Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam
Evaporation of micro-droplets in an arc plasma flow under the action of a self-consistency electron beam and the condition of direct heating of micro-droplets by fast electrons are considered. It is shown that the plasma is heated under the influence of the beam, even taking into account the fact th...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1958932023-12-08T12:34:21Z Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam Goncharov, A.A. Maslov, V.I. Litovko, I.V. Ryabtsev, A. Low temperature plasma and plasma technologies Evaporation of micro-droplets in an arc plasma flow under the action of a self-consistency electron beam and the condition of direct heating of micro-droplets by fast electrons are considered. It is shown that the plasma is heated under the influence of the beam, even taking into account the fact that the electrons and ions of the plasma lose energy for the evaporation of micro-droplets. It is demonstrated that small micro-droplets evaporate more intensively. It is shown that the plasma electron density should be optimal. For the destruction of macro-particles in a plasma with a higher concentration, more powerful beams are required than in a case of plasma with a lower concentration. Розглядаються випаровування мікрокрапель у потоці дугової плазми під дією самоузгодженого електронного пучка та умова прямого нагріву мікрокрапель швидкими електронами. Показано, що плазма гріється під дією пучка, навіть з урахуванням того, що електрони і іони плазми втрачають енергію на випаровування мікрокрапель. Встановлено, що дрібні мікрокраплі випаровуються інтенсивніше. Показано, що має бути оптимальна щільність електронів плазми. Для руйнування макрочастинок у плазмі з більшою концентрацією потрібні потужніші пучки, ніж у плазмі з меншою концентрацією. 2022 Article Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam / A.A. Goncharov, V.I. Maslov, I.V. Litovko, A. Ryabtsev // Problems of Atomic Science and Technology. — 2022. — № 6. — С. 89-94. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 29.17.+w; 41.75.Lx DOI: https://doi.org/10.46813/2022-142-089 http://dspace.nbuv.gov.ua/handle/123456789/195893 en Problems of Atomic Science and Technology Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Low temperature plasma and plasma technologies Low temperature plasma and plasma technologies |
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Low temperature plasma and plasma technologies Low temperature plasma and plasma technologies Goncharov, A.A. Maslov, V.I. Litovko, I.V. Ryabtsev, A. Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam Problems of Atomic Science and Technology |
| description |
Evaporation of micro-droplets in an arc plasma flow under the action of a self-consistency electron beam and the condition of direct heating of micro-droplets by fast electrons are considered. It is shown that the plasma is heated under the influence of the beam, even taking into account the fact that the electrons and ions of the plasma lose energy for the evaporation of micro-droplets. It is demonstrated that small micro-droplets evaporate more intensively. It is shown that the plasma electron density should be optimal. For the destruction of macro-particles in a plasma with a higher concentration, more powerful beams are required than in a case of plasma with a lower concentration. |
| format |
Article |
| author |
Goncharov, A.A. Maslov, V.I. Litovko, I.V. Ryabtsev, A. |
| author_facet |
Goncharov, A.A. Maslov, V.I. Litovko, I.V. Ryabtsev, A. |
| author_sort |
Goncharov, A.A. |
| title |
Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam |
| title_short |
Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam |
| title_full |
Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam |
| title_fullStr |
Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam |
| title_full_unstemmed |
Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam |
| title_sort |
evaporation of micro-droplets in cathode arc plasma coating under formed electron beam |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2022 |
| topic_facet |
Low temperature plasma and plasma technologies |
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http://dspace.nbuv.gov.ua/handle/123456789/195893 |
| citation_txt |
Evaporation of micro-droplets in cathode arc plasma coating under formed electron beam / A.A. Goncharov, V.I. Maslov, I.V. Litovko, A. Ryabtsev // Problems of Atomic Science and Technology. — 2022. — № 6. — С. 89-94. — Бібліогр.: 8 назв. — англ. |
| series |
Problems of Atomic Science and Technology |
| work_keys_str_mv |
AT goncharovaa evaporationofmicrodropletsincathodearcplasmacoatingunderformedelectronbeam AT maslovvi evaporationofmicrodropletsincathodearcplasmacoatingunderformedelectronbeam AT litovkoiv evaporationofmicrodropletsincathodearcplasmacoatingunderformedelectronbeam AT ryabtseva evaporationofmicrodropletsincathodearcplasmacoatingunderformedelectronbeam |
| first_indexed |
2025-07-17T00:09:44Z |
| last_indexed |
2025-07-17T00:09:44Z |
| _version_ |
1837850673379540992 |
| fulltext |
ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142).
Series: Plasma Physics (28), p. 89-94. 89
https://doi.org/10.46813/2022-142-089
EVAPORATION OF MICRO-DROPLETS IN CATHODE ARC PLASMA
COATING UNDER FORMED ELECTRON BEAM
A.A. Goncharov1, V.I. Maslov2, I.V. Litovko3, A.V. Ryabtsev1
1Institute of Physics NASU, Kyiv, Ukraine;
2National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine;
3Institute for Nuclear Research NASU, Kyiv, Ukraine
E-mail: gonchar@iop.kiev.ua;
ryabtsev@iop.kiev.ua
Evaporation of micro-droplets in an arc plasma flow under the action of a self-consistency electron beam and the
condition of direct heating of micro-droplets by fast electrons are considered. It is shown that the plasma is heated
under the influence of the beam, even taking into account the fact that the electrons and ions of the plasma lose energy
for the evaporation of micro-droplets. It is demonstrated that small micro-droplets evaporate more intensively. It is
shown that the plasma electron density should be optimal. For the destruction of macro-particles in a plasma with a
higher concentration, more powerful beams are required than in a case of plasma with a lower concentration.
PACS: 29.17.+w; 41.75.Lx
INTRODUCTION
In many cases, modern industrial technological
processes require the transition of production means to
work with nanometer scales. Such trends can be seen not
only in the production of integrated circuits and devices
on a chip but also in cases of application of modifying
surface layers for various purposes. Fabrication quality
and precision should also be complemented by maximum
productivity.
It is known that ion plasma sources of the MEVVA
type based on a vacuum-arc discharge are reliable and
well-tested generators of ion plasma flow for creating
high-current (several amperes) beams of heavy metal
ions with particles of medium energy (1...100 keV) [1].
They are widely used in science and industry for
deposition of various protective and functional coatings,
and also to modify the surface properties of structural and
decorative materials. However, the presence of micro-
droplets of cathode material (size from ≈ 0.01 μm to tens
of μm) in the ion-vapor flow of erosion plasma sources
restricted their use for creating high-quality coatings,
especially with uniformity at the nano level. Existing
filters and methods for eliminating the micro-droplet
phase from the working flow are based on various
methods of separation (removal) of the droplet phase
from the ion-vapor flow [2]. Various mechanical and
electrophysical filters are usually used to reduce the
concentration of droplets. These filters effectively
remove micro-droplets larger than 1 μm without
significant loss of ion-vapor flow particles. At the same
time, the use of existing filters to remove micro-droplets
with smaller sizes leads to a significant (several times)
decrease in the density of the metal plasma on the
processed products. That is, modern filtration methods
limit it the effective use of the high rate of ion-vapor flow
generation inherent to an erosive plasma source. The
Institute of Physics of the NAS of Ukraine has many
years of experience in proposing and implementing ideas
for the creation of axially symmetrical cylindrical
plasma-dynamic systems based on the fundamental
principles of medium-energy plasma optics. Such
systems are a well-developed means of focusing and
manipulating ion beams in cases where the problem of
compensation of the space charge of the beam is
important [3]. The use of these systems in MEVVA-type
sources creates new opportunities for predetermined
manipulation of a low-energy ion-plasma beam
propagating in the direction of the substrate (in the case
of sputtering) or to the emission grid (in the case of a
plasma source).
In previous works [4-7], a new approach to the
elimination of micro-droplets from a dense flow of metal
plasma using plasma-dynamic systems such as an
electrostatic plasma lens (PL) and systems based on a
discharge with a hollow cathode (HC) was proposed and
investigated. These systems generate a beam of energetic
electrons, which is formed in a self-consistent manner
due to secondary ion-electron emission from the inner
surface of the central electrode of the lens in the wall
layer (in the case of PL) or from the surface of the
cathode itself (in the case of HC). Preliminary
evaluations and experiments have shown that this
electron beam can provide effective evaporation and
elimination of micro-droplets. However, for the most
effective application of the proposed systems, an actual
task is to determine the fundamental physical
mechanisms of influence on the state of microinclusions
in a dense dusty plasma during its passing through
plasma-dynamic systems with fast electrons.
The article is devoted to the determination of the
fundamental mechanisms of the action of fast electrons
on the flow of a dense metallic dusty plasma.
1. EQUATIONS OF THE EVAPORATION OF
MICRO-DROPLETS IN A PLASMA FLOW
Micro-droplets worsen the properties of the films that
are formed. But the simple mechanical separation of
mailto:gonchar@iop.kiev.ua
90 ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142)
micro-droplets from the flow dramatically reduces the
rate of film growth. Therefore, it is necessary to
evaporate the micro-drops. Micro-droplets fly in an
expanding cone [6] (Fig. 1). The flow of micro-droplets
passes through the diaphragm, on which part of the
micro-droplets remains. After the diaphragm, the flow
spreads into the plasma-optical system. Therefore, for the
evaporation of micro-droplets, it is enough to influence
the near-surface hollow cylindrical flow layer of a certain
thickness. We consider a plasma-optical system [4-8], in
which, for the evaporation of micro-droplets, additional
energy is pumped by a self-consistent electron beam. The
electron beam is formed as a result of secondary ion-
electron emission during the bombardment of the inner
surface of the cylinder by peripheral plasma ions [4-8].
This group of high-energy electrons (electron beam) is
accelerated by the applied potential difference
approximately along the radius in the direction of the axis
of the system.
It is possible to formulate a system of two equations
that describe the evaporation of micro-droplets in the
plasma flow under the action of a self-consistent electron
beam. So, we obtain:
𝑑𝑇𝑑𝑟
𝑑𝑡
=
3
𝑟𝑑𝑟𝜌𝑑𝑟𝑐
{𝑘𝑛𝑛𝑉𝑛𝑡ℎ(𝑇𝑛 − 𝑇𝑑𝑟) − 𝛼𝜎𝑇𝑑𝑟
4 +
𝑛0𝑘𝑇𝑒√
2𝑘𝑇𝑒
𝜋𝑚𝑒
exp (
−𝑒𝜑𝑑𝑟
𝑘𝑇𝑒
) + 0.15𝑛0√
𝑘𝑇𝑒
𝑚𝑖
[4√2(1 +
𝑟𝑑
2 𝑟𝑑𝑟
2⁄ ) + 𝛾𝑒𝜑0]} , (1)
𝑑𝑇𝑒
𝑑𝑡
= 0.6 𝛾 𝑒 𝜑0√
𝑘 𝑇𝑒
𝑚𝑖
(
2
𝑅
− 𝑛𝑑𝑟 𝜋 𝑟𝑑𝑟
2). (2)
The first equation is for the micro-droplet temperature
Tdr. The second equation is for the plasma electron
temperature Te.
Here 0 – wall jump of electric potential; dr – electric
potential of a micro-drop which is equals
(Te/e) ln[0.6(2me/mi)1/2]); i = e dr is the energy of ions
bombarding a micro-drople; c – heat capacity of the
micro-droplet substance; mdr, rdr ‒ mass and radius of
micro-droplet; rd ‒ Debye radius; k ‒ Boltzmann's
constant; Tdr ‒ temperature of micro-droplet; Tn ‒
temperature of neutral particles; nn, Vnth ‒ density and
thermal velocity of neutral particles; jb = nbVb; nb, Vb ‒
the beam density and its speed; dr ‒ the density of the
substance of the micro-droplet; γ ‒ the secondary
emission coefficient; α ‒ the emissivity of the micro-
droplet; σ ‒ the Stefan-Boltzmann constant.
In (1), the coefficient (1+rd
2/rdr
2) approximately in the
vicinity of rd ≥ rdr demonstrates the fact that the surface
from which the ions are accelerated to the micro-drop can
be largerthan the surface of the micro-drop. Indeed, the
ions are accelerated from the surface where the field of
the micro-droplet penetrates. Then, if the Debye radius rd
of plasma electrons is less than the size of the micro-drop
rd < rdr, then the surface that collects ions is
approximately equal to the surface of the micro-drop. If
the Debye radius of plasma electrons is larger than the
size of a micro-droplet, then the surface that collects ions
is approximately equal to the Debye surface rd
2. For
typical experimental parameters (Te = 3 eV,
ne = 1012 cm-3), the Debye radius of electrons is equal to
rd
= (Te/4nee2)½ ≈ 13 m.
It can be seen that only for the largest micro-droplets
rd → rdr may be achieved. That is, the class of large
micro-droplets rd < rdr includes only a small part of the
largest micro-droplets. Thus, practically all micro-
droplets belong to the class of small micro-droplets with
rd >> rdr; as will be shown, they evaporate efficiently.
Fig. 1. Scheme of a system with a self-consisted electron
beam for evaporation of droplets in an arc plasma flow
in vacuum-arc coating deposition technology. Solid
arrows show the directions of movement of micro-
droplets. Dashed arrows show the direction of self-
consistent electron beam injection
It is clear that the contribution to the heating of
macroparticles from multi-charged ions is greater than
from singly-charged ones. However, in this article, we
consider the approximation of singly charged ions only.
It should be noted that the coefficient (1+(rd/rdr)2) is
obtained in the approximation of the absence of
competition of micro-droplets. That is, it is obtained in
approximation ndr
-1/3
> rd.
It can be seen from equation (1) that the influence of
the mass of the droplet Mdr remains in the denominator
as the product of the density of the substance of the
micro-droplet dr and its radius rdr.
2. CONDITION OF MORE INTENSIVE
EVAPORATION OF SMALL MICRO-
DROPLETS
From equation (1), it can also be seen those small
micro-droplets whose radius rdr is smallerthan the Debye
radius of plasma electrons rdr << rd are most likely
evaporate. The rate of energy pumping into a small
micro-droplet (that is, the rate dTdr/dt of its temperature
Tdrincrease) is proportional to
𝑑𝑇𝑑𝑟
(𝑠)
𝑑𝑥
~
1
𝑟𝑑𝑟
3 . (3)
While for large micro-droplets, the size of which is not
less than the Debye radius of plasma electrons, we have
𝑑𝑇𝑑𝑟
(𝑙)
𝑑𝑥
~
1
𝑟𝑑𝑟
. (4)
From the comparison (3) and (4), we can conclude
that energy pumping is more intense for small micro-
droplets.
ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142) 91
3. CONDITION FOR DIRECT HEATING OF
LARGE DROPLETS BY HIGH-ENERGY
ELECTRONS
Under certain conditions, direct heating of micro-
droplets by high-energy electrons exceeds indirect
heating of micro-droplets, when high-energy electrons
first heat plasma electrons. The plasma electrons then
heat the micro-droplets directly, as well as by
accelerating the plasma ions on the micro-droplets. This
condition has the form 𝛾𝑒𝜑0 > 4√2 𝜀𝑖(1 + 𝑟𝑑
2 𝑟𝑑𝑟
2⁄ ).
Based on the fact that this condition is more easily
fulfilled for the largest micro-droplets, the radius of
which is not less than the Debye radius of electrons. From
this condition for the largest micro-droplets, we have
𝛾𝑒𝜑0 > 4√2 𝜀𝑖. If i ≈ 40 eV and γ ≈ 0.1, then
at 𝑒𝜑0 > 2260 eV direct heating of micro-droplets by
high-energy electrons exceeds heating of micro-droplets
by electrons and plasma ions.
4. CONDITION FOR SIMULTANEOUS
EVAPORATION OF MICRO-DROPLETS
AND PLASMA HEATING
Now we will show that even taking into account the
fact that plasma electrons and ions lose energy due to the
evaporation of micro-droplets, the plasma heats up
instead of cooling down. To do this, let's compare the
flow of energy pumped into the plasma by the beam and
the flow to all micro-droplets of plasma ion energy Ii in
the approximation that the volume of all micro-droplets
is a small part of the plasma volume,
𝐼𝜀𝑏 = 2𝜋𝑅𝐿𝛾𝑛𝑖𝑉𝑠𝑒𝜑0,
𝐼𝜀𝑖 = 𝜋(𝑟𝑑𝑟
2 + 𝑟𝑑
2)𝑛𝑖𝑉𝑠𝜋𝑅2𝐿𝑛𝑑𝑟𝜀𝑖,
𝐼𝜀𝑏 𝐼𝜀𝑖 = 2𝛾𝑒𝜑0(𝑟𝑑𝑟
2 + 𝑟𝑑
2)𝜋𝑅𝑛𝑑𝑟𝜀𝑖 .⁄ (5)
For i ≈ 40 eV, b ≈ 1500 eV, ≈ 0.1 and large micro-
droplets from (5) we obtain
𝐼𝜀𝑏
𝐼𝜀𝑖
=
2.4
𝑟𝑑𝑟
2 𝑅 𝑛𝑑𝑟 > 1. (6)
Condition (6) is fulfilled if the intersection of all micro-
droplets is less than the surface area of the beam
injection.
Thus, the plasma is heated under the influence of the
beam, and not cooled, even taking into account the fact
that the electrons and ions of the plasma lose energy for
the evaporation of micro-droplets.
5. HEATING SPEEDS OF PLASMA
ELECTRONS AND MICRO-DROPLETS
In order to clearly see the ratio of heating rates of
plasma electrons and micro-droplets, consider small
times after the start of the electron beam exposure, when
the change in the radii of the micro-droplets can be
neglected. Consider equations (1), and (2) in the simplest
case of neglecting the contribution to the evaporation of
micro-droplets by gas and the direct evaporation of
micro-droplets by high-energy electrons. We also neglect
the contribution to the evaporation of micro-droplets of
plasma electrons compared to the contribution of plasma
ions. This is determined by the fact that the currents of
electrons and ions per micro-drop are equal (Fig. 2), but
the energy edr >> Te which is brought by the ions to the
micro-drop is greater than that of the electrons.
Fig. 2. Flows of high-energy electrons, as well as
plasma electrons and ions on a micro-droplet
Then there remains the evaporation of micro-droplets
only by plasma ions when they are accelerated in the
electric potential of the micro-droplet created by plasma
electrons. And so, the equations (1), (2) take the form
𝑑𝑇𝑑𝑟
𝑑𝑡
=
3
𝑟𝑑𝑟𝜌𝑑𝑟𝑐
{− 𝛼𝜎𝑇𝑑𝑟
4 + 0.6√2𝑛0𝜀𝑖√
𝑇𝑒
𝑚𝑖
(1 +
𝑟𝑑
2 𝑟𝑑𝑟
2⁄ )}, (7)
𝑑𝑇𝑒
𝑑𝑡
= 1.2 𝛾 𝑒 𝜑0√
𝑘 𝑇𝑒
𝑚𝑖
/𝑅. (8)
Integrating (8), it can be obtained that Te grows
approximately according to
𝑇𝑒 = 𝑡2 ∙ 0.36 (
𝛾𝑒𝜑0
𝑅
)
2
/𝑚𝑖 (9)
that is, proportional to the square of time Te t2.
From equation (7) at a short time after the beginning
of the impact of the electron beam, when the change in
the radii of the micro-droplets can be neglected, it
follows, taking into account the expression
i = Te ln[0.6(2me/mi)1/2], that the temperature Tdr
(l) of
largemicro-droplets grow according to
𝑇𝑑𝑟
(𝑙) ≈ 0.1√2𝑛0𝑡4 (
𝛾𝑒𝜑0
𝑅
)
3 ln[0.6 (2𝑚𝑒/𝑚𝑖)1/2]
𝑟𝑑𝑟𝜌𝑑𝑟𝑐 𝑚𝑖
2 . (10)
The temperature Tdr
(s) of small micro-droplets increases
even faster
𝑇𝑑𝑟
(𝑠) ≈ 0.01𝑡6 (
𝛾𝑒𝜑0
𝑅
)
5 ln[0.6 (2𝑚𝑒/𝑚𝑖)1/2]
√2𝜋𝑒2𝑟𝑑𝑟
3𝜌𝑑𝑟𝑐 𝑚𝑖
3. (11)
That is according to (10) Tdr
(l) grows proportional to t4,
whereas from (11) Tdr
(s) is proportional to t6.
When the thermal radiation of micro-droplets
becomes significant, their temperature stabilizes. The
temperature Tdr
(l) of large micro-droplets stabilizes at
𝑇𝑑𝑟
(𝑙) = 𝑇𝑒
3/8 {
0.6√2𝑛0ln[0.6 (2𝑚𝑒 𝑚𝑖⁄ )1/2]
𝛼𝜎√𝑚𝑖
}
1/4
. (12)
92 ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142)
The temperature Tdr
(s) of small micro-droplets stabilizes
at
𝑇𝑑𝑟
(𝑠) = 𝑇𝑒
5/8 {
0.3𝑛0ln[0.6 (2𝑚𝑒 𝑚𝑖⁄ )1/2]
√2𝜋𝑛0𝑒2𝑟𝑑𝑟𝛼𝜎√𝑚𝑖
}
1/4
. (13)
The stationary temperature of small micro-droplets is
greater than that of large micro-droplets because, as it is
seen from (12) and (13)
𝑇𝑑𝑟
(𝑠)
𝑇𝑑𝑟
(𝑙) ≈ √
𝑟𝑑
𝑟𝑑𝑟
(𝑠) ≫ 1. (14)
Thus, from (14) it can be concluded that small micro-
droplets evaporate more intensively. So, the largest
micro-drops can be separated, and the small micro-drops
can be evaporated by the new device.
6. TWO APPROXIMATIONS FOR THE
ENERGY BALANCE EQUATION OF THE
ELECTRON BEAM AND PLASMA
ELECTRONS
Until now, we have used the approximation that all
the energy of the beam is transmitted to the plasma
electrons, except for that transmitted by the beam directly
to the drops:
𝑑𝑇𝑒
𝑑𝑡
(1)
= 0.6𝛾𝑒𝜑0√
𝑘𝑇𝑒
𝑚𝑖
(
2
𝑅
− 𝑛𝑑𝑟𝜋𝑟𝑑𝑟
2). (15)
Equation (15) for the plasma electron temperature Te.
One may look at the less favorable case when the
beam transfers energy to plasma electrons only due to
collisions:
𝑑𝑇𝑒
𝑑𝑡
(2)
=
𝑑𝑇𝑒
𝑑𝑡
(1)
𝜈𝑒𝑏
𝑅
3𝑉𝑏
; eb – frequency of
beam collisions with plasma electrons. The model using
collisions can lead to a lower intensity of energy
exchange of the beam with plasma electrons, that is
𝑑𝑇𝑒
𝑑𝑡
(1)
>
𝑑𝑇𝑒
𝑑𝑡
(2)
only in the weakly collisional case
𝜈𝑒𝑏 <
𝑅
3𝑉𝑏
.
However, the reality is somewhere in between.
Namely, beam-plasma instability can play a big role in
the transfer of electron beam energy to plasma electrons.
7. OPTIMUM PLASMA DENSITY
It should be noted that the higher the plasma electron
density, the faster the droplets evaporate, and the energy
absorbed by the plasma from high-energy electrons is
also proportional to the plasma electron density. Then
less energy reaches the drops, especially taking into
account the fact that plasma electrons have a finite
lifetime. That is, plasma electrons take energy, and then
it is not used for evaporation of microdroplets. Therefore,
there must be an optimal density of plasma electrons.
That is, for the destruction of macro-particles in a plasma
with a higher concentration, more powerful beams are
needed than in a plasma with a lower concentration.
For effective droplet evaporation, a significant
density of the energy reserve neTe, which is proportional
to the plasma density ne, must be transferred to the
plasma. But if the density of the plasma is too low, then
ions must be collected from the larger volume of the
droplet's plasma so that the droplet evaporates. If the
competition starts, then the drop cannot get more ions.
That is, the maximum number of ions that can fall on a
droplet is determined by the size of ndr
-1/3. That is, the
energy of the ions collected from the volume with a
radius of ndr
-1/3 should be sufficient for the evaporation of
the drop
4𝜋
3
𝑅3𝑛𝑖𝑒𝜑0 >
4𝜋
3
𝑛𝑠𝑠𝑟𝑑𝑟
3𝜀𝑒𝑣, 𝑅 = 𝑛𝑑𝑟
−1/3.
The plasma density necessary for evaporation follows
from the latter
𝑛𝑖
𝑛𝑑𝑟
𝑒𝜑0 > 𝑛𝑠𝑠𝑟𝑑𝑟
3𝜀𝑒𝑣 that is
𝑛𝑖
𝑛𝑑𝑟
>
𝑛𝑠𝑠𝑟𝑑𝑟
3𝜀𝑒𝑣
𝑒𝜑0
. This is the optimal plasma density and
the optimal droplet density
𝑛𝑖
(𝑜𝑝𝑡)
𝑛𝑑𝑟
>
𝑛𝑠𝑠𝑟𝑑𝑟
3𝜀𝑒𝑣
𝑒𝜑0
. That is,
the plasma density should be no less than optimal, and
the droplet density should be no more optimal.
8. EVAPORATION RATE OF DROPLETS
Depending on the parameters, two droplet
evaporation modes can be implemented. During droplet
evaporation, one mode can change the other. One mode
is realized if the density of drops ndr in the arc plasma
flow is sufficiently large. This mode is implemented if
the next inequality holds 𝑛𝑑𝑟
−1/3 ≪ 𝑟𝑑 (rd – the Debye
radius of plasma electrons), This condition corresponds
to the fact that there are many drops in a sphere whose
radius is equal to the Debye radius 𝑛𝑑𝑟𝑟𝑑
3 ≫ 1. In this
case, the drops are not independent. Their competition for
plasma ions, which accelerate to the droplets and
determine their evaporation, becomes particularly
significant. The size of the volume from which plasma
ions are accelerated to drops is approximately equal to
ndr
-1/3. Then the energy flow per drop, which is
determined by plasma ions, is proportional to the large
parameter ndr
-1/3 or rdr, where rdr is the radius of the
droplet.
Initially or in the process of evaporation of droplets
in the arc plasma flow, the density of droplets ndr may
become small. Then the first condition may become
invalid. If the inequality holds 𝑛𝑑𝑟
−1/3 > 𝑟𝑑 in a sphere
whose radius is equal to the Debye radius, there is no
more than one drop. In this case, the drops are
independent and they do not compete. The radius of the
sphere from which plasma ions are accelerated to drops
is equal to rd. Then the energy flow per drop, which is
determined by the plasma ions, is proportional to the
larger parameter from rd and rdr.
When passing from the first regime (first condition)
to the second regime (second condition) as the droplets
evaporate, the flow of ions per drop increases, and their
energy also increases (because the potential of the
droplets increases (without taking into account thermal
emission and secondary electron-electron emission)). It
should be noted that as the temperature of the droplets
increases, thermal emission from the droplets increases,
and secondary electron-electron emission also grows
when the droplets evaporate. So, the second regime may
be the most favorable for the evaporation of droplets. In
this mode, the highest droplet temperature and the most
significant rate of their evaporation are achieved.
ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142) 93
9. SIMULATION RESULTS
As was shown earlier, when a plasma flow passes
through a plasma lens with a magnetic field, if a negative
potential of 2...3 kV is applied to its central electrode, a
radial beam of fast electrons is self-consistently formed,
the energy of which is sufficient to destroy the drop [8].
Also, this beam heats the plasma electrons, which in turn
interact with the drops, this is described by bursts (1), (2).
Numerical results of solving these equations for the case
of a plasma lens with a magnetic field are presented in
Figs. 3, 4. During the calculation, it was assumed that the
potential of the drop is calculated according to the
expression: dr=(Te/e) ln[0.6(2me/mi)1/2].
Fig. 3. Heating of plasma electrons by a beam of fast
electrons
Fig. 4. Heating of the drop by electrons over time
depending on the size of the drop:
1 – rdr = 0.5 m; 2 – rdr = 1 m; 3 – rdr = 5 m
On Fig. 3 shows the heating of plasma electrons over
time. It can be seen that, taking into account the heating
of the plasma by a beam of fast electrons, the temperature
of the plasma electrons also rapidly increases and they
also begin to heat the droplets.
On Fig. 4 shows the change over time in the
temperature of the drop depending on the radius. As you
can see, for small drops it grows rapidly and reaches the
boiling point, and they begin to evaporate. For large
drops, its temperature rises more slowly and it takes more
time to reach the boiling point, and evaporation has
begun.
CONCLUSIONS
A system of two equations describing the evaporation
of micro-droplets in an arc plasma flow under the action
of a self-consistent electron beam and the condition of
direct heating of micro-droplets by fast electrons was
formulated. It was shown that the plasma is heated under
the influence of the beam, even taking into account the
fact that the electrons and ions of the plasma lose energy
for the evaporation of micro-droplets. The heating rates
of plasma electrons and micro-droplets at short exposure
times of a self-consistent electron beam were obtained. It
was shown that small micro-droplets evaporate more
intensively.
It should be noted that the higher the plasma electron
density, the faster the droplets evaporate, and the energy
absorbed by the plasma from high-energy electrons is
also proportional to the plasma electron density. Then
less energy reaches the droplets, especially taking into
account the fact that plasma electrons have a finite
lifetime. That is, plasma electrons take energy, and then
it is not used for evaporation of microdroplets. Therefore,
there must be an optimal density of plasma electrons.
That is, for the destruction of macro-particles in a plasma
with a higher concentration, more powerful beams are
needed than in a plasma with a lower concentration.
The magnitude of the energy contribution of the fast
electrons flow to the overall heating of micro-droplets
depends on the temperature of the plasma electrons and
the potential of the droplet. As the calculations show,
during the stay of the drop in the volume of the plasma
device, it has the opportunity to heat up to the
temperature necessary for its evaporation.
It is clear that the contribution to the heating of
macroparticles from multi-charged ions is greater than
from singly-charged ones.
ACKNOWLEDGEMENTS
This work is supported partly by the. grant
1.4.B/191 and grant 2/2022-D from the Presidium
of the NAS of Ukraine and, especially, by support
project № PL-20-22 in frame of target program scientific
research NASU “Plasma Physics and Plasma
Electronics: fundamental research and applications for a
2020-2022”.
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Article received 09.10.2022
ФОРМУВАННЯ ЕЛЕКТРОННОГО ПУЧКА І ЙОГО РОЛЬ У НОВІЙ ПЛАЗМООПТИЧНІЙ
СИСТЕМІ ДЛЯ ВИПАРОВУВАННЯ КРАПЕЛЬ У ДУГОВІЙ ПЛАЗМІ
О.A. Гончаров, В.І. Маслов, І.В. Літовко, А.В. Рябцев
Розглядаються випаровування мікрокрапель у потоці дугової плазми під дією самоузгодженого
електронного пучка та умова прямого нагріву мікрокрапель швидкими електронами. Показано, що плазма
гріється під дією пучка, навіть з урахуванням того, що електрони і іони плазми втрачають енергію на
випаровування мікрокрапель. Встановлено, що дрібні мікрокраплі випаровуються інтенсивніше. Показано,
що має бути оптимальна щільність електронів плазми. Для руйнування макрочастинок у плазмі з більшою
концентрацією потрібні потужніші пучки, ніж у плазмі з меншою концентрацією.
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