximately
␩ ⬇ 30%. Additional voltage oscillations occurring after the
main pulse would somewhat increase the net incident pulse
energy and therefore further reduce energy coupling efficiency. At these conditions, direct measurements of the
coupled pulse energy may well be problematic.
Figure 11 compares pulse energy per unit volume
coupled to the plasma and energy coupling efficiency, predicted by the numerical model and given by the analytic
solution, Eqs. 共34兲 and 共35兲, as functions of peak voltage and
reduced pressure, pⴱ = p共300/ T兲. It can be seen that the
coupled pulse energy increases with both peak voltage and
pressure 共more accurately, with the number density兲 due to
breakdown voltage increase 关see Eq. 共34兲兴. Since the coupled
energy increases approximately linearly with reduced pressure 共see Fig. 11兲, the energy coupled per molecule, approxi-
mately 0.25 meV/molecule, remains nearly independent of
the number density. This shows that pulse energy coupled to
the plasma during a burst of nanosecond pulses decreases
approximately inversely proportional to temperature,
Qcoupled ⬃ 1 / T, due to temperature rise in the plasma and
breakdown voltage reduction. The bulk of the energy is
coupled to the plasma during the breakdown pulse, which is
represented by the first term in Eq. 共34兲, Qbreak. However,
reducing the pulse duration to ␶ ⬃ 1 – 2 ns would considerably increase the energy coupled after the breakdown, given
by the second term in Eq. 共34兲, Qafter. Finally, both increasing
pulse peak voltage and reducing pulse duration would result
in increase in the breakdown voltage, given by Eq. 共26兲. This
demonstrates that using very short duration, high overvoltage
pulses could significantly increase coupled pulse energy.
As discussed above, the present analytic model remains
accurate if the initial electron density 共i.e., residual electron
density remaining from the previous discharge pulse兲 is
much less than the peak electron density in the plasma generated during the pulse, i.e., if n0 Ⰶ n⬁. Comparison of the
analytic model to the numerical calculations has shown that
FIG. 11. 共Color online兲 Pulse energy per unit volume coupled to the plasma 共left兲 and energy coupling efficiency 共right兲 as functions of peak voltage and
discharge pressure predicted by numerical and analytic discharge models. Pulse energy is given by Eqs. 共34兲 and 共26兲, energy coupling efficiency by Eq. 共35兲.
Solid lines, numerical model; dashed lines, analytic model; symbol, experimental value inferred from O atom density measurements in dry air 共Ref. 5兲.
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-10
Phys. Plasmas 16, 113505 共2009兲
Adamovich et al.
the coupled pulse energy dependence on the initial electron
density in the range n0 = 107 – 1011 cm3 is relatively weak
共within about 20%, see Fig. 12兲. In Fig. 12, coupled pulse
energy predicted by the numerical model is calculated with
and without taking into account energy coupled in the nonuniform region between the plasma and the sheath 共see Fig.
7兲. This was done by 共a兲 integrating coupled power density
both over time and the entire discharge region or 共b兲 integrating centerline power density over time and multiplying it by
the discharge volume. The difference between these two results decreases for larger electrode gaps.
From Fig. 12, it can be seen that the analytic model
agrees well with the numerical calculations if n0
⬍ 1011 cm3, i.e., is n0 / n⬁ ⬍ 0.1. At higher initial electron
densities, coupled pulse energy decreases significantly since
plasma shielding occurs very rapidly. In nitrogen, this effect
imposes an upper limit on the highest pulse repetition rates at
which the analytic model is accurate, above which the residual electron density would become comparable to the
peak electron density reached during the pulse 共i.e., the
plasma would not have time to decay between the pulses兲.
For n⬁ ⬇ 1012 cm3 共see Fig. 8兲, the estimated decay time due
to recombination is ␶rec ⬃ 1 / ␤n⬁ ⬃ 30 ␮s, where ␤ ⬃ 3
⫻ 10−8 cm3 / s is the electron-ion recombination rate coefficient in nitrogen 共assuming Te ⬃ 1 eV兲. Therefore the
present analytic model is no longer accurate in nitrogen at
pulse repetition rates above ␯limit ⬃ 10 kHz. In air, typical
plasma decay time between the pulses is much shorter due to
three-body electron attachment to oxygen, ␶att ⬃ 1 / ␣att共nO2兲2,
where ␣att ⬃ 2.5⫻ 10−30 cm6 / s is the attachment rate
coefficient,3 ␶att ⬃ 2.5 ␮s at P = 60 torr, giving ␯limit
⬃ 100 kHz.
Although the present calculations have been conducted
for nitrogen, model predictions for electronegative air
plasma turn out to be very similar. For this, an additional
equation for the number density of dominant negative ions
共O2−兲 has been added to the system of Eqs. 共1兲–共4兲 with
Townsend ionization coefficient for air taken the same as in
Ref. 11. Basically, at these strong electric fields and short
pulse durations, the effect of electron attachment to oxygen
and negative ion formation is fairly insignificant. Therefore,
as expected, coupled pulse energies in nitrogen, given by
Eqs. 共34兲 and 共26兲 and plotted in Fig. 11, are in good
agreement with our previous measurements of a single
nanosecond pulse energy coupled to the plasma in dry air
at P = 60 torr and T = 300 K, Qcoupled = 0.76 mJ 共i.e.,
0.076 mJ/ cm3兲.7 At these conditions, the present analytic
model predicts specific coupled pulse energy of
0.08 mJ/ cm3 共see Fig. 11兲 or Qcoupled = 0.80 mJ. Based on
the results summarized in Fig. 11, in plasma chemistry modeling calculations discussed in Sec. III below the coupled
pulse energy are approximated as follows:
冋 册冋 册
Qcoupled = 0.76
p共torr兲
60
300
共mJ/pulse兲,
T共K兲
with pulse energy loading per molecule being constant.
共36兲
III. AIR PLASMA CHEMISTRY MODEL
The kinetic model of the air plasma used in the present
work incorporates a set of equations for number densities of
neutral species N, N2, O, O2, O3, NO, NO2, N2O, and NO3;
charged species e−, N+, N2+, N3+, N4+, O+, O2+, O4+, NO+,
NO2+, N2O+, N2O2+, N2NO+, O2NO+, NONO+, O−, O2−,
O3−, NO−, NO2−, NO3−, and N2O−; and excited species
N2共A 3⌺兲, N2共B 3⌸兲, N2共C 3⌸兲, N2共a⬘ 1⌺兲, O2共a 1⌬兲,
O2共b 1⌺兲, O2共c 1⌺兲, N共 2D兲, N共 2 P兲, and O共 1D兲 produced in
the plasma.12 The dominant neutral species in air plasma are
excited electronic states of nitrogen A 3⌺, B 3⌸, C 3⌸, and
a⬘ 1⌺ formed by electron impact excitation, O atoms, generated both by electron impact dissociation of oxygen and during excited nitrogen quenching by oxygen, as well as ozone.
The species concentration equations are coupled with the
two-term expansion Boltzmann equation13 for the energy distribution function of plasma electrons. The Boltzmann equation solver is using electron impact cross sections14,15 and
calculates the electron energy distribution function, as well
as the rate coefficients of electron impact ionization, dissociation, and electronic excitation used by the rest of the
model. Rate coefficients of remaining kinetic processes in
the air plasma, such as electron recombination, attachment,
and detachment, ion-molecule reactions, and reactions of excited electronic species are taken from Ref. 12. The list of air
plasma processes incorporated into the model and their rates
is summarized in our previous paper.2
At the conditions of the experiments discussed in Sec. I,
energy loss from a slow plasma flow heated by a repetitively
pulsed nanosecond discharge is primarily due to conduction
to the quartz channel walls. In the present zero-dimensional
FIG. 12. 共Color online兲 Pulse energy coupled to the plasma vs initial electron density predicted by numerical and analytic 关Eqs. 共34兲 and 共26兲兴
discharge models. Numerical results are shown with and without taking
into account pulse energy dissipated in the nonuniform sheath region 共see
Fig. 7兲.
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-11
Phys. Plasmas 16, 113505 共2009兲
Energy coupling to the plasma…
FIG. 13. 共Color online兲 O atom mole fraction vs time after a single high-voltage pulse in air, in methane-air 共equivalence ratio ⌽ = 1.0兲, and in ethylene-air
共⌽ = 0.5兲 mixtures at P = 60 torr 共Ref. 5兲. Pulse energy used in the modeling calculations Qcoupled = 0.76 mJ.
共0D兲 plasma chemistry model, this is described by incorporating a conduction heat transfer term into the energy
equation
dT
␭共T兲 T − Tw
= q̇ −
dt
␳c p 共L/␲兲2
= q␯rep −
冉 冊
1
dn
␭共T兲 T − T
兺 hi dti − ␳cp 共L/␲兲w2 ,
␳c p i
共37兲
where q = QcoupledAL / ␳c p is the coupled pulse energy per unit
mass, ␯rep is the pulse repetition rate in a burst, hi are enthalpies of chemical and excited species, dni / dt are the rates of
species concentrations 共in kmole/ m3兲 change in chemical reactions, ␭共T兲 is the thermal conductivity, Tw = 300 K is the
wall temperature, L is the channel height, and L / ␲ is the
spatial scale for conduction heat transfer with uniform generation in the rectangular geometry. Applicability of a 0D
approximation for time-dependent conduction heat transfer
was validated by solving separately a one-dimensional unsteady heat transfer equation
冉
冊
1 ⳵
⳵T
⳵T
= q̇ +
␭共T兲
,
⳵t
␳c p ⳵ x
⳵x
共38兲
with the same initial and boundary conditions, and the rate
of temperature rise q̇ = q␯rep − 共1 / ␳c p兲兺ihi共dni / dt兲 the same
as predicted by the plasma chemistry model. Comparing
the time-dependent plasma temperature T共t兲 obtained by
the plasma chemistry model using Eq. 共37兲 with the spatially
averaged temperature obtained by solving Eq. 共38兲,
T̄共t兲 = 共1 / L兲兰Lo T共x , t兲dt, shows that the difference between
them does not exceed a few per cent. The use of this approach allows prediction of the time-dependent temperature
distribution in the gap between the electrodes, T共x , t兲, without expanding the full air plasma chemistry model to two
dimensions 共i.e., time and distance between the electrodes兲,
which would require substantial computational resources.
The high voltage pulse shape used by the plasma chemistry model is a Gaussian fit to the experimentally measured
voltage pulse shape given by Eq. 共5兲, such as shown in Fig.
2. However, as discussed in Sec. II, due to strong shielding
of the applied voltage caused by charge accumulation on the
quartz channel walls after breakdown, the field in the plasma
will be much lower than the applied field 关see Eq. 共21兲 and
Fig. 8兴. In our previous work,2,7,8 the pulse peak reduced
electric field 共E / N兲peak was considered an adjustable parameter chosen to reproduce the peak O atom concentration density measured in a single-pulse nanosecond discharge air
plasma at P = 60 Torr, T = 300 K.7 This occurred at the effective reduced electric field of 共E / N兲peak = 250 Td, at which
the coupled pulse energy was Qcoupled = 0.76 mJ.7 The
present air plasma chemistry model, expanded to incorporate
hydrocarbon-air plasma chemical processes, has been validated by comparing with time-resolved O atom concentration measurements in single-pulse nanosecond discharges in
air, methane-air, and ethylene-air mixtures,7 showing very
good agreement 共see Fig. 13兲. This shows that electron impact and plasma chemical processes in room temperature air
plasmas are well understood and adequately described by the
present model.
As discussed in Sec. II, theoretical single pulse coupled
energy predicted for the experimental conditions in Ref. 7 by
the analytic discharge model, Qcoupled = 0.80 mJ 共see Fig. 11兲,
is very close to the value inferred from the experiment.7 To
model plasma chemical processes in air excited by a burst of
nanosecond pulses, i.e., at the conditions when the plasma
temperature during the burst changes significantly, the model
needs to account for the coupled pulse energy variation with
the number density. In the present work, this was done by
adjusting the peak reduced electric field pulse to pulse, such
that the coupled pulse energy Qcoupled varied according to the
prediction of the nanosecond pulse discharge model described in Sec. II, shown in Fig. 11 and approximated by Eq.
共36兲. This simple coupling of the nanosecond pulsed discharge and the plasma chemistry models incorporates the
most essential effect of pulse energy variation proportional to
the number density in the plasma. Note that full coupling of
the two models predicting both nanosecond pulsed discharge
kinetics and plasma chemistry is completely unnecessary and
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-12
Adamovich et al.
FIG. 14. 共Color online兲 Comparison of experimental spatially averaged
共emission兲 temperatures in a repetitively pulsed nanosecond discharge in air
to the plasma chemistry model prediction at different pressures.
would require massive computational resources. Indeed,
pulsed discharge breakdown and sheath formation occur on
the time scale of pulse duration 共a few nanoseconds兲, while
plasma chemical processes responsible for heating occur on
time scales of the pulse repetition rate and pulse burst duration 共a few tens of microseconds and a few milliseconds,
respectively兲.
The present air plasma chemistry model, coupled with
the nanosecond pulse discharge model discussed in Sec. II,
was used to predict the temperature rise in nanosecond pulse
burst discharge in air. The model predictions were compared
with the results of experiments discussed in Sec. I, where
spatially averaged rotational temperature of the plasma was
inferred from nitrogen second positive system emission spectra, and with the results of purely rotational coherent antiStokes Raman spectroscopy 共CARS兲 temperature measurements at the center of the plasma at the same conditions.16
Figure 14 compares experimental spatially averaged
time-resolved temperatures, determined by N2 emission
spectroscopy, in a repetitively pulsed nanosecond discharge
in air at ␯ = 40 kHz and P = 40– 100 torr with the model prediction. In the experiments, each N2 emission spectrum
共0 → 0 band of the second positive band system at 337 nm兲
has been accumulated over a period of 0.5 ms 共20 discharge
pulses兲 using an optical multichannel analyzer with a 0.5 m
spectrometer, 1200 lines/mm grating, and a gated PI-MAX
ICCD camera. The rotational temperature was inferred from
the experimental spectra using best fit synthetic spectra. This
approach has been calibrated previously by comparing the
temperature inferred in preheated air flows excited by a
nanosecond pulsed discharge operated at a low pulse repetition rate with thermocouple measurements without the
discharge.17 From Fig. 13, one can see that after initial rise
Phys. Plasmas 16, 113505 共2009兲
FIG. 15. 共Color online兲 Comparison of experimental spatially averaged
共emission兲 and centerline 共CARS兲 temperatures 共Ref. 15兲 in a repetitively
pulsed nanosecond discharge in air at P = 40 torr to the plasma chemistry
model prediction.
over 10–20 ms 共400–800 pulses兲 the plasma temperature
nearly levels off due to increasing heat transfer losses which
balance energy loading by the discharge. Since discharge
energy loading per molecule is nearly independent of the
number density 共see Fig. 11兲, higher temperatures reached at
higher pressures indicate slower heat transfer at these conditions, consistent with Eq. 共37兲. It can be seen that the agreement of the present model with the experimental measurements is very good, accurately reproducing both the
temperature rise time and near steady-state temperatures,
which provides experimental validation of the nanosecond
pulse discharge model and of the plasma chemistry model.
Figure 15 compares time-resolved air plasma temperatures inferred from N2 emission spectra and from rotational
CARS spectra16 with the plasma chemistry model predictions at P = 40 torr and ␯ = 40 kHz. It can be seen that both
emission 共i.e., spatially averaged兲 and CARS 共centerline兲
temperatures are very close to each other until approximately
7.5 ms 共300 pulses兲. After this, the emission temperature rise
becomes slower and the temperature starts leveling off, indicating significant heat transfer losses, while the CARS temperature continues to increase up to 17 ms 共680 pulses兲. At
t = 15 ms 共600 pulses兲, the difference between CARS and
emission temperature is about 100 ° C. In the modeling calculations, the spatially averaged temperature is given by Eq.
共37兲 and the centerline temperature is given by Eq. 共38兲. One
can see that the average temperature predicted by the model
agrees well with the emission temperature, while the model
prediction for the centerline temperature is in good agreement with the CARS data 共see Fig. 15兲. This demonstrates
that the present air plasma chemistry/heat transfer model,
coupled with the nanosecond pulse discharge model dis-
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
113505-13
Phys. Plasmas 16, 113505 共2009兲
Energy coupling to the plasma…
cussed in Sec. II, adequately describes transverse temperature distribution in the plasma.
IV. SUMMARY
A new analytic quasi-one-dimensional model of energy
coupling to nanosecond pulse discharge plasmas in plane-toplane geometry has been developed. The use of a onedimensional approach is based on images of repetitively
pulsed nanosecond discharge plasmas in dry air, demonstrating that the plasma remains diffuse and uniform on a nanosecond time scale over a wide range of pressures. The two
key processes affecting the pulse energy coupled to the
plasma and incorporated by the model are 共i兲 nanosecond
pulse breakdown and 共ii兲 charge accumulation on dielectric
surfaces covering the electrodes. Plasma decay processes,
such as electron-ion recombination and electron attachment,
are insignificant on the nanosecond time scale. The model
provides analytic expressions for the time-dependent electric
field and electron density in the plasma, electric field in the
sheath, sheath boundary location, and coupled pulse energy.
The analytic model predictions are in very good agreement
with the numerical model of the discharge in drift-diffusion
approximation. The model demonstrates that 共i兲 the energy
coupled to the plasma during an individual nanosecond discharge pulse is controlled primarily by the capacitance of the
dielectric layers and by the breakdown voltage and 共ii兲 the
pulse energy coupled to the plasma during a burst of nanosecond pulses decreases as a function of the pulse number in
the burst. This occurs primarily because of plasma temperature rise and resultant reduction in breakdown voltage, such
that the coupled pulse energy varies approximately proportionally to the number density. The energy loading per molecule for the same pulse voltage waveform remains nearly
independent of the number density.
Analytic expression for coupled pulse energy scaling
with the number density has been incorporated into the air
plasma chemistry model, which was validated previously by
comparison with atomic oxygen number density measure-
ments in nanosecond pulse discharges. The results of kinetic
modeling using the modified air plasma chemistry model are
compared with time-resolved temperature measurements in a
repetitively pulsed nanosecond discharge in air, by emission
spectroscopy and purely rotational CARS, showing good
agreement.
1
A. A. Kirpichnikov and A. Yu. Starikovskii, IEEE Trans. Plasma Sci. 36,
898 共2008兲.
2
M. Uddi, N. Jiang, I. V. Adamovich, and W. R. Lempert, J. Phys. D 42,
075205 共2009兲.
3
Yu. P. Raizer, Gas Discharge Physics 共Springer-Verlag, Berlin, 1991兲.
4
I. Choi, M. Uddi, Y. Zuzeek, I. V. Adamovich, and W. R. Lempert, 47th
AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL, 5–8 January
2009 共unpublished兲.
5
D. S. Nikandrov, L. D. Tsendin, V. I. Kolobov, and R. R. Arslanbekov,
IEEE Trans. Plasma Sci. 36, 131 共2008兲.
6
L. D. Tsendin and D. S. Nikandrov, Plasma Sources Sci. Technol. 18,
035007 共2009兲.
7
M. Uddi, N. Jiang, E. Mintusov, I. V. Adamovich, and W. R. Lempert,
Proc. Combust. Inst. 32, 929 共2009兲.
8
I. V. Adamovich, I. Choi, N. Jiang, J.-H. Kim, S. Keshav, W. R. Lempert,
E. Mintusov, M. Nishihara, M. Samimy, and M. Uddi, Plasma Sources
Sci. Technol. 18, 034018 共2009兲.
9
S. O. Macheret, M. N. Shneider, and R. C. Murray, Phys. Plasmas 13,
023502 共2006兲.
10
S. T. Surzhikov and J. S. Shang, J. Comput. Phys. 199, 437 共2004兲.
11
S. O. Macheret, M. N. Shneider, and R. B. Miles, IEEE Trans. Plasma Sci.
30, 1301 共2002兲.
12
I. A. Kossyi, A. Yu. Kostinsky, A. A. Matveyev, and V. P. Silakov, Plasma
Sources Sci. Technol. 1, 207 共1992兲.
13
L. G. H. Huxley and R. W. Crompton, The Diffusion and Drift of Electrons
in Gases 共Wiley, New York, 1974兲.
14
Y. Itikawa, M. Hayashi, A. Ichimura, K. Onda, K. Sakimoto, K.
Takayanagi, M. Nakamura, M. Nishimura, and T. Takayanagi, J. Phys.
Chem. Ref. Data 16, 985 共1986兲.
15
Y. Itikawa, A. Ichimura, K. Onda, K. Sakimoto, K. Takayanagi, Y. Hatano,
M. Hayashi, H. Nishimura, and S. Tsurubichi, J. Phys. Chem. Ref. Data
18, 23 共1989兲.
16
Y. Zuzeek, A. Montello, I. Choi, M. Nishihara, I. V. Adamovich, and
W. R. Lempert, 40th AIAA Plasmadynamics and Lasers Conference,
San Antonio, TX, 22–25 June 2009 关“Energy coupling and heat release in
air and ethylene-air nanosecond pulse discharge plasmas,” J. Phys. D:
Appl. Phys. 共to be published兲兴.
17
A. Bao, Yu. G. Utkin, S. Keshav, G. Lou, and I. V. Adamovich, IEEE
Trans. Plasma Sci. 35, 1628 共2007兲.
Downloaded 20 Nov 2009 to 164.107.168.82. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp