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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 4, AUGUST 1998
A Simplified Fluid Model of the Metallic
Plasma and Neutral Gas Interaction
in a Multicathode Spot Vacuum Arc
Héctor Kelly, Adriana Márquez, and Fernando O. Minotti
Abstract— A stationary fluid model with spherical symmetry
is presented to describe the interaction between metallic plasma
ions with neutral gas in the outer region of a multicathode spot
vacuum arc operated with a neutral background gas. It is found
that the neutrals penetrate into the metallic plasma with density
values smaller than the initial gas density values, but higher than
the metallic ion densities. The neutrals are also strongly heated
during the transient expansion stage of the metallic plasma. As a
consequence, the ion kinetic energy is gradually delivered to the
neutral gas so that the mean free path for ion-neutral elastic
collisions is larger than the visible plasma ball radius which
surrounds the arc.
Index Terms—Fluid model, ion-gas interaction, vacuum arc.
I. INTRODUCTION
T
HE application of cathodic vacuum arcs to the deposition
of thin films has mainly arisen from the attractive characteristics of the ion emission at the cathode spots for film growth
[1]–[2]. In these devices a high current electrical discharge is
generated between conducting electrodes immersed in a vacuum (or low pressure) chamber, the current being conducted by
a plasma consisting of ionized vapor of the cathode material.
The emission of electrons and vapor comes from minute and
mobile 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 away
from 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 the
production of coatings by placing a substrate in a location
where it intercepts the plasma jet. Furthermore, if a reactive
gas such as nitrogen or oxygen is also present in the discharge
chamber, metal oxide and nitride films can be produced.
A typical example of cathodic vacuum arc reactive coating
is the production of TiN films using a Ti cathode in a low
pressure N ambient. Although this coating process is an
established industrial technology, little is known about the
interaction between the ion flux emitted from the cathode and
Manuscript received July 24, 1997; revised March 2, 1998. This work was
supported by grants from Buenos Aires University and the Consejo Nacional
de Investigaciones Cientı́ficas y Tecnológicas (CONICET).
The authors are with the Instituto de Fı́sica del Plasma (CONICET),
Departamento de Fı́sica, 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 of
a neutral gas produces an attenuation of the ion current with
respect to that measured under vacuum conditions [6]–[9], the
magnitude of such attenuation being dependent on the type
of gas, gas pressure, and on the distance between the ion
collecting probe and the electrodes.
A surrounding gas will limit the expansion of the plasma at
some distance from the spot center. This effect was experimentally investigated by Meunier and Drouet [10], who measured
the size of the expanding luminous plasma as a function of
the gas pressure. After a short transient lasting a few s,
for the plasma which
they found an equilibrium radius
constant for a given gas
followed the scaling law
species ( is the gas pressure and the total discharge current).
Boxman and Goldsmith [11] presented a zero-order model
for the plasma jet-background gas interaction, by equating
the plasma jet momentum flux with the background gas
pressure. In this case, the theoretical curves relating , ,
constant scaling law. Later on,
and followed a
Meunier [12] proposed a snowplow fluid model to study the
transient expansion of the plasma jet. The main assumption
of this model is the existence of an impermeable moving
barrier (shock wave) separating two gas-like volumes, the
metallic plasma, and the neutral gas. The shock wave, driven
by the metallic plasma, sweeps the neutral gas until the barrier
vanishes because it has slowed down below the sound velocity
of the gas, thus reaching an equilibrium radius. By equating
the total work made by the plasma to the variation of the
kinetic energy of the plasma and the gas, Meunier obtained an
equation for the velocity of the barrier in terms of , , the
erosion rate , the initial ion velocity, and several parameters
of the background gas. Finally, he derived a scaling law for
of the type:
where
is the gas mass density
and and are constants depending on the gas species. The
obtained 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 the
metallic plasma and the neutral gas has been undertaken from
a kinematics point of view, and hence: a) no local balance of
either momentum or energy has been attempted; b) no account
of the mixing between the metallic plasma and neutral gas has
been taken; and c) heating and expansion of the surrounding
gas has not been taken into account. That one should expect
some 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 high
current (500–2000 A), short inter-electrode length pulsed arcs
where TiN films were produced at a very high deposition
rate, operating with a Ti cathode in a N background gas
at relatively high pressure (up to 10 mbar) with the anode
acting at the same time as the substrate to be coated [13],
[14]. It should be noted that these particular experiments did
not follow the experimental general trend to produce reactive
coatings.
The purpose of this paper is to present a fluid-type simplified model locally describing the metallic plasma-neutral gas
interaction. Instead of studying the fast transient expansion of
the metallic plasma (as was done in [12]) we will look for
steady 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 processes
of the neutral gas and metallic vapor will not be included
in this model because, for typical values of the metallic
plasma density and electron temperature ( 10 –10 cm
and 2–10 eV, respectively), the probability of these processes
is relatively low, so they can be obtained afterwards once the
zero-order plasma-gas structure has been established from the
model.
II. STATEMENT
OF THE
MODEL
In this section, the equations of a stationary fluid model with
spherical symmetry are presented to describe the outer region
of the arc where the main interaction between metallic plasma
and neutral gas occurs. These equations are applied to the socalled multicathode spot vacuum arc (MCS) [15] because arcs
operating in this mode have a well-developed inter-electrode
plasma region.
The application of a fluid model requires that the velocity
distribution function of each species involved be sufficiently
close to Maxwellian, which is justified when the particle mean
free path is smaller than the typical scale lengths of the
problem, and when the collision frequency is higher than any
other characteristic frequency of the problem. The condition
on the frequency is well-satisfied in arcs of duration larger
than the ion transit time along the discharge chamber, about
10 s. For the typical values of the plasma quantities in the
3 eV
inter-electrode space of a MCS vacuum arc (
10 –10 cm ), the electron-electron and ionand
ion mean free paths for elastic collisions is about 1 mm,
while the typical lengths of the discharge chamber are several
centimeters. Although the ions have relatively high energies
eV), as their mass is much larger than the electron
(
mass, the average relative electron-ion velocity is essentially
the electron thermal velocity, and hence the electron-ion mean
free path is of the same order of the electron-electron mean
free path. This condition allows the use of classical relaxation
rates for the exchange of momentum and energy between ions
and electrons. On the other hand, the fact that the ions have
kinetic energies much higher than their thermal energies results
in a relatively long mean free path for collisions with neutrals
when 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 main
arc is much smaller than the initial gas density).
Other simplifications are as follows.
1) Assumption of a neutral plasma, which is justified by
, which for the
the smallness of the Debye length
cm, well
plasma conditions considered is
below any other characteristic length.
2) Spherical symmetry, supported by the experimental
observation of remarkably spherical distributions of
luminosity in the space around the electrodes, found in
experiments performed at relatively high background
gas pressures [12].
3) Neglect of the electron inertia in the electron momentum equation.
4) Although the cathode emission involves several ions
differing in their kinetic energy and charge; in order to
determine the plasma-gas structure a single ion species
is assumed, averaged in charge and energy. In addition,
as a first approximation consistent with the assumed
spherical symmetry, the angular dependence of the ion
flux [16] will be neglected.
5) No metal vapor is considered in the region outside the
electrodes.
A final, important consideration of the model concerns the
behavior of the charge carriers in the outer region of the arc.
Let us consider the electrodes-chamber configuration, shown
in Fig. 1, consisting in two electrodes facing each other and
enclosed in a discharge chamber whose walls and cathode
are grounded. Consider first a situation in which vacuum
or low gas pressure surrounds the arc, so that ion-electron
recombination is negligible in the outer region. It has been
experimentally found [5] that metallic shields biased to the
grounded cathode potential collect a purely positive (ion)
current, thus indicating that a positive sheath must exist in
the vicinities of the collector, represented in this case by
the chamber walls. If electrons were continuously dragged
along by the ions leaving the arc, they would accumulate in
the vicinities of the sheath (with the consequent increase of
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 4, AUGUST 1998
charge in this region), and no stationary state would exist.
Consequently, in the steady state, an electrostatic potential
( being the spherical coordinate) must develop outside
the electrodes in order to force a zero electron drift velocity
. It will be shown later, however, that the role of
in
this problem is much more complex than just slowing down
the electrons. In the high neutral gas pressure case (when
plasma recombination is important), the general description
above quoted remains valid (in fact, metallic shields biased to
the grounded cathode potential also collect a positive current
in this situation [6]), but with a slight difference: when ion
velocity is sufficiently reduced by interactions with the neutral
gas, recombination with the electrons is more likely (probably
not directly but with the mediation of a charge exchange
process with the gas [7]). Consequently, some electrons are
dragged from the arc in order to compensate for the electron
is different
losses due to ion-electron recombination and
from zero. However, we will show later that the effect of the
ensuing electronic current is negligible in all practical cases.
With all these considerations, we start with the equations for
the low pressure case (zero electron drift velocity), but later
on the effects on the plasma structure introduced by a nonzero
electron drift velocity will be discussed.
We consider the discharge region as represented by an
( is the
spherical “black box” of radius at a potential
anode potential, which is expected to be close to the plasma
potential in the arc) acting as an ion and thermal source for
the outer region. The latter is filled with a neutral plasma and
at temperature
,
with neutral gas particles of mass
. The plasma electrons have temperature
and density
and density
, and the plasma ions have mass
,
kinetic energy
and density
.
charge
As the kinetic ion energy is much larger than their thermal
energy, it is not necessary to consider the ion temperature
equation (when the ions are slowed down to energies of the
order of the neutral temperature both species are considered
to be in thermal equilibrium). An outline of the problem is
shown in Fig. 1.
The equations that determine the plasma structure in the
outer region of the discharge are
—ion continuity
kinetic energy as [18]
(1)
is the fraction of the ion current leaving the arc
where
—charge neutrality
(2)
—ion momentum
(3)
is the
where
friction force of electrons on ions ( is the plasma conductivity
is Braginskii’s thermoelectric coefficient as used in
and
is the cross section for elastic scattering between
[17]), and
is expressed as a function of the ion
ions and neutrals.
where
,
, and
are constants with typical values
cm ,
eV,
eV.
—electron momentum
(4)
—neutral momentum
(5)
—electron energy
(6)
is (minus) the electron thermal
is the electron thermal conductivity [17]),
is the energy equipartition frequency
between electrons and neutrals (the corresponding one between
is a nondimensional
electrons and ions is negligible),
parameter which measures the electron-neutral energy transfer
for monoatomic gases, while
per collision (
for diatomic gases [19]),
is the electron
is the cross section for elastic
thermal velocity, and
scattering between electrons and neutrals.
—neutral energy
where
flux (
(7)
is (minus) the neutral thermal flux
where
is the neutral thermal conductivity [20]) and
is the
(
fraction 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 values
, and
of
. Of all these magnitudes, only the temperature
derivatives are not given by the characteristics of the arc.
is taken as the same value corresponding to a
vacuum arc (without filling gas) determined by requiring that
approaches zero at infinity, while
is taken as
zero (see discussion below). The integration starts at
and ends at the value of
where the energy of the
ions is comparable to the neutral gas temperature. From this
point on, the ions are assumed to be stopped and thermalized
with the neutral gas, and so there is no more exchange of
either energy or momentum between both species. In this
way, a neutral gas heat conduction equation (without energy
with the boundary conditions
sources) is solved for
at
and room temperature at the discharge
given by
. Since in this external region the ions
chamber radius
have already delivered their kinetic energy, the pressure of the
neutral gas remains constant and equal to the prescribed value
of the filling pressure.
is chosen
The neutral gas density at the electrodes
so as to reach the desired value of the filling pressure at
KELLY et al.: SIMPLIFIED FLUID MODEL
1325
(a)
(b)
Fig. 2. Profiles of several quantities derived from the model for an arc operating in Ar gas at a filling pressure p = 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) = 1:8 1014 cm03 ,
Ei (a) = 51:8 eV, and rw = 25 cm. (a) ' and Ei and (b) ne and Te .
, whereas the neutral temperature at the electrodes
is selected so as to reach the ambient temperature at
. If
the temperature derivatives at
are changed,
and
change accordingly to fit the unperturbed gas conditions
far away, and a solution very similar to the original one is
obtained, the difference being in a small region near
(in this sense the exact values of the temperature derivatives
are not important).
different from zero, some
When considering the case of
modifications must be incorporated to the equations of the
model. For this it is convenient to quantify the electron drift
velocity in terms of the ratio of the electron to ion currents
, in terms of which,
in the outer region of the arc:
using the continuity equation of electrons and the neutrality
is
times the ion velocity. In this way,
condition (2),
the appearing in the expressions of
and
in (3) and
to represent the relative
(4) must be multiplied by
ion-electron drift velocity. Furthermore, a term of the form
must be subtracted from the right-hand side
of (4) to account for the electron-neutral drag, and a term of the
must be added
form
to the right-hand side of (6) to account for the divergence of
the 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 pressure
mbar with a copper cathode
. The values
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 4, AUGUST 1998
(c)
Fig. 2. (Continued.) Profiles of several quantities derived from the model for an arc operating in Ar gas at a filling pressure p = 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) = 1:8 1014
cm03 , 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
value
corresponds to the average value of the kinetic energies of
copper ions for an arc with 100 A of discharge current,
experimentally determined by Davis and Miller [21] once the
corrections of the plasma potential of the arc [4] have been
performed. The total arc current corresponding to this example
is not directly related to the value of . If all the ion current
produced in the cathode spots had escaped from the arc, the
(with
– ) would be valid. Since
relation
part of the ion current is collected by the anode, only a fraction
of the total ion current penetrates into the surrounding space.
In practice, this fraction can be evaluated in each case from
the knowledge of the electrodes radii and the inter-electrode
gap. In the example considered, for a inter-electrode gap of
A corresponds to an arc current
0.5 cm,
A (the geometrical factor was calculated following [5]). From
Fig. 2(a) it can be seen that the electrostatic potential decreases
is determined by the
by 7 V in the first 2 cm. Since
,
electron momentum (4) and of the two terms forming
is very small compared
the electron-ion drag
to the electron pressure gradient in this equation (it results
is dictated by the
10 –10 -times smaller), the profile of
(mainly) and , which for small values of
behavior of
are both decreasing functions of [see Fig. 2(b)]. For
cm, remains constant, following the behavior of .
The ions are subjected to an accelerating electrical force
due to the potential drop and to a decelerating force due to the
elastic collisions with the neutrals. In the example considered,
decreases continuously
the last force dominates, and so
cm.
with until the ions are stopped at
[Fig. 2(b)] is determined by
The decreasing profile of
the almost zero divergence of the electron thermal flux in
the electron energy equation because the energy transfer rate
between electrons and neutral gas (or ions) is much smaller
(by a factor 10 –10 ) than any term in the expansion of
the thermal flux divergence.
is proportional to
.
According to (1) and (2),
is in general a decreasing quantity and hence proWhile
produces a
duces an increase in , the geometrical factor
decrease in , so there are two competing effects. In general,
the geometrical factor will dominate in situations for which
. In the example considered, the geometrical factor
cm), and hence
is
dominates for small values of (
reduced by a factor of seven in the first 1.5 cm. Then, an almost
develops in the region
cm
,
flat profile of
where the ions are strongly slowed down.
The neutral density [Fig. 2(c)] continuously increases (starting from an initial value of 8.5 10 cm ) as a consequence of
a decrease in its temperature (because of neutral heat conducdue to the impulse transferred by
tion) and an increase in
the ions. Similarly, the neutral pressure continuously increases
to a value of 2 mbar
from a value of 1.75 mbar at
. The neutral gas is still strongly perturbed at this
at
cm
and
eV),
point
so that its state is relaxed at constant pressure by thermal
cm
conduction until unperturbed values (
and
eV) are reached at . The gas temperature
profile is not shown in Fig. 2. In this last region, diffusion
and absorption of metallic ions and electrons into the neutral
gas should occur at a rate equal to the ion flux leaving
the arc. It must be noted that in the described example the
effects of the ion-electron recombination in the presence of
can be estimated as follows:
the neutral gas for
assuming values of the absorption cross sections such as that
cm ), the corresponding mean
reported in [7] (
0.15 cm,
free path calculated with the filling density is
KELLY et al.: SIMPLIFIED FLUID MODEL
1327
TABLE I
SEVERAL NOTICEABLE CHARACTERISTICS OF THE PLASMA-GAS STRUCTURE OBTAINED FOR TWO DIFFERENT VALUES OF THE
ARC RADIUS (a) AND SEVERAL VALUES OF THE FILLING PRESSURES. THE VALUES CORRESPOND TO AN ARC WITH
COPPER CATHODE (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 considerable
for a discharge chamber dimension of several cm. However,
calculations performed with the corrected equations to allow
different from zero showed practically the same plasma
for
) as that previously obtained
structure (for the region
, even in cases for which ion recombination was
with
, that is, the electron current
considered almost complete (
dragged from the arc completely balanced the ion current).
Other calculations for filling gas pressures up to 10 mbar have
had practically no influence on the resulting
showed that
plasma structure.
In Table I we present other noticeable characteristics of
the plasma-gas structure obtained for two different values of
and several values of the filling pressures.
the arc radius
As in the previous example, a copper cathode (
eV), Ar gas,
A,
eV, and
V are considered. Note that the ion density in
the boundary of the arc
, once the value of
is fixed,
decreases
depends only on the value of . The value of
as the pressure increases (as expected) and depends slightly
on the value of . The potential drop in the region of the
decreases for
ion energy deposition
increasing values of
or decreasing values of
(see the
discussion in the next section), while the neutral density in
shows an opposite behavior. The
the arc boundary
values are only a small fraction (typically 0.5–2%)
of the neutral filling densities, but they are higher than the
arc plasma densities. The neutral temperature in the boundary
increases with pressure, and this fact seems
of the arc
to indicate a better thermal energy transfer from the metallic
plasma to the neutrals during the transient plasma expansion
stage. For comparison purposes, the ratio of the ion, neutral
density, and electron and neutrals temperatures between their
and
are also given in the Table I. Note
values at
that the variations of the ion density (or the electron density)
increases and decreases, as a consequence
are larger as
of geometrical effects (see the discussion in the next section).
In Fig. 3 values of
are plotted as a function of the
for several gases: helium, nitrogen, argon,
parameter
and SF . In this figure the filling gas pressure was varied
in the range 0.5–6 mbar with
A, the rest of the
arc parameters being equal to those presented in Fig. 2. For a
, other values of the parameter
produce
fixed value of
only small changes in the curves (for instance, variations of
Fig. 3. Values of r3 as a function of the parameter p=Ir , for several gases:
helium, nitrogen, argon, and SF6 . Ir = 18 A and the rest of the arc parameters
are equal to those used in Fig. 2.
in the range 7–40 A produce changes in the value of
smaller than 10%). Fig. 3 also shows that different gases lie
on different curves as a consequence of the dependence on
the neutral mass of the ion momentum transfer fraction per
] showing that the ions are
head-on collision [
more efficiently stopped in a heavy gas than in a lighter one.
IV. DISCUSSION
The model presented predicts the existence of an electrostatic voltage profile in the outer region of the arc. The voltage
drop in this region is mainly controlled by the electron pressure
[(4) of the model] because the friction force of electrons on
)
ions (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 that
(
the fraction of the voltage drop which is required to stop
is very small: as the
the electrons in cases where
should be of the order of the ion directed
maximum
velocity, the maximum electron kinetic energy will be less
than one hundredth of eV. As in general the spatial gradient
is much larger than that of
, it follows from (4)
of
term and the temperature gradient) that
(neglecting the
, and hence the voltage drop
in a distance
can be estimated as
.
is proportional to
, when the
Considering that
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 4, AUGUST 1998
values of and are low
,
strongly decreases in
because geometrical effects dominate
the region
over the ion slowing down. Hence large voltage drops are
cm and
expected in this case (see the case
mbar in Table I). For high values of and ,
the decrease of
(and consequently that of
) is very
small (indeed, it is possible to find situations of enough high
and
where
increases in the outer region
values of
cm and
mbar in
of the arc, see the case
Table I).
The model also predicts that some penetration of the neutral
gas into the metallic plasma is necessary in order to satisfy
the boundary conditions of the steady-state problem. The
situation is different from the transient expansion phase of
the plasma studied in [12], where two not mixed gas-like
volumes (metallic plasma and neutral gas) exist. Although the
neutral densities in the vicinities of the arc are only a small
fraction (typically 0.5–2%) of the neutral filling densities, they
are higher than the arc plasma densities, thus accounting for
those experiments with the same geometry of Fig. 1 where
reactive films were efficiently produced on an anode acting at
the same time as the substrate to be coated [13], [14].
It should be noted that the gas temperature in regions close
to the arc remains quite high (1–2 eV) as a consequence of
the strong heating of the gas during the transient stage, which
is sustained in the steady state by the ion energy deposition.
As a consequence of the heating and rarefaction of the neutral
gas in the zone where the ion kinetic energy is delivered,
the ion range corresponding to ion-neutral collisions are much
larger than that calculated with the initial gas neutral filling
density.
For a given gas species, it is found from Fig. 3 that the
constant dependence
slopes of the curves follow a
which resembles Boxman and Goldsmith predictions [11] and
also Meunier calculations [12]. However, the independent
, so
parameter in those references was proportional to
the comparison with our calculations is not straightforward
for all
because it requires the knowledge of the ratio
possible values of the arc parameters. On the one hand, it must
be noted that the ion production at the cathode spots is pressure
dependent due to redeposition or cathode poisoning effects;
and on the other hand, it must be taken into account that only a
fraction of the total cathode ion emission leaves the main arc,
this fraction being dependent on the geometrical parameters
of the electrodes in each particular experiment. In addition,
was determined
as the steady size of the plasma “ball”
using a filtered framing camera [10] which registered visible
light coming mainly from the metallic plasma and vapor (and
was defined in this work
not from the neutral gas), while
as the radius at which the ions are stopped, it is by no means
obvious that and have the same physical meaning. In any
case, if in order to compare these two quantities one takes the
(as was done in [11]), one finds the correct
ratio
dependence on filling pressure and arc current (see Fig. 3),
2–2.5 times larger than the corresponding
but values of
in terms of
values of . We tried alternative definitions of
other physical quantities derived from the model, such as the
ion momentum flux, electron density, etc., but we were not
able to obtain ’s closer to the observed values of , thus
concluding that a complete modeling of the light emission
from the plasma is required to compare with the experiments.
V. CONCLUSION
The model presented above for the interaction between the
metallic plasma and the neutral gas has allowed us to obtain
the plasma-gas structure attained under steady state conditions
in a multi-cathode spot arc. The general structure derived from
the model for the steady state seems to match well with the
mechanism proposed by Meunier [12] to explain the expansion
of the metallic plasma during the transient state, because it
is expected that during the first stages of the discharge the
neutral gas density remains quite high (close to the initial
filling value) so that the ion energy is delivered onto a thin
superficial layer producing a shock wave which sweeps the
neutral gas. However, this layer of strongly heated neutral
gas begins to expand with the consequent decrease of its
density and enlargement of the ion-neutral mean free path.
Hence, the strength of the shock continuously decreases until
an equilibrium radius is reached in the steady state.
The main features of the model are: 1) the existence of
an electrostatic voltage gradient which balances the electron
pressure gradient; 2) a penetration of the neutrals into the
metallic plasma of the arc with neutral density values small
with respect to the filling density values but higher than
the metallic ion densities in the arc. The neutrals are also
strongly heated during the transient expansion stage of the
metallic plasma; 3) as a consequence of the previous point,
the ion kinetic energy is delivered to the neutral gas with a
characteristic spatial scale much larger than that calculated
with the initial neutral gas filling density; and 4) the ion range
are larger than the visible plasma radius [10] or the theoretical
predictions based on the balance of the plasma-neutral pressure
at 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.
REFERENCES
[1] H. Randhawa, “Cathodic arc plasma deposition technology,” Thin Solid
Films, vol. 167, pp. 175–185, Feb. 1988.
[2] R. L. Boxman and S. Goldsmith, “Principles and applications of vacuum
arc coatings,” IEEE Trans. Plasma Sci., vol. 17, no. 7, pp. 705–712,
1989.
[3] J. E. Daalder, “Components of cathode erosion in vacuum arcs,” J. Phys.
D, Appl. Phys., vol. 9, no. 8, pp. 2379–2395, 1976.
[4] J. Kutzner and H. C. Miller, “Integrated ion flux emitted from the
cathode spot region of a diffuse vacuum arc,” J. Phys. D, Appl. Phys.,
vol. 25, pp. 686–693, Mar. 1992.
[5] C. W. Kimbling, “Erosion and ionization in the cathode spot regions of
vacuum arcs,” J. Appl. Phys., vol. 44, no. 7, pp. 3074–3081, 1973.
[6] C. W. Kimbling, “Cathode spot erosion and ionization phenomena in
the transition from vacuum to atmospheric pressure arcs,” J. Appl. Phys.,
vol. 45, no. 12, pp. 5235–5244, 1974.
[7] J. L. Meunier and M. D. de Acevedo, “Carbon cathode spot plasma
flux distributions in low pressures of hydrogen: Some evidence for the
C + H2
CH+ H reaction,” IEEE Trans. Plasma Sci., vol. 20,
no. 6, pp. 1053–1059, 1992.
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KELLY et al.: SIMPLIFIED FLUID MODEL
[8] S. Anders and B. Juttner, “Influence of residual gases on cathode spot
behavior,” IEEE Trans. Plasma Sci., vol. 19, no. 5, pp. 705–712, 1991.
[9] D. Grondona, H. Kelly, and A. Márquez, “Ion-gas interaction in a
vacuum arc operated at intermediate pressures,” in Proc. 7th LatinAmerican Workshop in Plasma Physics., P. Martin and J. Puerta, Eds.
The Netherlands: Kluwer, in press.
[10] J. L. Meunier and M. G. Drouet, “Experimental study of the effect of
gas pressure on arc cathode erosion and redeposition in He, Ar and SF6
from vacuum to atmospheric pressure,” IEEE Trans. Plasma Sci., vol.
15, no. 5, pp. 515–519, 1987.
[11] R. L. Boxman and S. Goldsmith, “Momentum interchange between
cathode-spot plasma jets and background gases and vapors and its
implications on vacuum-arc anode-spot development,” IEEE Trans.
Plasma Sci., vol. 18, no. 2, pp. 231–236, 1990.
[12] J. L. Meunier, “Pressure limits for the vacuum arc deposition technique,”
IEEE Trans. Plasma Sci., vol. 18, no. 6, pp. 904–910, 1990.
[13] R. L. Boxman, S. Goldsmith, S. Shalev, H. Yaloz, and N. Brosh, “Fast
deposition of metallurgical coatings and production of surface alloys
using a pulsed high current vacuum arc,” Thin Solid Films, vol. 139,
pp. 41–52, Jan. 1986.
[14] H. Bruzzone, H. Kelly, A. Márquez, D. Lamas, A. Ansaldi, and C.
Oviedo, “TiN coatings generated with a pulsed plasma arc,” Plasma
Sources Sci. Technol., vol. 5, no. 5, pp. 582–587, 1996.
[15] S. Goldsmith, “The interelectrode plasma,” in Handbook of Vacuum Arc
Science and Technology, R. L. Boxman, P. J. Martin, and D. M. Sanders
Eds. Park Ridge, NJ: Noyes Publications, 1995, pp. 283–284.
[16] J. Kutzner and H. Craig Miller, “Ion flux from the cathode region of
a vacuum arc,” IEEE Trans. Plasma Sci., vol. 17, no. 5, pp. 688–694,
1989.
[17] E. M. Epperlein and M. G. Haines, “Plasma transport coefficients in
a magnetic field by direct numerical solution of the Fokker-Planck
equation,” Phys. Fluids, vol. 29, no. 4, pp. 1029–1041, 1985.
[18] V. E. Golant, A. P. Zilinskij, and S. E. Sacharov, Fundamentals Plasma
Physics (in Italian). Moscow, Russia: MIR, 1983, pp. 65–69.
[19] Y. P. Raizer, Gas Discharge Physics, Berlin, Germany: Springer-Verlag,
1991, pp. 14–18.
[20] Y. S. Touloukian, “Thermophysics,” in Physics Vade Mecuun, H. L.
Anderson, Ed. New York: American Inst. Phys., 1981, pp. 314–325.
[21] W. D. Davis and H. C. Miller, “Analysis of the electrode products
emitted by dc arcs in a vacuum ambient,” J. Appl. Phys., vol. 40, no.
5, pp. 2212–2221, 1969.
[22] S. Goldsmith and R. L. Boxman, “Excited-state densities in a
multicathode-spot Al vacuum arc. II. Theoretical approach,” J. Appl.
Phys., vol. 51, no. 7, pp. 3649–3656, 1980.
1329
Héctor Kelly was born in Mendoza, Argentina, on February 14, 1948. He
received the M.S. degree and Ph.D. degree in physics from Buenos Aires
University, Argentina, in 1972 and 1979, respectively.
Since 1973, he has worked as Researcher at the Plasma Physics Laboratory
of the Science Faculty of Buenos Aires University. His current research
interests are in powerful electrical discharges and plasma coatings based on
electrical discharges.
Dr. Kelly is a member of the National Research Council of Science of
Argentina.
Adriana Márquez was born in Buenos Aires, Argentina, on August 4, 1964. She received the M.S.
degree and Ph.D. degree in physics from Buenos
Aires University, Argentina, in 1989 and 1994,
respectively.
She joined the Plasma Physics Laboratory of
the Science Faculty of Buenos Aires University in
1987. Her current research interests are in powerful
electrical discharges and plasma coatings based on
electrical discharges.
Dr. Márquez is a Fellow of the National Research
Council of Science of Argentina.
Fernando O. Minotti was born in Buenos Aires,
Argentina, on January 27, 1960. He received the
M.S. degree and the Ph.D. degree in physics from
Buenos Aires University, Argentina, in 1984 and
1990, respectively.
Since 1984 he has worked as Researcher at the
Plasma Physics Laboratory of the Science Faculty
of Buenos Aires University. His current research
interests are in Fluid Mechanics and Turbulence.
Dr. Minotti is a member of the National Research
Council of Science of Argentina.