A simplified fluid model of the metallic plasma and neutral gas interaction in a multicathode spot...

1322 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 4, AUGUST 1998

A Simplified Fluid Model of the MetallicPlasma and Neutral Gas Interactionin a Multicathode Spot Vacuum Arc

Hector Kelly, Adriana Marquez, and Fernando O. Minotti

Abstract—A stationary fluid model with spherical symmetryis presented to describe the interaction between metallic plasmaions with neutral gas in the outer region of a multicathode spotvacuum arc operated with a neutral background gas. It is foundthat the neutrals penetrate into the metallic plasma with densityvalues smaller than the initial gas density values, but higher thanthe metallic ion densities. The neutrals are also strongly heatedduring the transient expansion stage of the metallic plasma. As aconsequence, the ion kinetic energy is gradually delivered to theneutral gas so that the mean free path for ion-neutral elasticcollisions is larger than the visible plasma ball radius whichsurrounds the arc.

Index Terms—Fluid model, ion-gas interaction, vacuum arc.

I. INTRODUCTION

T HE application of cathodic vacuum arcs to the depositionof thin films has mainly arisen from the attractive charac-

teristics of the ion emission at the cathode spots for film growth[1]–[2]. In these devices a high current electrical discharge isgenerated between conducting electrodes immersed in a vac-uum (or low pressure) chamber, the current being conducted bya plasma consisting of ionized vapor of the cathode material.The emission of electrons and vapor comes from minute andmobile sites on the cathode surface, known as cathode spots.The cathode material is emitted in the form of a highly ionized(30–100%) plasma jet [3]. This jet contains ions flowing awayfrom the cathode spots with energies in the range 10–100 eV,with a mean charge state depending on the cathode material[4] and carrying in vacuum 7–10% of the total arc current[5]. Such cathodic phenomena have attracted attention to theproduction of coatings by placing a substrate in a locationwhere it intercepts the plasma jet. Furthermore, if a reactivegas such as nitrogen or oxygen is also present in the dischargechamber, metal oxide and nitride films can be produced.

A typical example of cathodic vacuum arc reactive coatingis the production of TiN films using a Ti cathode in a lowpressure N ambient. Although this coating process is anestablished industrial technology, little is known about theinteraction between the ion flux emitted from the cathode and

Manuscript received July 24, 1997; revised March 2, 1998. This work wassupported by grants from Buenos Aires University and the Consejo Nacionalde Investigaciones Cient´ıficas y Tecnol´ogicas (CONICET).

The authors are with the Instituto de Fısica del Plasma (CONICET),Departamento de Fısica, Facultad de Ciencias, Exactas y Naturales (UBA),Ciudad Universitaria Pab. I, 1428 Buenos Aires, Argentina (e-mail:[email protected]).

Publisher Item Identifier S 0093-3813(98)05346-6.

the neutral gas. It is a well-known fact that the presence ofa neutral gas produces an attenuation of the ion current withrespect to that measured under vacuum conditions [6]–[9], themagnitude of such attenuation being dependent on the typeof gas, gas pressure, and on the distance between the ioncollecting probe and the electrodes.

A surrounding gas will limit the expansion of the plasma atsome distance from the spot center. This effect was experimen-tally investigated by Meunier and Drouet [10], who measuredthe size of the expanding luminous plasma as a function ofthe gas pressure. After a short transient lasting a fews,they found an equilibrium radius for the plasma whichfollowed the scaling law constant for a given gasspecies ( is the gas pressure andthe total discharge current).Boxman and Goldsmith [11] presented a zero-order modelfor the plasma jet-background gas interaction, by equatingthe plasma jet momentum flux with the background gaspressure. In this case, the theoretical curves relating, ,and followed a constant scaling law. Later on,Meunier [12] proposed a snowplow fluid model to study thetransient expansion of the plasma jet. The main assumptionof this model is the existence of an impermeable movingbarrier (shock wave) separating two gas-like volumes, themetallic plasma, and the neutral gas. The shock wave, drivenby the metallic plasma, sweeps the neutral gas until the barriervanishes because it has slowed down below the sound velocityof the gas, thus reaching an equilibrium radius. By equatingthe total work made by the plasma to the variation of thekinetic energy of the plasma and the gas, Meunier obtained anequation for the velocity of the barrier in terms of, , theerosion rate , the initial ion velocity, and several parametersof the background gas. Finally, he derived a scaling law for

of the type: where is the gas mass densityand and are constants depending on the gas species. Theobtained values of the exponent (from [12, Fig. 5]) were:

1.3 (for He gas), 1.7 (for Ar gas), and 2.0 (for SF gas).It must be noted that in the above quoted theoretical

developments, the description of the interaction between themetallic plasma and the neutral gas has been undertaken froma kinematics point of view, and hence: a) no local balance ofeither momentum or energy has been attempted; b) no accountof the mixing between the metallic plasma and neutral gas hasbeen taken; and c) heating and expansion of the surroundinggas has not been taken into account. That one should expectsome penetration of the neutral gas into the otherwise pure

0093–3813/98$10.00 1998 IEEE

KELLY et al.: SIMPLIFIED FLUID MODEL 1323

metallic plasma comes from some experiments with highcurrent (500–2000 A), short inter-electrode length pulsed arcswhere TiN films were produced at a very high depositionrate, operating with a Ti cathode in a Nbackground gasat relatively high pressure (up to 10 mbar) with the anodeacting at the same time as the substrate to be coated [13],[14]. It should be noted that these particular experiments didnot follow the experimental general trend to produce reactivecoatings.

The purpose of this paper is to present a fluid-type simpli-fied model locally describing the metallic plasma-neutral gasinteraction. Instead of studying the fast transient expansion ofthe metallic plasma (as was done in [12]) we will look forsteady state solutions with the aim of addressing points a), b),and c) mentioned in the previous paragraph. For simplicity,atomic physics describing excitation and ionization processesof the neutral gas and metallic vapor will not be includedin this model because, for typical values of the metallicplasma density and electron temperature (10 –10 cmand 2–10 eV, respectively), the probability of these processesis relatively low, so they can be obtained afterwards once thezero-order plasma-gas structure has been established from themodel.

II. STATEMENT OF THE MODEL

In this section, the equations of a stationary fluid model withspherical symmetry are presented to describe the outer regionof the arc where the main interaction between metallic plasmaand neutral gas occurs. These equations are applied to the so-called multicathode spot vacuum arc (MCS) [15] because arcsoperating in this mode have a well-developed inter-electrodeplasma region.

The application of a fluid model requires that the velocitydistribution function of each species involved be sufficientlyclose to Maxwellian, which is justified when the particle meanfree path is smaller than the typical scale lengths of theproblem, and when the collision frequency is higher than anyother characteristic frequency of the problem. The conditionon the frequency is well-satisfied in arcs of duration largerthan the ion transit time along the discharge chamber, about10 s. For the typical values of the plasma quantities in theinter-electrode space of a MCS vacuum arc ( 3 eVand 10 –10 cm ), the electron-electron and ion-ion mean free paths for elastic collisions is about 1 mm,while the typical lengths of the discharge chamber are severalcentimeters. Although the ions have relatively high energies( eV), as their mass is much larger than the electronmass, the average relative electron-ion velocity is essentiallythe electron thermal velocity, and hence the electron-ion meanfree path is of the same order of the electron-electron meanfree path. This condition allows the use of classical relaxationrates for the exchange of momentum and energy between ionsand electrons. On the other hand, the fact that the ions havekinetic energies much higher than their thermal energies resultsin a relatively long mean free path for collisions with neutralswhen the densities of the latter are not too high (as it will be

Fig. 1. Outline of the geometry employed to develop the model.

shown later, the neutral density in the surrounding of the mainarc is much smaller than the initial gas density).

Other simplifications are as follows.

1) Assumption of a neutral plasma, which is justified bythe smallness of the Debye length , which for theplasma conditions considered is cm, wellbelow any other characteristic length.

2) Spherical symmetry, supported by the experimentalobservation of remarkably spherical distributions ofluminosity in the space around the electrodes, found inexperiments performed at relatively high backgroundgas pressures [12].

3) Neglect of the electron inertia in the electron momen-tum equation.

4) Although the cathode emission involves several ionsdiffering in their kinetic energy and charge; in order todetermine the plasma-gas structure a single ion speciesis assumed, averaged in charge and energy. In addition,as a first approximation consistent with the assumedspherical symmetry, the angular dependence of the ionflux [16] will be neglected.

5) No metal vapor is considered in the region outside theelectrodes.

A final, important consideration of the model concerns thebehavior of the charge carriers in the outer region of the arc.Let us consider the electrodes-chamber configuration, shownin Fig. 1, consisting in two electrodes facing each other andenclosed in a discharge chamber whose walls and cathodeare grounded. Consider first a situation in which vacuumor low gas pressure surrounds the arc, so that ion-electronrecombination is negligible in the outer region. It has beenexperimentally found [5] that metallic shields biased to thegrounded cathode potential collect a purely positive (ion)current, thus indicating that a positive sheath must exist inthe vicinities of the collector, represented in this case bythe chamber walls. If electrons were continuously draggedalong by the ions leaving the arc, they would accumulate inthe vicinities of the sheath (with the consequent increase of


charge in this region), and no stationary state would exist.Consequently, in the steady state, an electrostatic potential

( being the spherical coordinate) must develop outsidethe electrodes in order to force a zero electron drift velocity

. It will be shown later, however, that the role of inthis problem is much more complex than just slowing downthe electrons. In the high neutral gas pressure case (whenplasma recombination is important), the general descriptionabove quoted remains valid (in fact, metallic shields biased tothe grounded cathode potential also collect a positive currentin this situation [6]), but with a slight difference: when ionvelocity is sufficiently reduced by interactions with the neutralgas, recombination with the electrons is more likely (probablynot directly but with the mediation of a charge exchangeprocess with the gas [7]). Consequently, some electrons aredragged from the arc in order to compensate for the electronlosses due to ion-electron recombination and is differentfrom zero. However, we will show later that the effect of theensuing electronic current is negligible in all practical cases.

With all these considerations, we start with the equations forthe low pressure case (zero electron drift velocity), but lateron the effects on the plasma structure introduced by a nonzeroelectron drift velocity will be discussed.

We consider the discharge region as represented by anspherical “black box” of radius at a potential ( is theanode potential, which is expected to be close to the plasmapotential in the arc) acting as an ion and thermal source forthe outer region. The latter is filled with a neutral plasma andwith neutral gas particles of mass at temperature ,and density . The plasma electrons have temperature

and density , and the plasma ions have mass,charge kinetic energy and density .As the kinetic ion energy is much larger than their thermalenergy, it is not necessary to consider the ion temperatureequation (when the ions are slowed down to energies of theorder of the neutral temperature both species are consideredto be in thermal equilibrium). An outline of the problem isshown in Fig. 1.

The equations that determine the plasma structure in theouter region of the discharge are

—ion continuity

(1)

where is the fraction of the ion current leaving the arc—charge neutrality

(2)

—ion momentum

(3)

where is thefriction force of electrons on ions (is the plasma conductivityand is Braginskii’s thermoelectric coefficient as used in[17]), and is the cross section for elastic scattering betweenions and neutrals. is expressed as a function of the ion

kinetic energy as [18]

where , , and are constants with typical valuescm , eV, eV.

—electron momentum

(4)

—neutral momentum

(5)

—electron energy

(6)

where is (minus) the electron thermalflux ( is the electron thermal conductivity [17]),

is the energy equipartition frequencybetween electrons and neutrals (the corresponding one betweenelectrons and ions is negligible), is a nondimensionalparameter which measures the electron-neutral energy transferper collision ( for monoatomic gases, while

for diatomic gases [19]), is the electronthermal velocity, and is the cross section for elasticscattering between electrons and neutrals.

—neutral energy

(7)

where is (minus) the neutral thermal flux( is the neutral thermal conductivity [20]) and is thefraction of energy transfer per ion-neutral collision,

.The above set of equations was solved numerically using

a standard Runge-Kutta fourth–fifth order, for given valuesof , and

. Of all these magnitudes, only the temperaturederivatives are not given by the characteristics of the arc.

is taken as the same value corresponding to avacuum arc (without filling gas) determined by requiring that

approaches zero at infinity, while is taken aszero (see discussion below). The integration starts atand ends at the value of where the energy of theions is comparable to the neutral gas temperature. From thispoint on, the ions are assumed to be stopped and thermalizedwith the neutral gas, and so there is no more exchange ofeither energy or momentum between both species. In thisway, a neutral gas heat conduction equation (without energysources) is solved for with the boundary conditionsgiven by at and room temperature at the dischargechamber radius . Since in this external region the ionshave already delivered their kinetic energy, the pressure of theneutral gas remains constant and equal to the prescribed valueof the filling pressure.

The neutral gas density at the electrodes is chosenso as to reach the desired value of the filling pressure at


(a)

(b)

Fig. 2. Profiles of several quantities derived from the model for an arc operating in Ar gas at a filling pressurep = 2 mbar with a copper cathode(zr = 1:85). The values of the other discharge parameters are:Ir = 10 A, a = 0:5 cm, Te(a) = 3 eV, '(a) = 20 V, ne(a) = 1:81014 cm�3,Ei(a) = 51:8 eV, and rw = 25 cm. (a) ' and Ei and (b) ne and Te.

, whereas the neutral temperature at the electrodesis selected so as to reach the ambient temperature at. Ifthe temperature derivatives at are changed, and

change accordingly to fit the unperturbed gas conditionsfar away, and a solution very similar to the original one isobtained, the difference being in a small region near(in this sense the exact values of the temperature derivativesare not important).

When considering the case of different from zero, somemodifications must be incorporated to the equations of themodel. For this it is convenient to quantify the electron driftvelocity in terms of the ratio of the electron to ion currentsin the outer region of the arc: , in terms of which,using the continuity equation of electrons and the neutralitycondition (2), is times the ion velocity. In this way,

the appearing in the expressions of and in (3) and(4) must be multiplied by to represent the relativeion-electron drift velocity. Furthermore, a term of the form

must be subtracted from the right-hand sideof (4) to account for the electron-neutral drag, and a term of theform must be addedto the right-hand side of (6) to account for the divergence ofthe electron enthalpy flux.

III. RESULTS

In Fig. 2 we present the profiles of and [Fig. 2(a)],and [Fig. 2(b)], and and the neutral pressure

[Fig. 2(c)] for an arc operating in Ar gas at a filling pressurembar with a copper cathode . The values


(c)

Fig. 2. (Continued.)Profiles of several quantities derived from the model for an arc operating in Ar gas at a filling pressurep = 2 mbar with a coppercathode(zr = 1:85). The values of the other discharge parameters are:Ir = 10 A, a = 0:5 cm, Te(a) = 3 eV, '(a) = 20 V, ne(a) = 1:81014

cm�3, Ei(a) = 51:8 eV, and rw = 25 cm. (c) Nn and pn.

of the other discharge parameters are: A,cm, eV, V, cm ,

eV, and cm. The valuecorresponds to the average value of the kinetic energies ofcopper ions for an arc with 100 A of discharge current,experimentally determined by Davis and Miller [21] once thecorrections of the plasma potential of the arc [4] have beenperformed. The total arc current corresponding to this exampleis not directly related to the value of . If all the ion currentproduced in the cathode spots had escaped from the arc, therelation (with – ) would be valid. Sincepart of the ion current is collected by the anode, only a fractionof the total ion current penetrates into the surrounding space.In practice, this fraction can be evaluated in each case fromthe knowledge of the electrodes radii and the inter-electrodegap. In the example considered, for a inter-electrode gap of0.5 cm, A corresponds to an arc currentA (the geometrical factor was calculated following [5]). FromFig. 2(a) it can be seen that the electrostatic potential decreasesby 7 V in the first 2 cm. Since is determined by theelectron momentum (4) and of the two terms forming ,the electron-ion drag is very small comparedto the electron pressure gradient in this equation (it results10 –10 -times smaller), the profile of is dictated by thebehavior of (mainly) and , which for small values ofare both decreasing functions of[see Fig. 2(b)]. Forcm, remains constant, following the behavior of .

The ions are subjected to an accelerating electrical forcedue to the potential drop and to a decelerating force due to theelastic collisions with the neutrals. In the example considered,the last force dominates, and so decreases continuouslywith until the ions are stopped at cm.

The decreasing profile of [Fig. 2(b)] is determined bythe almost zero divergence of the electron thermal flux in

the electron energy equation because the energy transfer ratebetween electrons and neutral gas (or ions) is much smaller(by a factor 10 –10 ) than any term in the expansion ofthe thermal flux divergence.

According to (1) and (2), is proportional to .While is in general a decreasing quantity and hence pro-duces an increase in , the geometrical factor produces adecrease in , so there are two competing effects. In general,the geometrical factor will dominate in situations for which

. In the example considered, the geometrical factordominates for small values of( cm), and hence isreduced by a factor of seven in the first 1.5 cm. Then, an almostflat profile of develops in the region cm ,where the ions are strongly slowed down.

The neutral density [Fig. 2(c)] continuously increases (start-ing from an initial value of 8.5 10 cm ) as a consequence ofa decrease in its temperature (because of neutral heat conduc-tion) and an increase in due to the impulse transferred bythe ions. Similarly, the neutral pressure continuously increasesfrom a value of 1.75 mbar at to a value of 2 mbarat . The neutral gas is still strongly perturbed at thispoint cm and eV),so that its state is relaxed at constant pressure by thermalconduction until unperturbed values ( cmand eV) are reached at . The gas temperatureprofile is not shown in Fig. 2. In this last region, diffusionand absorption of metallic ions and electrons into the neutralgas should occur at a rate equal to the ion flux leavingthe arc. It must be noted that in the described example theeffects of the ion-electron recombination in the presence ofthe neutral gas for can be estimated as follows:assuming values of the absorption cross sections such as thatreported in [7] ( cm ), the corresponding meanfree path calculated with the filling density is 0.15 cm,


TABLE ISEVERAL NOTICEABLE CHARACTERISTICS OF THEPLASMA-GAS STRUCTURE OBTAINED FOR TWO DIFFERENT VALUES OF THE

ARC RADIUS (a) AND SEVERAL VALUES OF THE FILLING PRESSURES.THE VALUES CORRESPOND TO ANARC WITH

COPPERCATHODE (zr = 1:85; Ei(a) = 51:8 eV), AR GAS FILLING, Ir = 10 A, Te(a) = 3 eV, AND '(a) = 20 V

thus indicating that recombination in this case is considerablefor a discharge chamber dimension of several cm. However,calculations performed with the corrected equations to allowfor different from zero showed practically the same plasmastructure (for the region ) as that previously obtainedwith , even in cases for which ion recombination wasconsidered almost complete ( , that is, the electron currentdragged from the arc completely balanced the ion current).Other calculations for filling gas pressures up to 10 mbar haveshowed that had practically no influence on the resultingplasma structure.

In Table I we present other noticeable characteristics ofthe plasma-gas structure obtained for two different values ofthe arc radius and several values of the filling pressures.As in the previous example, a copper cathode (

eV), Ar gas, A, eV, andV are considered. Note that the ion density in

the boundary of the arc , once the value of is fixed,depends only on the value of. The value of decreasesas the pressure increases (as expected) and depends slightlyon the value of . The potential drop in the region of theion energy deposition decreases forincreasing values of or decreasing values of (see thediscussion in the next section), while the neutral density inthe arc boundary shows an opposite behavior. The

values are only a small fraction (typically 0.5–2%)of the neutral filling densities, but they are higher than thearc plasma densities. The neutral temperature in the boundaryof the arc increases with pressure, and this fact seemsto indicate a better thermal energy transfer from the metallicplasma to the neutrals during the transient plasma expansionstage. For comparison purposes, the ratio of the ion, neutraldensity, and electron and neutrals temperatures between theirvalues at and are also given in the Table I. Notethat the variations of the ion density (or the electron density)are larger as increases and decreases, as a consequenceof geometrical effects (see the discussion in the next section).

In Fig. 3 values of are plotted as a function of theparameter for several gases: helium, nitrogen, argon,and SF. In this figure the filling gas pressure was variedin the range 0.5–6 mbar with A, the rest of thearc parameters being equal to those presented in Fig. 2. For afixed value of , other values of the parameter produceonly small changes in the curves (for instance, variations of

Fig. 3. Values ofr� as a function of the parameterp=Ir, for several gases:helium, nitrogen, argon, and SF6. Ir = 18 A and the rest of the arc parametersare equal to those used in Fig. 2.

in the range 7–40 A produce changes in the value ofsmaller than 10%). Fig. 3 also shows that different gases lieon different curves as a consequence of the dependence onthe neutral mass of the ion momentum transfer fraction perhead-on collision [ ] showing that the ions aremore efficiently stopped in a heavy gas than in a lighter one.

IV. DISCUSSION

The model presented predicts the existence of an electro-static voltage profile in the outer region of the arc. The voltagedrop in this region is mainly controlled by the electron pressure[(4) of the model] because the friction force of electrons onions (or electrons on neutrals in the case of non zero)is small for the whole range of arc parameters investigated( – mbar, – A). It is worth noting thatthe fraction of the voltage drop which is required to stopthe electrons in cases where is very small: as themaximum should be of the order of the ion directedvelocity, the maximum electron kinetic energy will be lessthan one hundredth of eV. As in general the spatial gradientof is much larger than that of , it follows from (4)(neglecting the term and the temperature gradient) that

, and hence the voltage drop in a distancecan be estimated as .

Considering that is proportional to , when the


values of and are low , strongly decreases inthe region because geometrical effects dominateover the ion slowing down. Hence large voltage drops areexpected in this case (see the case cm andmbar in Table I). For high values of and ,the decrease of (and consequently that of ) is verysmall (indeed, it is possible to find situations of enough highvalues of and where increases in the outer regionof the arc, see the case cm and mbar inTable I).

The model also predicts that some penetration of the neutralgas into the metallic plasma is necessary in order to satisfythe boundary conditions of the steady-state problem. Thesituation is different from the transient expansion phase ofthe plasma studied in [12], where two not mixed gas-likevolumes (metallic plasma and neutral gas) exist. Although theneutral densities in the vicinities of the arc are only a smallfraction (typically 0.5–2%) of the neutral filling densities, theyare higher than the arc plasma densities, thus accounting forthose experiments with the same geometry of Fig. 1 wherereactive films were efficiently produced on an anode acting atthe same time as the substrate to be coated [13], [14].

It should be noted that the gas temperature in regions closeto the arc remains quite high (1–2 eV) as a consequence ofthe strong heating of the gas during the transient stage, whichis sustained in the steady state by the ion energy deposition.As a consequence of the heating and rarefaction of the neutralgas in the zone where the ion kinetic energy is delivered,the ion range corresponding to ion-neutral collisions are muchlarger than that calculated with the initial gas neutral fillingdensity.

For a given gas species, it is found from Fig. 3 that theslopes of the curves follow a constant dependencewhich resembles Boxman and Goldsmith predictions [11] andalso Meunier calculations [12]. However, the independentparameter in those references was proportional to, sothe comparison with our calculations is not straightforwardbecause it requires the knowledge of the ratio for allpossible values of the arc parameters. On the one hand, it mustbe noted that the ion production at the cathode spots is pressuredependent due to redeposition or cathode poisoning effects;and on the other hand, it must be taken into account that only afraction of the total cathode ion emission leaves the main arc,this fraction being dependent on the geometrical parametersof the electrodes in each particular experiment. In addition,as the steady size of the plasma “ball” was determinedusing a filtered framing camera [10] which registered visiblelight coming mainly from the metallic plasma and vapor (andnot from the neutral gas), while was defined in this workas the radius at which the ions are stopped, it is by no meansobvious that and have the same physical meaning. In anycase, if in order to compare these two quantities one takes theratio (as was done in [11]), one finds the correctdependence on filling pressure and arc current (see Fig. 3),but values of 2–2.5 times larger than the correspondingvalues of . We tried alternative definitions of in terms ofother physical quantities derived from the model, such as theion momentum flux, electron density, etc., but we were not

able to obtain ’s closer to the observed values of, thusconcluding that a complete modeling of the light emissionfrom the plasma is required to compare with the experiments.

V. CONCLUSION

The model presented above for the interaction between themetallic plasma and the neutral gas has allowed us to obtainthe plasma-gas structure attained under steady state conditionsin a multi-cathode spot arc. The general structure derived fromthe model for the steady state seems to match well with themechanism proposed by Meunier [12] to explain the expansionof the metallic plasma during the transient state, because itis expected that during the first stages of the discharge theneutral gas density remains quite high (close to the initialfilling value) so that the ion energy is delivered onto a thinsuperficial layer producing a shock wave which sweeps theneutral gas. However, this layer of strongly heated neutralgas begins to expand with the consequent decrease of itsdensity and enlargement of the ion-neutral mean free path.Hence, the strength of the shock continuously decreases untilan equilibrium radius is reached in the steady state.

The main features of the model are: 1) the existence ofan electrostatic voltage gradient which balances the electronpressure gradient; 2) a penetration of the neutrals into themetallic plasma of the arc with neutral density values smallwith respect to the filling density values but higher thanthe metallic ion densities in the arc. The neutrals are alsostrongly heated during the transient expansion stage of themetallic plasma; 3) as a consequence of the previous point,the ion kinetic energy is delivered to the neutral gas with acharacteristic spatial scale much larger than that calculatedwith the initial neutral gas filling density; and 4) the ion rangeare larger than the visible plasma radius [10] or the theoreticalpredictions based on the balance of the plasma-neutral pressureat a sharp boundary [11], [12].

A detailed theoretical study of the size of the plasma “ball”cannot be performed at this stage of the model development,because to give account of excitation, de-excitation, ionization,and recombination of the neutral gas and metallic vapor,atomic physics must be included in the model equations.Efforts on these improvements are currently under way.

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[22] S. Goldsmith and R. L. Boxman, “Excited-state densities in amulticathode-spot Al vacuum arc. II. Theoretical approach,”J. Appl.Phys., vol. 51, no. 7, pp. 3649–3656, 1980.

Hector Kelly was born in Mendoza, Argentina, on February 14, 1948. Hereceived the M.S. degree and Ph.D. degree in physics from Buenos AiresUniversity, Argentina, in 1972 and 1979, respectively.

Since 1973, he has worked as Researcher at the Plasma Physics Laboratoryof the Science Faculty of Buenos Aires University. His current researchinterests are in powerful electrical discharges and plasma coatings based onelectrical discharges.

Dr. Kelly is a member of the National Research Council of Science ofArgentina.

Adriana M arquez was born in Buenos Aires, Ar-gentina, on August 4, 1964. She received the M.S.degree and Ph.D. degree in physics from BuenosAires University, Argentina, in 1989 and 1994,respectively.

She joined the Plasma Physics Laboratory ofthe Science Faculty of Buenos Aires University in1987. Her current research interests are in powerfulelectrical discharges and plasma coatings based onelectrical discharges.

Dr. Marquez is a Fellow of the National ResearchCouncil of Science of Argentina.

Fernando O. Minotti was born in Buenos Aires,Argentina, on January 27, 1960. He received theM.S. degree and the Ph.D. degree in physics fromBuenos Aires University, Argentina, in 1984 and1990, respectively.

Since 1984 he has worked as Researcher at thePlasma Physics Laboratory of the Science Facultyof Buenos Aires University. His current researchinterests are in Fluid Mechanics and Turbulence.

Dr. Minotti is a member of the National ResearchCouncil of Science of Argentina.

A simplified fluid model of the metallic plasma and neutral gas interaction in a multicathode spot...

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