An Air Breakdown Kinetic Model

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An air breakdown kinetic model

A. E. Rodrfguez, W. L. Morgan, K. J. Touryan, and W. M. MoenyTetra Corporation, 3701 Hawkins Street NE, Albuquerque, New Mexico 87109

T. H. MartinSandia National Laboratories, Albuque rque, New Mexico 87185

(Received 16 November 1990; accepted or publication 13 May 1991)

A kinetic model is developed or the glow phaseof a spark gap in air. The mode l includes

heavy-particle collisions leading to thermal ionization and electron-collision processes.Rates for the latter are calculated by a computer program that solves he electron Boltzmannequation, verified using empirical measurements. hese rates include attachment andionization, excitation followed by ionization of the excited species,and dissociation followedby ionization of the atomic species;molecular-impact detachment was found to benecessary o match experimental closing times. With those processes, he model predictsclosing times well, over a wide range of voltages above and below the dc breakdown voltage.

I. INTRODUCTION

A. Purpose

Spark gapshave beenused or decades s pulsed-powerclosing switches or to sharpen pulse peaks.A spark break-down has at least two phases: the glow phase of the airspark breakdown follows the fast precursor streamer,which leaves a medium-level ionization channel and cul-minates in electrical closure of the gap. As argued byHumphries and Martin,’ the spark-gap closing time isdominated by the time it takes ohmic heating to bring thegas temperature to a point where thermal ionization takesover (several housand K) . This paper describesa kinetics

model of the air spark “cooking,” i.e., under the influenceof ohmic heating, including all important chemicalchanges.

Martin2 noted that spark-gap closing times for a widevariety of gases and electrode geometries can be fitted(within factors of 2) by a power-law dependence n aver-age (not maximum) electric field E scaledby massdensityp rather than number density N. The closing time 7 issurprisingly insensitive to gap length and also to externalcircuit parameters or precise %losing” criteria (e.g., byvoltage or current). For air or SF6, Martin’s empirical fit(in MKS) is

p=7.4x 10’4(p/E)3.4? (1)

At 1 atm, p = 1.29kg/m3, and the dc breakdown voltage sabout 3 MV/m (30 kV/cm), for which the closing timeshould be 72 ns. However, Martin’s fit includes data downto 4 kV/cm. In fact, triggered spark-gap devices outinelyfire well below the dc breakdown voltage. For instance,Wells3 observedstreamers ollowed by breakdown for 100kV acrossa 6-cm gap at 620 Torr, which is equivalent o 25kV/cm at 1 atm (760 Torr) .

An acceptable model of the glow phase must repro -duce the following salient observed features: ( 1) The

closing time must be well defined; (2) the simulated clos-ing times must match measurementsat least as well as

Martin’s empirical fit over a wide range of E/p; and (3)closure must be ach ievedeven or undervolted gaps (belowdc breakdown).

B. Basis for a zero-dimensional kinetic model

The problem is approximately zero dimensional be-cause any local field enhancements long the filament arequickly shunted by the faster ionization, and the radius isapproximately constant. The radius is also immaterial, ex -cept as a multiplier of the current density, since ohmicheating dependson volumetric densities /cm3), not lineardensities. Based on Wells’ current measurements, con-firmed by full two-dimensional (2D) simulations by Wuand Kunhardt4 plus our own 1D simulations,5we took the

initial electron density to be about 10’4/cm3.The chemical kinetics model is basedon that of Park.”

However, Park was modeling reentry vehicle ion sheaths,where the dominant power flow is. rom the neutrals to theelectrons. In a high-voltage discharge, he power flows theother way around, and the electron temperature is muchhigher. Therefore, we took two exceptions to Park’smodel: ( 1) We use the gas temperature instead of thegeometric mean with vibrational temperature for calculat-ing the dissociation rates, since Park’s ustification doesnotseem o apply to our case. (2) We calculate the electron-impact dissociation rates by solving the Boltzmann equa-tion as describedbelow, since t is closer to first principles.The thermal ionization rates have an Arhenius formCT” exp( - A?3/kT), where Ag is an energystep and k isthe Boltzmann constant. Becauseof the sensitivity of theBoltzmann factor to temperature T, there will be a well-defined closing time when thermal ionization takes over,increasing he electron density, which in turn increases heheating rate. This explains the insensitivity of closing timeto the external circuit or precise definitions of the closingpoint.

These data are used in our kinetics code, which alsocalculates he gas emperature Tg by integrating the ohmicheating term

Tg = aE2/pC,, (2)

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0~ Temperature-Dependent Dissociative Attachment DRY AIR with IO4 NZ A3Z; Tv=300-3000K

P4 -3OOK

1 o-5

0 2 4 6

Energy (eV)

8 10

FIG. 1. O2 attachment cross sections vs ekctron energy at several tem-peratures.

where T s conductivity, and uses he resulting temperatureto calculate he chemical ate coefficients. he specificheatat constantvolume, C,, can be shown o be appropriate at1 atm for an ionization channel radius > 10 ,um. Wells3measured he diameter at about 1 mm, and Wu and Kun-hardt’s simulations4 n pure N2 show a near-constant10t4e/cm3 hannelof about 0.2 mm radius. We a ssumedcurrent-collecting area of 0.01 cm’. (It is not a criticalparameter.

II. BOLTZMANNANALYSIS

The remaining kinetics data are the electron-impactreactionsand their rates. These ates are obtainedby solu-tion of the Boltzmann equation. Our Boltzmann analysisuses he ELENDIF code by Morgan and Penetrante.7 h ecross-section etsare primarily from Phelpsand Pitchford*for N2, Phelps’ or 02, and Kieffer’s compilation” for Ar.Dissociation data from Kieffer were added o the N2 andO2 data sets, as were N,(A) ionization data from Thon-That and Flannery.” Kieffer’s temperature-dependentcross sections or attachment o O 2 (Fig. 1) were ncludedto determinehow muc h difference he temperaturedepen-dencewould make. Its effect on closing ime turned out to

be negligible.Dissociative onization was nitially separatedfrom the total ionization and dissociation,but w e found itto be negligible.

The vibrational temperature affects the Boltzmannanalysis.The vibrational manifold population heatsup theelectron-energydistribution function (EEDF) by super-elastic collisions. The ELENDIF code will calculate his ef-fect, given the vibrational temperature or population ofeach vibrational level. Assuming u-u ransfers redistributethe energy,we can define a single vibrational temperatureT,. The power flow from the electron swarm to the vibra-tional manifold can be calculated from electron-number

density t, the vibrationa l excitation ratesKu, calculatedby

0 5 IO 15 20 25 30 35 40

Energy (eV)

FIG. 2. Electron-energy distribution functions from Boltzmann analysi s.

ELENDIF, and the energy evels gs’,as n,&fY$“,. The errorfrom truncating the sum at u==8 or N2 was estimatedbyextrapolating he v=7,8 trend to be 2%-3%. Normalizingto the electron mean energy (8) = 1.5kT,, we can definea vibrational “energy-transfer” rate coefficient as

)1 Ku%,

Ku= (&$J) ’ (3)

which defines he time scaleon which the vibrational tem-perature reachesequilibrium with the electron tempera-

ture. From theELENDIF

output for N2:02:Ar::79:20:1 (dryair), Kt, evaluates o about 3 X 1 0 9 cm3/s for E/N= 50-100 Td. For the initial electron-number density iz,=: 10t4e/cm3,t would take about 3000ns for T, to becomecomparable o electron temperature T,, which is muchlonger than the expectedclosing times ~100 ns. By de-tailed balancing, he loss rate must be such that

Tu=n&u(Te- T,)-*-, (4)

where the ellipses (. * *> stand for power flow to transla-tional motion a nd other neglected erms. From Lacina’sFig. 3,12 he v-t deactivation time for N2 at 1 atm is> 10 ’ s, much too long to be of any consequence. hus

the aboveequation s a reasonable asis or calculating thevibrational temperature.

Sincewe run the kinetics simulation until the gas em-perature reachesT,, we repeated he Boltzmann calcula-tions for T, = 300, 1000,and 3000K. Since he NZ domi-nates, we follow only its vibrational temperature. TheassumedT, for O2 is based on the observation that itsvibrational energy ransfer rate is ten times slower than forN2. Th us all electron-transport and rate coefficients arefunctions of both T, an d T,,.

Figures 2-10 show the results of the ELENDIF runs,plus experimental data for comparison.t3Note that the

distribution functions (Fig. 2) for a vibrational tempera-

2016 J. Appl. Phys., Vol. 70, No. 4, 15 August 1991 Rodrfguez et al. 2016




DRY AIR with IO4 NP A3X; Tv=300-3000K

108

iii

g

,x.j IO7

>"YE

cl

IO" ,

.A.i

_ , , . . ;

experimental

-- I____%!

W6 lo-‘5 lo-”

E/N (V-cm’)

FIG. 3. Calculated vs experimental electron drift velocity. FIG. 5. Calculated vibrational energy-transfer-rate coefficients.

ture of 300 K (solid lines) and 1000K (dashed ines) areindistinguishableexcept at low E/N, and even he 3000-Kcurves (dotted lines) are indistinguishableabove 200 Td.The calculated drift velocities (Fig. 3) are in good agree-ment with experimenta l data. Not surprisingly, electronmean energies (Fig. 4) are significantly higher only for3000 K at low E/N. The vibrational energy-transfer ates(defined above, shown in Fig. 5) are nearly constant be-tween 60 and 80 Td, and insensitive o T,.

III. ELECTRON -IMPACTPROCESSES

A. Attachment and ionization

The attachment cross sections (Fig. 1) shift with tem-perature, and consequently he rates are functions, in prin-

DR Y AIR with IO’ NP A3X; Tv=300-3 000K

mean / /

-3OOK‘.-.. 000

-3000

ICI-‘6 IO-l5 IO"4

E/N (V-cm’)

,045 I : & , ,

10-1" 10-15 IO-'4

E/N (V-cm2)

FIG. 4. Mean and characteristic energies from Bol tzmann analysis. FIG. 6. Calculated vs experimental attachment rates.

IO”I

7.z,\Es2

b5 IO”

s!l-

6k

15p 10.‘0

.g

E.Lz>

lo-” _

DRY AIR lo4 N2 A3Z; Tv=300-3000K

-3OOK_‘. 1000

-3000.

----I02

\/’

E/N (V-cm’)

ciple, of three temperatures:T,( E/N), T,, and Tg- SinceT, and Tg are very close, we avoided a third interpolationparameterby assumingTg =: T,,. For T, = 1000K, we in-terpolated between he 300- and 1030-K calculated rates;for T, = 3000 K, we initially extrapolated rom the 1475-and 1930-K calculated rates. However, the kinetics simu-lations (described below) show T, z 2000 K when T,

z 3000 K, and so we used the 1930-K attachment data.Figure 6 shows he calculated attachment rates comparedto experimentalattachment ratesI The calculated “total”

ionization rates (0.2X@ rate + 0.8 XNz rate) shown inFig. 7 came within a factor of 2 of the experimentaldata.Therefore, we scaled he calculated onization and attach-

DRY AIR with IO4 NZ A3X; Tv=300-3000K

IO-9 E-

lo-‘0 :

ST-> 10-i’ 7EZLc$ lo-i2 r

s

g 10-13 -

IO.14 7





DRY AIR with ItI4 N:! A%; Tv=300-3000K DRY AIR with lo4 N2 A38; T,=300-3000K

106

10-7

"? :5

5 lo-'0

6' 10-l'

'ZJg lo-l2

$IO“3

IO.14

10-15

lo-‘6-IO-‘6 -lo-‘5 IO-l4

E/N (V-cm2)

FIG. 7. Partial, total, and experimental i onization rates.

ment rates o match the experimental ates at 300 K, usingthe Bo ltzmann analysis to determine he temperature de-pendence.

For an undervolted gap, the ionization rate is less hanthe attachment ate. If it were not so, t would not hold offthe voltage. That net loss rate (electron removal fasterthan production) is what makes he gas a good insulator.In 1 atm air, the attachment rate is - 3 x lO’/s just belowbreakdown, and the ionization rate decays quickly as Edecreases. n initial lOI e/cm3 would increase he tem-peratureonly -200 K before hey were ost to attachment.

Yet triggered undervolted spark gaps do break down.Therefore, there must be an electron-sourcemechanismcoming into play which is normally negligible at dc, lowcurrents. This suggests wo-step onization, where he pop-ulation of an intermediate excited speciesmust rise signif-icantly be fore osses ike attachment quench he discharge.

6. Excitation/ionization

One class of two-step ionization mechani sms s elec-tronic excitation to a metastable evel, followed by ioniza-tion of the excited species.The dominant metastablestatein NZ s the A38 state at 6.17 eV, denotedNZ(A>. Figure 8shows electronic excitation rates to three vibrationalgroupsof N&4 ) : 0 < v < 4 at 6.17 eV, 5~~9 at 7.00 eV,and v)lO at 7.80 eV. We took the total N&4) excitationrate to be the s um of these direct excitation rates plusexcitation to the B311and dII levels (shown in Fig. 9),since hesedecayvery quickly to theA32 state according oBoeuf and Kunhardt.i4

N2(A) is quenchedby N and NO (Ref. 15) and byitself.14The self-quenchingproduces N2(B) and Nz( C);following Boeuf and Kunhardt, we assume hey decay n-stantaneously o Nx(A).

Thus calculated, N2 two-step ionization is strong

enough o be significant. However, Ar two-step onization

jp L--.--> .I il I.8

10-16 10-15 1,0“4

E/N (V-cm2)

FIG. 8. Electronic excitation to N?(A) by vibrational groups.

is definitely too weak to contribute to replacing the10t4e/cm3 ost to attachment in tens of ns. At 80 Td, thetotal gains of Ar* are

lo- i* cm3/s X 10i4e/cm3 3X 1017Ar/cm3

=3X 10’g/(cm3 s). (5)

In 30 ns, that would barely produce OI Ar*/cm3, and theproduction of Ar** is weaker still. Therefore, we decidedto build a N2/02 kinetics model, gnoring Ar (except n theBoltzmann analysis).

The 8.40-eV a’Z state, denoted N2a, is neededsinceBrunet and Rocca Serra16 laim the reaction

N2a + N2a-N2 + N2+ + e-, at lo-*’ cm3/s,

DRY AIR with lo4 N2 A32; Tv=300-300OK

Io-l;aF ;B--J---- i- ’ ’ - ’ 110-1s IO.‘4

E/N (V-ch2)

FIG. 9. Electronic excitation to N2 states beyond A3Z.

2018 J. Appl. Phys., Vol. 70, No. 4, 15 August 1991 Rodriguez et al. 2018




is a dominan t source of ionization. It turned out to be asecondary source, ess mportant than the N*(A) path, butnot negligible.

On the chance that superelastic collisions withNz(A> might heat the EEDF, we included populations of10 - 5 to 10m3 of N*(A) in the Boltzmann analysis. Fur-thermore, since the created N*(A) is highly excited vibra-tionally, we assumeda population distribution among the

three groupings corresponding to 15 000 K (approxi-mately T, for 80 Td). It turned out not to affect the Bolt-zmann results significantly ( -0.1% ) . However, such ahigh vibrational temperature will affect the ionization rate.

Cacciatore, Capitelli, and Gorse17 eport cross sectionsfrom vibrational statesof N2. The most obvious effect s theshifting of the energy threshold by the vibrational energy.We can ignore the effect of vibrational populations for thehigh-energy onization and dissociation processes, ince hevibrational temperature is small compared to the thresholdenergies or the bulk of the closing time. However, it couldsignificantly increase the effective ionization rate ofN,(A). A Maxwellian distribution would populate the vi-brational levels by factors proportional toexp ( - $?JkT,) . Assuming the ionization rates for thehigher levels are related to the calculated ionization ratefrom the 6.17-eV level by the Boltzmann factor

exp[ - ( gi-- 8,)/kT,]/exp( - ?Z ’/kT,)

=exp(gdkT,), (6)

where SYi s the ionization threshold, we can calculate ascaling factor for the effect of the vibrational cross-sectionshifting as

1 -I- %, exp[g,( l/kT, - l/kT,)]1 -I 2, exp( - $JkT,,) ’

From the excitation rates of Phelps’ three levels at 6.17,7.00, and 7.80 eV, it is. clear that the N,(A) is createdvibrationally “hot,” with a vibrational temperature z T,.

That simplifies the above scaling factor to 3/[1 + exp( - 0.83 eV/kTJ f exp( - 1.63 eV/kT,)].

A detailed Boltzmann calculation with three energy-shifted N,(A) ionization cross sections or T, = 15 000 Kgave the same results within three significant figures.Therefore, the calculated scaling factor with the sums trun-cated at v=8 (about 1.6 at 80 Td) was used to correct all

Nz(A) ionization rates. We also calculated detailed scalingfactors for all N2 + e - reactions, which were negligible for300 K and at most 1.17 for 3000 K at 60 Td.

C. Dissociation/ionization

Atomic 0 and N are very reactive and can lead tosignificant ionization once their populations become signif-icant. Therefore, we computed electron-impact ionizationrates by Boltzmann analysis, rather than using Park’s for-mulas, which are more appropriate for thermal dissocia-

tion. We used the calculated rates for N2 shown in Fig. 10,but for O2 t is lessstraightforward. The O2 statesabove he

DRY AIR with 10” N2 A3X; Tv=300-3000K

c

10‘8 F

IO” 5

,q 1o“O r

E -lo-” r

.g lo-= 5m

‘8 ‘O-l3 :

$j q&4 ;

10-1s 5

IO-= 1o-15 IO-‘4

E/N (V-cm’)

FIG. 10. N2 and O2 dissociation rates (compared to ionization).

dissociation energy (4.5 eV) dissociate virtually immediately; we assumed 100% efficiency, following Itikawa e

aI.‘*

The subsequent atomic ionization rates were takefrom Park’s data?

0$-e--+0+ +e- +e-,

at 6.476~ 10gT,-3.gexp( - 158 500/T,),

N+e--+N+ +e- +e-,

at 6 .476X 10’“T-3.82 exp( - 168600/T,).In addition, the associative onization

O+N-+NO’ +e-,

at 2.541X lo- I5 p;37 exp( - 3200/T,)

has such a low threshold it is likely to be a significansource of thermal ionization.

D. Other processes

Recombination increases slightly the electron losseover those due to attachment. We used the recombinatio

data of Biondi. lg Attached electrons are recovered (freedby several detachment processes. We used the data oMcDaniel et al.:*’

O- + N2-+N20 + e-, at lo-‘* cm3/s-+-2X 107/s,

O- +02+03+e-, at lo- ‘* cm3/s--+ 5X 106/s,

O- + O-O2 + e-, at 1.9X 10-l’ cm3/s+ -2X 107

The last numbers above are nominal detachment rates, obtained by multiplying each rate coefficient by the density othe second reactant ( - lOi O/cm3 in midglow phase, ashown below). These rates contribute to decreasing th

0 - lifetime and can be used to judge the importance oeach reaction.





Other processes onsidered but not included were

O- + N+NO + e-, at 2.2~ lo-” cm3/s+-106/s,

O-+NO-NO,+e-,

at3 X 10-r’ cm3/s--t < 104/s,

O- +N2(A)-+NO+N+e-,

at lo- ‘o-1O-9 cm3/s+ < lO’/s,

O- + O,(a’A) -03 + e-,

at 3 X 10 lo cm3/s-+ - 106/s,

O- -I-03+202+e-, at 5 X 10 lo cm3/s+ - 103/s.

Photoionization rate data are given by Penney andHummert.2* Integrating their rate function over all ranges(0 to C O, we found that the ratio of photoions (anywhere)to electron-impact ions is about 10 -s. Since photons es-caping the filament radially reduce that even further,photoionization was deemed negligible.

We also considered the possibility of trace impuritiescontributing to ionization. Traces of long-chain hydrocar-

bons like tri-propyl-amine (TPA) are unavoidable n gas-discharge devices, despite careful precautions. TPA’s ion-ization potential of 7 eV is typical. Penning onization from-3x 1015Nz(A)/cm3 of -3~10’~ TPA/cm” (10 ppm)could produce 10z2e/cm3/s f the rate constant is - 10 ’cm3/s. However, that would require almost complete ion-ization of the impurity. Furthermore, a Boltzmann calcu-lation showed that the ionization rate of TPA in air at 80Td is about 10 - lo cm3/s; a noticeable effect on the totalionization rate would require 1000 ppm (much too highfor an “impurity”). Since we cannot estimate mpurity lev-els or composition, we included none.

IV. KINETIC SIMULATIONS AND RESULTS

Sinceall electron-impact reaction rates are functions ofT, and T,, two-parameter interpolation is used among thetabulated data on log-log scales (so linear interpolationcorresponds to piecewise power-law fits). We scaled allelastic collision rates by the- Boltzmann factorexp( - A%‘/T,), which accounts for the first-order behav-ior, in order to improve interpolation accuracy.

The variable time step is controlled by the LivermoreSolver of Ordinary Differential Equations ( LSODE).*' A

simple C-L external circuit was charged o 20,25, or 30 kV(corresponding to 74, 93, and 112 Td at 1 atm). The sim-ulations were at STP density, but Tg and T,, were initial-ized to 300 K to account for the heating from the faststreamer phase.

The simulations were stopped when the gas tempera-ture exceeded he electron temperature. This was taken tobe the closing time. Figure 11 showsclosing times versusE

field, comparing the calculated results of our model toMartin’s fit formula and to empirical data collected byMartin.’ Note the good agreement, even at the limits ofapplicability o f our model. At very low vol tages ( < 10kV/cm), the simulation ran too long for the computer to

handle it. At very high overvoltages (1000 kV/cm), the

2020 J. Appl. Phys., Vol. 70, No. 4, 15 August 1991

Data and Theory for Breakdown Delays

Points = measurement3compiled by T.H.Mxti n

-Martin’s Fit --Kinetic Modal

IO3IO”5 IO-

,o.13 ,042 10’ 10”-110

7

Reduced Closing Time U J (s-gm/cm3)

FIG. 11. Calculated closing times compared to experimental data and tobest power-law fit (Martin’s formula).

simulated closing time increased instead of decreasing.This indicates a breakdown of the model, probably due tounjustified extrapolation of the kinetic coefficients beyondthe region of validity of the Boltzmann analysis. Even so,the glow-phasekinetics model we have derived clearly hasa wide range of applicability (one order of magnitudeabove and below the dc breakdown voltage).

The detailed behavior of the model is exemplified bythe seriesof plots for the 25-kV/cm simulation (Figs. 12-15). (Others produced similar plots, but with the timescale stretched or compressed.) Figure 12 shows the pop-ulations of all species n the kinetics set versus ime; Fig. 13

shows he gas and vibrational temperatures; Fig. 14 and 15are voltage and current traces from the simple externalcircuit model. The general behavior is ust what we expect-ed: Ohmic heating raises he temperature evenly as long

Air Spark Glow Phase (25 kV/cm)

0 30 80 90 120 150

TIME (ns)

FIG. 12. Population densities vs time for 25-kV/cm simulation.

Rodriguez et al. 2020




Air Spark Glow Phase (25kV/cm)

lo51

0 30 60 90 120 150

TIME (ns)

FIG. 13. Temperatures vs time for 2%kV/cm simulation.

as the electron density is nearly constant; then thermalionization kicks in, and both n, and Tg rise sharply, eachaccelerating he other in a strong nonlinear coupling, clos-ing the gap at a well-defined ime.

A simpler graphical summary (Fig. 16) is electron(only) density versus ime for the 30-, 25-, and 20-kV/cmsimulations, compared to Martin’s fit benchmarks. Sincethe assumed nitial electron density representsa criticaluncertainty, the 20-kV/cm simulation was repeatedwithdouble he initial electronsand the 30-kV/cm one with halfthe initial electrons. The agreement s acceptable,consid-ering that Martin’s fo rmula is only accurate to within afactor of 2.

The calculated nitial rates mply that significant pop-ulations of N, 0, and Nz(A) would be created n the faststreamer phase and would be more appropriate as initialconditions for the glow phase.A simulation with nonzero


1: --- ..___...._.__._...____420000

fn

gj 15000>

10000 I

FCl

5000

1

-$&L---~-

q ; DJChl-.T-

L- - ..-..-.-A

\Cd

0 t---b

0 30 so 90 ml 150

TIME (ns)

FIG. 14. Calculated voltage traces for 25-kV/cm simulation.


10-l ’ ,L I ’ ’ t t ’ ’ k10 30 60 90 120 150

TIME (ns)

FIG. 15. Calculated current traces for 25-kV/cm simulation.

estimated nitial populations of thesespecies an only a fewns faster. Note in Fig. 12 that quasiequilibrium evels arereached n the first 10% of the closing time, and so theinitial conditions do not matter much, except for electrondensity. Clearly, the electron density left by the faststreamer is an important initial condition for the glow-phasemodeling.

A detailed analysis by reaction at selected timesshowed hat the dominant sink of electrons s attachmentand the dominant source s direct ionization, as expected.Below 30 kV/cm, attachment is higher, and so secondary

ionization processesmust make up the differenceand pro-duce the net electron gain rate. However, neitherexcitation/ionization nor dissociation/ionization are suffi-cient; the secondary onization mechanismsprimarily re-

Air Spark Glow Phase Simulations

Martin’s Fit Closing Times 1

10'4

10'3 ' ' ' ' ' ' ' ' '

0 50 100 150 200 250 300 350 400 450

Time (ns)

FIG. 16. Electron-density traces compared to Martin’s fit closing times.





sponsible or making the net electron rate positive are thedetachment mechanisms, which free attached electrons.

v. CONCLUSlONS

Confidence in the model is supported by the goodagreementwith experimental closing times. The dominantphenomenology appears to be as follows: Electron at-tachment and ionization are nearly balanced,but detachedelectrons keep the discharge alive long enough for thermalionization to kick in and close the gap at a well-definedtime. Ignored phenomena, ncluding multidimensional ef-fects, must not have a significant effect on the growth ofconductivity after the fast streamers connect to the elec-trodes.

Elimination of negligible reactions and consolidationof related processes an simplify the model further. Incor-porating such a simplified model into a one-dimensionalfilamentary model (OD local calculation at grid points)will allow studying the transient effect of inhomogeneousinitial conditions, the most crucial of which is the electron

density left by the fast streamer.

ACKNOWLEDGMENTS

We wish to thank Dr. Art Phelps or many discussionson the physics ssuesand his valuable assistancen locatingmuch of the data used. We are also indebted to Dr. RussHumphries for sharing the results of his own literaturesearchand his tentative theory which inspired us to seek asimple theoretical model. This work was funded by SandiaNational Laboratories.

‘R. Humphries and T. H. Martin, Sandia National Laboratory report(unpublished).

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lished).9A. V. Phelps, JILA Report No. 28, 1985 (unpublished).

“L. J. Kieffer, JILA Report No. 13, 1973 (unpublished).

“D. Thon-That and M. R. Flannery, Phys. Rev. A 15, 517-526 (1977).“W. B. Lacina, Northrup Research 8r Technology Center Report No.

NRTC 75-25R, 1972 (unpublished).

13D. K. Dav ies and P. J. Chandry, Westinghouse Report Nos. T. N. 346and 352, 1984 (unpublished).

14J. P. Boeuf and E. E. Kunhardt, J. Appl. Phys. 60, 915 (1986).“R. A. Young and G. A. St. John, Chemical Reactions in Electrical

Discharges, edited by R. F. Gould (American Chemical Society, Wash-ington, DC, 1967), Chap. 8.

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