Veloсity dispersion of dust particles confined in a sheath
Velocity distribution of dust particles localized in a plasma sheath near an electrode was found in a number of experiments. Velocity dispersion indicated that the kinetic temperature of dust grains significantly exceeds the temperature of plasma environment. Consequently, the question arose about t...
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Zasenko, V.I. 2023-11-27T15:53:21Z 2023-11-27T15:53:21Z 2019 Veloсity dispersion of dust particles confined in a sheath / V.I. Zasenko // Problems of atomic science and technology. — 2019. — № 1. — С. 57-60. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 52.65.Cc https://nasplib.isofts.kiev.ua/handle/123456789/194590 Velocity distribution of dust particles localized in a plasma sheath near an electrode was found in a number of experiments. Velocity dispersion indicated that the kinetic temperature of dust grains significantly exceeds the temperature of plasma environment. Consequently, the question arose about the stochastic mechanisms of anomalous heating of grains. We propose the model in which the kinetic energy is due to the significant potential energy that grains have at the moment of their release from the crystalline structure on melting. Stochastic processes only modify the regular motion of dust grains, forming a velocity distribution similar to а Gaussian. У ряді експериментів визначався швидкісний розподіл порошинок, локалізованих в приповерхневому шарі плазми поблизу електрода. Дисперсія швидкості вказувала на те, що кінетична температура порошинок значно перевищує температуру плазмового оточення. Відповідно постало питання про стохастичні механізми аномального нагрівання порошинок. Запропоновано модель, в якій кінетична енергія порошинок обумовлена значною потенціальною енергією, яку частинки мають у момент вивільнення з кристалічної структури при плавленні. При цьому стохастичні процеси тільки модифікують регулярний рух частинок, формуючи розподіл за швидкостями, подібний до гаусового. В ряде экспериментов определялось скоростное распределение пылинок, локализованных в приповерхностном слое плазмы вблизи электрода. Дисперсия скорости указывала на то, что кинетическая температура пылинок значительно превышает температуру плазменного окружения. Соответственно возник вопрос о стохастических механизмах аномального нагрева пылинок. Предложена модель, в которой кинетическая энергия пылинок обусловлена значительной потенциальной энергией, которой частицы обладают в момент высвобождения из кристаллической структуры при плавлении. При этом стохастические процессы только модифицируют регулярное движение частиц, формируя распределение по скоростям, подобное гауссовому. The work is supported by the grant of the State Fund for Fundamental Research (project 33256). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Veloсity dispersion of dust particles confined in a sheath Дисперсія швидкості порошинок у приповерхневому шарі Дисперсия скорости пылинок в приповерхностном слое Article published earlier |
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| title |
Veloсity dispersion of dust particles confined in a sheath |
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Veloсity dispersion of dust particles confined in a sheath Zasenko, V.I. Basic plasma physics |
| title_short |
Veloсity dispersion of dust particles confined in a sheath |
| title_full |
Veloсity dispersion of dust particles confined in a sheath |
| title_fullStr |
Veloсity dispersion of dust particles confined in a sheath |
| title_full_unstemmed |
Veloсity dispersion of dust particles confined in a sheath |
| title_sort |
veloсity dispersion of dust particles confined in a sheath |
| author |
Zasenko, V.I. |
| author_facet |
Zasenko, V.I. |
| topic |
Basic plasma physics |
| topic_facet |
Basic plasma physics |
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2019 |
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English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Дисперсія швидкості порошинок у приповерхневому шарі Дисперсия скорости пылинок в приповерхностном слое |
| description |
Velocity distribution of dust particles localized in a plasma sheath near an electrode was found in a number of experiments. Velocity dispersion indicated that the kinetic temperature of dust grains significantly exceeds the temperature of plasma environment. Consequently, the question arose about the stochastic mechanisms of anomalous heating of grains. We propose the model in which the kinetic energy is due to the significant potential energy that grains have at the moment of their release from the crystalline structure on melting. Stochastic processes only modify the regular motion of dust grains, forming a velocity distribution similar to а Gaussian.
У ряді експериментів визначався швидкісний розподіл порошинок, локалізованих в приповерхневому шарі плазми поблизу електрода. Дисперсія швидкості вказувала на те, що кінетична температура порошинок значно перевищує температуру плазмового оточення. Відповідно постало питання про стохастичні механізми аномального нагрівання порошинок. Запропоновано модель, в якій кінетична енергія порошинок обумовлена значною потенціальною енергією, яку частинки мають у момент вивільнення з кристалічної структури при плавленні. При цьому стохастичні процеси тільки модифікують регулярний рух частинок, формуючи розподіл за швидкостями, подібний до гаусового.
В ряде экспериментов определялось скоростное распределение пылинок, локализованных в приповерхностном слое плазмы вблизи электрода. Дисперсия скорости указывала на то, что кинетическая температура пылинок значительно превышает температуру плазменного окружения. Соответственно возник вопрос о стохастических механизмах аномального нагрева пылинок. Предложена модель, в которой кинетическая энергия пылинок обусловлена значительной потенциальной энергией, которой частицы обладают в момент высвобождения из кристаллической структуры при плавлении. При этом стохастические процессы только модифицируют регулярное движение частиц, формируя распределение по скоростям, подобное гауссовому.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/194590 |
| citation_txt |
Veloсity dispersion of dust particles confined in a sheath / V.I. Zasenko // Problems of atomic science and technology. — 2019. — № 1. — С. 57-60. — Бібліогр.: 13 назв. — англ. |
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2025-11-24T11:50:10Z |
| last_indexed |
2025-11-24T11:50:10Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2019. №1(119)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2019, № 1. Series: Plasma Physics (25), p. 57-60. 57
VELOСITY DISPERSION OF DUST PARTICLES CONFINED
IN A SHEATH
V.I. Zasenko
Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine
E-mail: zasenko@bitp.kiev.ua
Velocity distribution of dust particles localized in a plasma sheath near an electrode was found in a number of
experiments. Velocity dispersion indicated that the kinetic temperature of dust grains significantly exceeds the
temperature of plasma environment. Consequently, the question arose about the stochastic mechanisms of
anomalous heating of grains. We propose the model in which the kinetic energy is due to the significant potential
energy that grains have at the moment of their release from the crystalline structure on melting. Stochastic processes
only modify the regular motion of dust grains, forming a velocity distribution similar to а Gaussian.
PACS: 52.65.Cc
INTRODUCTION
Already in the first experiments on melting of
plasma crystals [1-4] it was observed that dust grains
released from the crystalline structure move with
considerable velocities. The motion of particles was
recorded with high-speed cameras, and by processing
shots a velocity of the grains was calculated. A velocity
distribution function was built, and its form turned out
to be close to a Gaussian [1, 2]. Quite unexpected was
the fact that the "kinetic temperature", as a characteristic
of this motion, could exceed the temperature of the
surrounding plasma more than a thousand times.
From the first experiments, various statistical
processes that could lead to dust particle heating in the
transition from crystalline to gaseous phase were
proposed. Among the most typical processes are
electrostatic fluctuations of plasma environment, which
were studied on a basis of the Langevin equation [5],
and fluctuations of dust particle charge [6]. A discussion
of various models of random processes is given in the
recent paper [7]. The instability of dust grains due to
modification of an ion flux by their neighbors, and the
effects of negative friction caused by plasma particle
absorbtion [8-11] were considered as well. These and
other works show the complexity and versatility of
physical processes in dusty plasma. Nevertheless, the
consideration of various mechanisms did not distinguish
the dominant process, which would explain the
occurrence of anomalous "kinetic temperature" of
grains.
In our paper, we draw attention to the fact that the
considerable kinetic energy of particles can be caused
not by stochastic processes but due to significant
potential energy of grains in sheath and gravity fields at
the moment of their release from a crystalline structure
during melting. Stochastic processes only partially
modify dust particle motion. The experimental
evaluation of the fields [2] showed that a grain can gain
significant kinetic energy. As well in the works [1, 2], it
was not argued that high kinetic energy was obtained as
a result of stochastic heating.
The potential energy that leads to the intense
movement of particles can occur for two reasons. First,
in the state of a crystal, when it as a whole occupies the
lowest position in the potential well, each grain is in a
position that does not correspond to equilibrium state of
a free dust particle. Because of strong interaction with
neighbors it is shifted from the position of the local
equilibrium in sheath and gravity fields; and when
melting, it begins to oscillate around this position.
Secondly, the initial displacement from the state of the
local equilibrium of a free dust particle is determined
not only by neighbor particles but also by charge
variation of a grain. Grain charges differ by a few
percent [4], which means that their equilibrium
positions are shifted to each other.
Estimation of grain kinetic energy can be obtained
from the following considerations. For DC sheath the
floating potential is of the order of
1
ln
2
e e
i
T m
e m
,
where Te is an electron temperature, e-elementary
charge, me and mi are masses of an electron and ion;
here we neglect the effect of grains on the floating
potential. A depth of a potential well in which grains are
confined is of the order of this magnitude. A charge of
dust particle is four orders of magnitude higher than the
elementary charge. Thus a grain with kinetic energy five
orders of magnitude higher than electron temperature
still is confined in a sheath. Kinetic energy of grain
oscillations in RF sheath could be even greater. A
particular magnitude of grain energy depends on an
initial displacement of a grain from a position of its
local equilibrium. In general, it looks reasonable that a
kinetic energy of grain oscillations could be three orders
higher than the electron temperature.
Thus, we suppose that grain intense motion could be
caused by a potential energy of displacement from the
position of its local equilibrium at the moment of
crystalline structure melting. Along with this regular
oscillations are modified by effects of various stochastic
forces.
1. MODEL EQUATIONS
Based on these considerations, we shall take a
simple model of oscillations of grains with slightly
different charges in a parabolic potential well formed by
58 ISSN 1562-6016. ВАНТ. 2019. №1(119)
superposition of potentials of homogeneous
gravitational field and linearly increasing electric field
of a sheath. In addition, we assume that motion of grains
is influenced by either electrostatic fluctuations of
plasma environment, or fluctuations of grain charge,
and friction caused mainly by collisions with neutrals.
Pair interaction with neighboring grains is neglected.
The equations of one-dimensional motion of a grain
are of the form
'
0
,
(1 ) ( , ),
t
t
x v
v g E x v E x t
(1)
where x is a displacement from a position that
corresponds to the equilibrium of the grain with the
average charge, v is a grain velocity; g is the
gravitational acceleration, is a relative fluctuation of
a grain charge from the mean value, is a friction
coefficient. Acceleration of a dust particle is caused by
the electric field of a sheath which linearly depends on
grain displacement '
0E x . Electrostatic fluctuations
( , )E x t have the form of a superposition of harmonics
with random phases i [12, 13], which propagate along
the axis x with velocity с
1
( , ) co ( )s( )
N
i i i
i
E x t E k x ct . (2)
We assume that initially grains are placed on the x-
segment (-0.5, 0.5) and have zero velocity.
Our purpose is to find the velocity distribution
function of grains in a potential well under influence of
random force: either electrostatic fluctuations (2) or
charge fluctuation . Our consideration is qualitative,
and physical quantities are given in arbitrary units.
2. DISTRIBUTION FUNCTION OF GRAIN
VELOCITIES
We start this section with illustrations of the
individual motion of grains in absence of friction and
electrostatic fluctuations. Phase space trajectories of
arbitrarily ten particles with different initial charges are
shown in Fig. 1 during the half-period of oscillation in
the potential well (t=0, 10); the initial stage is
represented by black solid lines. Grains move with
different velocities, mainly to the center x = 0. But one
of particles moves in the opposite direction. Another
one, whose initial position turned out to be close to
equilibrium point determined by its charge, is hardly
moving.
The velocity distribution function of grains in the
potential well for t=2 is shown in Fig. 2. We shall trace
a change of its shape caused by turning on various
effects. The velocity distribution function of grains with
friction and equal charges is shown with gray points,
line 1. The distribution function of grains with different
initial charges is of triangular form (solid gray line 3),
the friction makes its narrower (solid black line 2).
Accounting for the electrostatic fluctuations (2) makes
this distribution closer to a Gaussian, Fig. 3.
Fig. 1. Ten particle orbits in phase space during a half
period of oscillation in the potential well,
t = (0, 10). Particles in the potential well with initial
charge variation (no electrostatic fluctuation, no
friction). Initial stage is shown with solid lines
Fig. 2. Distribution functions at t =2. Grains with equal
charges and friction (gray dot line 1); grains with
different charges and friction (black solid line 2); grains
with different charges without friction (gray solid
line 3). Electrostatic fluctuations are absent
Fig. 3. Distribution functions of grains with different
charges and friction. Electrostatic fluctuations are
absent (2), present (4)
In this simulation electrostatic fluctuations were
assumed to be a superposition of waves with a fixed set
of random phases. The averaging was made over
particle ensemble but not over random phases the only
one realization of the field was considered. In the next
section it is compared with additional averaging over
ISSN 1562-6016. ВАНТ. 2019. №1(119) 59
random phases. It can be expected that account for
random interaction with neighbors, as well as averaging
over random field realizations will further makes a form
of a distribution function closer to a Gaussian.
Other process that effects on velocity distribution of
grains is permanent fluctuations of their charge. The
velocity distribution function with charge fluctuations is
shown in Fig. 4 at two moments. Its form is rather close
to a Gaussian. Such form of a grain distribution over
velocities has been reported in the experimental works
[1, 2].
Fig. 4. Velocity distribution functions with charge
fluctuations and friction (1), and their approximations
with Gaussians (2); t = 1, 2
Thus, in the considered model a kinetic energy of
grains is caused mainly by their oscillations in a
potential well. Stochastic processes such as electrostatic
or charge fluctuations only modify a distribution
function bringing it closer to a Gaussian.
3. DIFFERENT ENSEMBLES OF
AVERAGING
In the previous section the electrostatic
fluctuations (2) were taken as a superposition of
harmonics with a fixed set of random phases ai (2).
Averaging was made over ensemble of particles which
are initially placed in different positions. However
additional averaging over various realizations of
electrostatic random fields could be done as well. In our
model it corresponds to averaging over random phases.
Snapshot processing to find the distribution function in
experiments may be closer to one or the other method of
averaging.
The difference between the results of averaging over
ensembles of grains in the electrostatic fields with fixed
and different sets of random phases is shown in
Figs. 5, 6. The ensemble of random phases leads to a
smooth dependence of an average velocity and velocity
dispersion on time (black lines in Figs. 5, 6). In this
approach, the dispersion of velocities may look like an
effective temperature.
For a fixed set of random phases such dependencies
(gray lines) are highly irregular. As far as they are not
well-defined characteristics, velocity dispersion cannot
serve as kinetic temperature of grains at all. Note, that
the Langevin approach, which is often applied to a
theoretical study of grain heating, is based on averaging
over an ensemble of random phases.
Fig. 5. Average velocity of grains in a potential well
with electrostatic field fluctuations and friction. Field
with fixed phases (gray line), additional averaging over
field realizations, i.e. random phases (black line)
Fig. 6. Velocity dispersion of grains in a potential well
with electrostatic field fluctuations and friction. Field
with fixed phases (gray line), additional averaging over
field realizations, i.e. random phases (black line)
CONCLUSIONS
We suggest that high velocities of dust particles that
were observed in a number of experiments on melting
of crystals are caused by a substantial initial potential
energy of grains. In a crystalline structure a grain
interacts strongly with neighbors which determine its
location. At the same time a crystal as a whole occupies
a position in which the gravitational force is balanced
by an electric field of a sheath. On melting, the
equilibrium positions of grains are determined by the
balance of these forces individually for each particle.
Since equilibrium positions of grains in a crystalline
structure and after release from it do not coincide, grains
start to oscillate. Motion of a grain displaced from an
equilibrium position in a potential well is mainly
regular. However, it is modified by stochastic processes,
such as fluctuations of its charge and electrostatic
fluctuations in the plasma environment.
Despite the stochastic forces may not be too strong,
grain velocities look somewhat like random. Stochastic
interactions make a velocity distribution function closer
to a Gaussian. Nevertheless, it is not appropriate to
characterize grain motion in terms of kinetic
temperature.
60 ISSN 1562-6016. ВАНТ. 2019. №1(119)
ACKNOWLEDGEMENTS
The work is supported by the grant of the State Fund
for Fundamental Research (project 33256).
REFERENCES
1. H.M. Thomas, G.E. Morfill. Melting dynamics of a
plasma crystal // Nature. 1996, v. 379, p. 806-09.
2. A. Melzer, A. Homann, A. Piel. Experimental
investigation of the melting transition of the plasma
crystal // Phys. Rev. E. 1996, v. 53, p. 2757-2766.
3. V.V. Zhakhovskii. Anomalous heating of a system of
dust particles in a gas-discharge plasma // Pis’ma Zh.
Eksp. Teor. Fiz. 1997, v. 66, p. 392-397.
4. R.A. Quinn, J. Goree. Experimental investigation of
particle heating in a strongly coupled dusty plasma //
Physics of Plasmas. 2000, v. 7, p. 3904-3911.
5. R.A. Quinn, J. Goree. Single-particle Langevin model
of particle temperature in dusty plasmas // Phys. Rev. E.
2000, v. 61, p. 3303-3041.
6. O.S. Vaulina et al. Charge-fluctuation-induced
heating of dust particles in a plasma // Phys. Rev. E.
1999, v. 60, p. 5959-5964.
7. O.S. Vaulina. Influence of inhomogeneous conditions
on the kinetic energy of dust macroparticles in plasma //
Zh. Eksp. Teor. Fiz. 2016, v.122, p. 193-202.
8. A.G. Zagorodny, P. Schram, S.A. Trigger. Stationary
velocity and charge distributions of grains in dusty
plasmas // Phys. Rev. Lett. 2000, v. 84 (16), p. 3594.
9. A.V. Filippov, A.G. Zagorodny, A.I. Momot, et al.
Screening of a moving charge in a nonequilibrium
plasma // Journal of Experimental and Theoretical
Physics. 2009, v. 108 (3), p. 497-515.
10. S.A. Trigger, A.G. Zagorodny. Negative friction in
dusty plasmas // Contr. Plasma Phys. 2003, v. 43,
p. 381-383.
11. I.L. Semenov, A.G. Zagorodny, I.V. Krivtsun. Ion
drag force on a dust grain in a weakly ionized
collisional plasma // Physics of Plasmas. 2013, v. 20(1),
p. 013701.
12. V. Zasenko, A. Zagorodny, J. Weiland. Stochastic
acceleration in peaked spectrum // Phys. Plasmas. 2005,
v. 12, p. 062311.
13. V. Zasenko, A. Zagorodny, J. Weiland. Particle
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Article received 25.11.2018
ДИСПЕРСИЯ СКОРОСТИ ПЫЛИНОК В ПРИПОВЕРХНОСТНОМ СЛОЕ
В.И. Засенко
В ряде экспериментов определялось скоростное распределение пылинок, локализованных в
приповерхностном слое плазмы вблизи электрода. Дисперсия скорости указывала на то, что кинетическая
температура пылинок значительно превышает температуру плазменного окружения. Соответственно возник
вопрос о стохастических механизмах аномального нагрева пылинок. Предложена модель, в которой
кинетическая энергия пылинок обусловлена значительной потенциальной энергией, которой частицы
обладают в момент высвобождения из кристаллической структуры при плавлении. При этом стохастические
процессы только модифицируют регулярное движение частиц, формируя распределение по скоростям,
подобное гауссовому.
ДИСПЕРСІЯ ШВИДКОСТІ ПОРОШИНОК У ПРИПОВЕРХНЕВОМУ ШАРІ
В.І. Засенко
У ряді експериментів визначався швидкісний розподіл порошинок, локалізованих в приповерхневому
шарі плазми поблизу електрода. Дисперсія швидкості вказувала на те, що кінетична температура порошинок
значно перевищує температуру плазмового оточення. Відповідно постало питання про стохастичні
механізми аномального нагрівання порошинок. Запропоновано модель, в якій кінетична енергія порошинок
обумовлена значною потенціальною енергією, яку частинки мають у момент вивільнення з кристалічної
структури при плавленні. При цьому стохастичні процеси тільки модифікують регулярний рух частинок,
формуючи розподіл за швидкостями, подібний до гаусового.
|