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A model for the basic plasma parameter profiles and the force exerted by fireballs with
non-isothermal electrons
J. Gruenwald, J. Kovačič, B. Fonda, and T. Gyergyek
Citation: Physics of Plasmas 25, 113508 (2018); doi: 10.1063/1.5054369
View online: https://doi.org/10.1063/1.5054369
View Table of Contents: http://aip.scitation.org/toc/php/25/11
Published by the American Institute of Physics
A model for the basic plasma parameter profiles and the force exerted
by fireballs with non-isothermal electrons
J. Gruenwald,
1,a)
J. Kovacˇicˇ,
2,3
B. Fonda,
3
and T. Gyergyek
2,3
1
Gruenwald Laboratories, Taxberg 50, 5660 Taxenbach, Austria
2
Faculty of Electrical Engineering, University of Ljubljana, Trza
ska cesta 25, SI-1000 Ljubljana, Slovenia
3
Jozef Stefan Institute, University of Ljubljana, Jamova 39, SI-1000 Ljubljana, Slovenia
(Received 31 August 2018; accepted 24 October 2018; published online 9 November 2018)
As discovered in recent work, plasma fireballs have the ability to exert considerable force onto ions
and neutrals and, hence, induce macroscopic gas flows. This property makes them interesting
objects for fundamental scientific research. Furthermore, there are also the possibilities for
applications in the field space propulsion. As there is a lack of fundamental understanding of these
plasma phenomena, this article aims to enhance the physical knowledge of fireballs by presenting a
mathematical model for the calculation of the force that can be provided by them. It will be shown
that all the main plasma parameters such as the plasma potential and the electron density can be
derived completely with the knowledge of the potential of the electrode and the radial electron
temperature profile. The calculations show very good agreement with the experimental data if two
species of electrons (i.e., fast and slow) are considered. Both electron populations have different
temperature profiles as is shown with measurements. Furthermore, it will be demonstrated that the
potential drop throughout the fireball is much larger than previously thought and that this larger
potential drop can considerably contribute to the acceleration of ions in the double layer. This
mechanism makes it more likely that the force exerted by the fireball is rather caused by heating of
the neutrals via collisions with those accelerated ions and the high energetic ions themselves than
by collisions between fast electrons and neutrals. Published by AIP Publishing.
https://doi.org/10.1063/1.5054369
I. INTRODUCTION
Fireballs (FBs) are spherical, highly luminous regions in
a comparably thin surrounding background plasma, which
are bounded by a double layer (DL). They were first reported
by Lehmann at the beginning of the last century.
1
It was dis-
covered some years ago by Stenzel et al.
2
that FBs are capa-
ble of producing macroscopic flows in the surrounding gas.
It was argued by the latter authors that these gas flows were
induced by fast electrons that heat the neutrals. This pioneer-
ing work was later supplemented by a theoretical model by
Makrinich and Fruchtman
3
along with the partial comparison
of the theoretical data with measurements. The aforemen-
tioned model is a good starting point for theoretical investi-
gations. However, there is still room for improvement.
Specifically, the assumption of isothermal electrons made in
Ref. 3is a simplification that is not congruent with experi-
mental data, obtained by different authors such as Rubens
and Henderson
4
who measured differences in T
e
of up to
250% between the interior of a FB and the bulk plasma. A
more recent experimental study, which clearly shows that
the assumption of isothermal electrons only holds within the
FB and outside the double layer (but with very different val-
ues), was conducted by Weatherford et al.
5
However, if only
isothermal electrons are considered, the behavior of the elec-
trons in the boundary region of the FB, which is formed by
the DL, is completely neglected. In Sec. III, the model by
Makrinich et al. is generalized to FBs with non-isothermal
electrons, and the results are compared with experimental
data. Additionally, it will be shown that the potential drop
throughout the fireball is much larger than it would be the
case with only one electron species and that this larger poten-
tial drop can considerably contribute to the acceleration of
ions in the double layer. The calculations presented in this
work are based on a simplified analytic model, which neglects
kinetic effects in the plasma but still yields results that are in
good agreement with the available experimental data.
II. EXPERIMENTAL SETUP
The experimental results, which were used to support
the following mathematical model, were obtained in a linear,
magnetized plasma machine that produces the plasma with a
hot wire cathode and has an axial magnetic field for
enhanced confinement. This device is described in more
detail elsewhere.
6
However, a schematic overview of the
machine is shown in Fig. 1.
The magnetic field was held as small as possible
(11 mT) to minimize anisotropy effects, and the radial
plasma parameter profiles were measured with Langmuir
and emissive probes. Both probe heads were made of thori-
ated tungsten with a length of 5 mm and a diameter of 75
lm. The probes were orientated perpendicular to the mag-
netic field lines, and the data evaluation was conducted
according to standard probe theory. The magnitude of the
magnetic field is also considered low enough not to influence
the probe measurements.
7
The experiments were conducted
a)
E-mail: jgruenwald@gmx.at
1070-664X/2018/25(11)/113508/7/$30.00 Published by AIP Publishing.25, 113508-1
PHYSICS OF PLASMAS 25, 113508 (2018)
in argon 5.0 at a pressure of 10
2
mbar. The discharge cur-
rent was 2.4 A at a discharge voltage of 50 V.
As depicted in Fig. 2within the FB, a second population
of electrons with considerably higher temperature evolves
inside the double layer surrounding the FB. The applied
potential on the anode was 30.9 V with respect to ground at a
maximum input current of 0.3 A.
This gives already a hint that the potential drop in the
double layer accelerates the ions produced in the FB, which
then heat the neutrals due to inelastic collisions rather than
the electrons. The magnitude of the temperature drop also
fits to the measurements of Stenzel et al. who observed bal-
listic ions leaving a pulsed FB with kinetic energies of 8.6 to
12.9 eV under very similar experimental conditions.
8
It has
to be noted that there are indeed isothermal electrons present
but at very low temperature which cannot account either for
the generation of fast ions that are observed in the FB or for
the neutral gas heating.
III. COMPARISON BETWEEN THE MODEL AND THE
EXPERIMENT
As can be seen in Fig. 2, the electron temperature profile
can be approximated with a logistic function that has the
general form
TeðrÞ¼AþLA
1þexpðkðrr0ÞÞ ;(1)
where A, L, k, and r
0
are usually fit parameters. However, in
this context, they refer to the following physical quantities: A
is the electron temperature in the bulk (far from the FB), L is
the maximum value of T
e
in the center (near the electrode), k
is the steepness, and r
0
is the midpoint of the curve. For the
sake of simplicity and without loss of generality, A is set to be
0 and L is from here on defined as the maximum electron tem-
perature with respect to T
e
in the bulk (L ¼9.1 eV). As shown
later, the hot electron population is less dense. This leads to a
larger scattering of the data points for the hot electrons
although 10 measurements were averaged for each acquisi-
tion. However, the overall accuracy of the Langmuir probe
measurements is about 10%. Both the plasma potential /
pl
and the electron temperature can be connected under the
assumption of quasi neutrality (ne;cþne;h¼neni¼n)and
a Maxwellian velocity distribution via
9,10
/pl ¼Te
21þln 2Mi
pme
/FB;(2)
where /
FB
is the potential on the FB anode inside the FB,
while its value is zero outside the FB where the electrode
potential is already shielded from the background plasma.
The indices “h” and “c” denote the contributions from the
hot and cold electrons, respectively. Due to the superposition
principle, this can be done separately for the hot and cold
electrons with their individual temperature profiles.
Two typical electron energy distribution functions
(EEDF) from the hot and cold population are shown in Fig. 3.
The distribution functions have been obtained by the
second derivative of the Langmuir probe traces. The probe
traces have been smoothed with a Savitzky–Golay filter with
the optimal parameters to minimize the error lined out in
Ref. 11. It can be seen that the EEDF for both electron spe-
cies are Maxwellian. However, there is some slight deviation
in the distribution function for the cold electrons, which can
be explained by the larger mean free paths at smaller elec-
tron energies.
FIG. 1. Sketch of the Ljubljana magnetized plasma machine with the addi-
tional anode and the FB.
FIG. 2. Double layer (pink) formed between 60 and 75mm and measured
radial temperature profiles of the cold (blue) and hot (red) electrons with a
logistic fit (black) as obtained with the listed fit parameters.
FIG. 3. Typical EEDF taken at positions 40 and 100 mm. Savitzky–Golay
parameters: polynomial order: 6 and data points: 191.
113508-2 Gruenwald et al. Phys. Plasmas 25, 113508 (2018)
The results for /
pl
are depicted in the graph (Fig. 4),
which shows a comparison between the plasma potential
profiles calculated according to Eq. (2) and the equation sug-
gested by Makrinich and Fruchtman.
3
The black squares in
Fig. 4denote the actually measured data. The plot contains
the data for the hot electrons (red line), the cold electrons
(blue line), and the sum of both contributions with the poten-
tial of the FB electrode (30.9 V) subtracted as described by
Eq. (2) (black line). It is clearly visible that both species of
electrons, i.e., the hot and the cold population, have to be
taken into account in order to obtain the correct plasma
potential profile. This is despite the fact that the majority of
electrons in the FB plasma are part of the cold, isothermal
population.
The best agreement between the experimental data and
the simulated curve can only be obtained if the cold and hot
electrons and the potential on the anode are taken into
account properly. Nevertheless, it has to be noted that the
temperature for the hot electrons was set to be zero outside
the FB as the only experimentally observed population in
this region is the one consisting of cold electrons. Hence, for
radial positions larger than 66 mm, the hot electrons are
omitted. However, for obtaining the blue line, isothermal
cold electrons were assumed, while for the red line, the two
different temperatures of the electrons within and outside the
FB have been regarded. Using Eq. (2) yields a potential drop
of around 30 V in the DL, which is around 6 times higher
than the potential drop calculated with the conventional for-
mula (4.8 V).
3
This again strengthens the claim that ions can
indeed gain enough kinetic energy in the potential drop,
which is then available for neutral gas heating via inelastic
collisions. The drop in the plasma potential of around 30 V
indicates the possibility of the creation of Ar
2þ
ions as the
second ionization potential of argon is 28.6 eV, but these
species are neglected here for simplicity. It has also to be
noted that the plasma potential inside the FB is slightly
higher than the potential on the electrode, which is explained
by the rapid loss of electrons on the anode surface along
with the efficient production of positively charged ions. The
derivative of Eq. (2) allows us to calculate the E-field for the
hot and cold electrons
Eh;c¼r/pl;h;c¼@
@r/pl;h;c:(3)
The contribution from each electron species to the E-
field was calculated from Eq. (3), and the results are depicted
in Fig. 5.
It can be seen that the cold electrons only have a mar-
ginal influence on the total electric field between the FB
anode and the background plasma since their contribution
only shows some minor fluctuations around the zero line.
The hot electrons, on the other hand, display E-field fluctua-
tions of several thousand V/m with a strong maximum value
of around 10 kV/m in the DL around the FB. However, it has
to be stressed that the large distortions of the E-field inside
the FB are not physical; they are due to the limitations in the
probe evaluation techniques in combination with the numeri-
cal derivation of the data curves. Since the electron tempera-
ture was determined from the semi-log plot of the probe
traces, even small errors may appear as large fluctuations.
Nevertheless, it can be concluded that the cold electrons
alone cannot be responsible for the large E-field variation
within the double layer as their contribution to the electrical
field is about three orders of magnitude too small compared
to the calculation from the measured plasma potential.
With the knowledge of the plasma potential and the
electron temperature profiles of both species, the electron
density profiles can be calculated in the following manner:
First, the general momentum equation for the electrons is
needed
eneE¼@ðneTeÞ
@r¼@ne
@rTeþ@Te
@rne:(4)
It is evident that the assumption of non-isothermal elec-
trons introduces an additional term into the momentum equa-
tion. Since the electric fields generated by the fast and slow
FIG. 4. Simulated radial plasma potential profile contributions of the hot
(red) and cold (blue) electrons and their sum minus the FB electrode poten-
tial (black line), calculated with Eq. (2). They are compared with the mea-
sured radial profiles (black squares) and with the theoretical predictions
from Ref. 3(green crosses).
FIG. 5. Simulated contributions to the electric field from the cold (blue) and
hot (red) electrons in comparison to the electric field calculated from the
experimental values (black).
113508-3 Gruenwald et al. Phys. Plasmas 25, 113508 (2018)
electrons can be superimposed, Eq. (4) can be treated sepa-
rately for both electron species
ene;cðrÞEcðrÞ¼@ne;c
@rTe;c(5)
and
ene;hEh¼@ne;h
@rTe;hþ@Te;h
@rne;h:(6)
It has to be noted that T
e,c
in Eq. (5) is constant in accor-
dance with the measured data, depicted in Fig. 2. Thus, Eq.
(5) yields the solution for the density of slow electrons
ne;c¼nc;0exp e
Te;cðr
1
EðfÞdfÞ
¼nc;0
exp e
Te;c
/pl;cð1Þ
exp e
Te;c
/pl;cðrÞ
;(7)
where n
c,0
denotes the density of cold electrons at the surface
of the FB electrode and /
pl,c
is the plasma potential contribu-
tion from the cold electrons. The hot electron density is
determined via Eq. (6). Using the general derivation rules for
the logistic function, the spatial derivative of the electron
temperature is then given by
@Te;h
@r¼Te;hð1Te;hÞ(8)
and
@2Te;h
@2r¼Te;hð1Te;hÞð12Te;hÞ:(9)
For the following mathematical treatment, also the
momentum equation for the ions
miCi¼nieE (10)
along with the continuity equation
1
r2
@
@rðr2CiÞ¼S(11)
is applied.
3
Here, C
i
denotes the ion particle flux density and
S is the source term that takes into account particle generation
via impact ionization within the FB. As the cold electrons
have too little energy to ionise, these calculations regard only
the hot electrons. For moderate gas and electron densities, the
source term is proportional to the neutral gas density N and
the electron impact ionization rate b(1016 m3=sforAr
12
)
S¼bNne;h:(12)
Additionally, a constant collision frequency of the neutrals
in the gas is assumed
¼rNvth;(13)
with the ion-atom cross section r(4 10
17
m
2
for Ar atoms
according to Phelps et al.
13
) and the thermal velocity of the
gas particles v
th
. Under the assumption of quasi neutrality,
(4) and (10) are combined to get
@ðne;hTehÞ
@r¼miCi;(14)
which yields in connection with Eqs. (11)–(13) the following
spherical diffusion equation:
mirbN2vthne;h¼Kne;h¼1
r2
@
@rr2@ðne;hTe;hÞ
@r
:(15)
Here, K was used as an abbreviation for all the constant pre-
factors in front of the hot electron density. It has to be noted
that strictly speaking, the criterion of quasineutrality is vio-
lated in the region of the double layer (DL) surrounding the
FB plasma, but the thickness of the DL is very small com-
pared to the spatial dimensions of the FB and the surround-
ing plasma that it can be neglected without introducing too
large errors in the calculations. Introducing the abbreviations
T’ and n’ for the spatial derivatives and solving the r.h.s. of
Eq. (15), the following differential equation is obtained for
the plasma density n:
n00Tr þn0ð2Tþ2T0rÞþnð2T0þrT00 KÞ¼0:(16)
Calculating K with parameters that are typical for FB dis-
charges (i.e., covering also the parameter range of the experi-
ments presented herein), namely, mi¼6:61026 kg for Ar
and N ¼2:41020m3for an ideal gas at room temperature
and a pressure of 10
2
mbar yields 61015m3kg=s2.The
thermal energy of the argon neutrals, which is needed for the
calculation of K, was obtained via
vth ¼ffiffiffiffiffiffiffiffi
8kT
pmi
r¼400 m=s:(17)
Hence, this term is neglected in the solution of Eq. (16).
Equation (16) indicates that
ðnTrÞ00 ¼0;(18)
which has the general solution for T;r6¼ 0
nðrÞ¼Br þC
Tr ¼B
TþC
Tr :(19)
Since the current continuity j¼enevehas to be ful-
filled for the whole plasma, the following condition is
satisfied:
nð0ÞTð0Þ¼nð1ÞTð1Þ:(20)
This can only hold if the solution for n(r) has no linear
dependence on r. Hence
B
Tr ¼
!0$B¼0:(21)
The particular solution is obtained by the definition of
suitable boundary conditions, i.e.,
113508-4 Gruenwald et al. Phys. Plasmas 25, 113508 (2018)
nð0Þ¼nh;0¼B
Te;hðr¼0Þ¼2
LB:(22)
Thus
nðrÞ¼nh;0L
21
Te;hðrÞ:(23)
The results of Eqs. (7) and (23) are shown in Fig. 6in
direct comparison with the measured density profiles, where
the red, blue, and black symbols depict the measured density
profiles for the different species. The dashed lines represent
the calculated values. The shapes of the profiles are in good
agreement except for the range in close vicinity to the FB
electrode where some more complicated physical processes
are possibly going on. However, most of the discrepancies
can be explained either by the accuracy of the probe data
evaluation or by the simplifications that were made during
the calculations [e.g., omitting the constant K in Eq. (15)].It
also has to be noted that there seems to be a divergence in
the cold electron density profile for larger radii. This is
purely due to the structure of Eq. (7) where even small inac-
curacies contribute exponentially to the density profile calcu-
lations. The same holds for the seemingly diverging n
e,c
profiles close to the electrode surface, where the distortions
of the plasma potential profile are relatively large. In com-
parison, the use of the equation presented by others
3
(green
dashed line) yields a profile that is only marginally depen-
dent on the radial position with a kind of average value for
the electron density. In fact, the results for n
e
calculated via
Eq. (11) from Ref. 3only vary in the 9th digit behind the
comma as depicted on the right hand side of Fig. 6.
With these results, the ion flux through the surface of
the FB can be written as
CR¼4pr2Ci¼ 4pr2
mirNvT
n0TþnT0
½
;(24)
while the total outward force on the ions is obtained by inte-
grating the momentum equation
Ftot ¼4pðr
0
n0TþnT0
½
~
r2d~
r:(25)
Figure 7displays the simulated total force exerted of a
FB with radius r along with the ion flux outwards the FB.
It can readily be seen that the ion flux reaches its maxi-
mum at the edge of the FB. Furthermore, the total force
exerted by the ions displays a very strong increase inside and
shortly outside the double layer, which surrounds the FB.
This is also a strong indication that at least some of the ion
thrust is due to electrostatic acceleration in the sheath. The
total force of the FB reaches a value of around 8 mN, which
seems astonishingly high; however, it has to be emphasized
that this is the value acting outwards the FB in all directions.
The area of a FB with a radius of 70 mm is 615.8cm
2
, which
yields a force per unit area of roughly 1.3 10
5
N/cm
2
.
This corresponds to a force of 48.8 lN on a 3.76 cm
2
pendu-
lum as it was used by Makrinich and Fruchtman.
3
under very
similar experimental conditions. This number is in excellent
agreement with the value obtained by those authors who
measured the force to be 46 65lN.
Consequently, the force exerted on a single ion at the
edge of the FB (where r ¼R) is given by
Ftot
CR
¼mirNvT
R2ðR
0
n0TþnT0
½
~
r2d~
r
n0TþnT0
½
:(26)
The value of the force per ion in the center of the DL,
which was numerically calculated from Eq. (26),is
9.1 10
18
N. However, the maximum force on a single ion
is 6 10
16
N/ion and is found to be about 1 cm outside the
DL, as shown in Fig. 8: One can see from the semi-log plot
of the force on a single ion that there is a strong increase in
force inside the DL from 2 10
18
to 3.7 10
17
. This indi-
cates that the main acceleration of the ions is indeed happen-
ing in the DL that surrounds the FB. It has to be noted at this
FIG. 6. Left: Simulated and measured
radial density profiles of the hot elec-
trons, calculated with Eq. (23) (red),
the cold electrons, calculated with Eq.
(7), and their sum in comparison to the
formula derived after equation (11) in
Ref. 3(green crosses). Right: A magni-
fication of the density profile simulated
with Markinich’s equation.
FIG. 7. Simulated total force exerted by the FB (black) and the simulated
ion flux in dependence of the radius (red).
113508-5 Gruenwald et al. Phys. Plasmas 25, 113508 (2018)
point that besides this, there is also another argument that
makes the neutral gas heating due to electron-neutral colli-
sions, which was suggested by former authors, very unlikely.
To elaborate this, the mean free path of electron-neutral col-
lisions has to be taken into account
14
kmfp ¼ve;th
Nhrvi;(27)
where v
e,th
is the average thermal velocity of the electrons
and hrviis the reaction rate coefficient for elastic neutral-
electron scattering. The former entity can be calculated anal-
ogous to Eq. (17) with the electron mass me¼9:11031
kg. This yields in accordance with the data from Ref. 15 an
electron mean free path (for T
e
¼8 eV) in Ar of around
10 cm for the experiments described herein. As this is on the
order of (or even larger than) the diameter of the fireball, it
can readily be concluded that the probability of electron-
neutral collisions within this plasma structure is very small.
Furthermore, the mean free path increases with decreasing
electron temperature, which makes also the energy transfer
between the cold electrons and neutrals even more unlikely.
The order of magnitude of the mean free path for electrons
with kinetic energies of around 1.5 eV lies on the order of
50 cm and more in our experiments.
15
Hence, the assumption
of isothermal electrons, which holds in our case only for the
low temperature electrons, is in direct contradiction with the
explanation of neutral gas heating via electron-neutral colli-
sions. The mean free path, on the other hand, of ion-neutral
collisions is much smaller, and thus, neutral gas heating just
outside the FB via inelastic ion-neutral collisions is far more
likely.
IV. CONCLUSION AND OUTLOOK
The theory of force exertion by plasma fireballs has
been generalized to FBs with non-thermal electrons. This
was done due to the discrepancy between the assumptions of
existing analytical models and the available experimental
data. The model proposed in this paper describes the shape
of the electron density and the radial plasma potential profile
very accurately. It has been shown that the potential drop,
which is predicted for the double layer around such a FB, is
considerably larger than expected before. Due to this finding,
it becomes possible to argue that the primary heating pro-
cesses of neutrals by the FB are rather induced by collisions
between accelerated ions that gain sufficient energy in the
potential drop of the FB anode. This claim is also corrobo-
rated by the fact that the large collision mean free path
between electrons and neutrals is too large to play a signifi-
cant role in the gas heating. The total force and the ion flux
through the FB surface were calculated based on the model
herein, and it was found that the force exerted by FB with
non-isothermal electrons is considerable. These results offer
an interesting possibility for the technical applications of
fireballs as it suggests their potential use for space propulsion
as this force was achieved with a total input power of 130 W
including the power for producing the background plasma in
the linear machine. Furthermore, most modern thruster sys-
tems working with gases with high atomic mass like krypton
or xenon are very cost intensive and require input powers on
the order of several kW.
16,17
It has been shown in this work
that plasma FBs are capable of producing substantial thrust
at very little input power and in low mass gases such as Ar.
The concept of using double layers for space propulsion is
not a new one. The so-called Hall double layer thrusters
(HDLTs) were described, for example, in the work of
Charles.
18
FB assisted thrusters offer, in principle, the same
advantages as HDLTs like the lack of movable parts which
leads to a longer lifetime or the possibility to operate the
device in the steady state and in the pulsed mode. Moreover,
FBs produce additional ions very efficiently within the rather
large potential drop of the surrounding sheath. Those ions
enhance the overall thrust, but it has to be mentioned that the
ion flow outwards the FB is spatially isotropic due to the
spherical geometry. Hence, future work should be dedicated
to improve the ion transport in a preferable direction in order
to enhance the achievable thrust even further. This could be
done by introducing suitable magnetic fields or even asym-
metric FB configurations. That the latter is feasible at least
for inverted FBs was shown in a previous paper.
19
However,
the available experimental data and theoretical modeling are
somehow scarce. Thus, an enhanced fundamental under-
standing of these phenomena is needed to lead the way to a
new generation of ion thrusters for space propulsion.
However, the model presented in this paper is just a first
attempt to generalize existing models of fireball dynamics,
and it is not fully complete. Further improvements are
expected by also taking kinetic effects into account, but this
was out of the scope of this work and is left to future
research.
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