Electrostatic Plasma Oscillations And The Kinetic Energy of a Charged Macroparticle In Weakly Ionized Plasma

O. S. Vaulina1, S. V. Vladimirov2, A. Yu. Repin1
1Institute for High Energy Density, RAS, Izhorskaya 13/19, Moscow 127412, Russia
2School of Physics, The University of Sydney, New South Wales 2006, Australia
Abstract. The stochastic energy acquired by an isolated charged macroparticle due to electrostatic fluctuations of a weakly
ionized plasma is investigated. Analytic relations are derived and numerical modeling of the problem for the conditions close to
those of typical laboratory experiments in a dusty plasma is done. The study demonstrates that the kinetic energy of a dust
particle, induced by the considered effect, can significantly exceed the temperature of the background gas.
Keywords: Dusty Plasma, Numerical Simulation.
PACS: 52.25. Ub, 52.25. Zb, 82.70. Db
INTRODUCTION
A dusty plasma is an ionized gas containing charged macroscopic particles (dust), typically of a micrometer size.
The majority of experimental studies of dusty plasmas are done in a weakly ionized plasma of a gas discharge under
the pressure P of the neutral gas within the range from 0.03 to 3 Torr when dissipation due to collisions with the gas
atoms/molecules is significant. The dust charge eZ can be significant ~ 102-104е, where e is the elementary charge.
These charged particles effectively interact with each other as well as with the background plasma and the electric
fields. The dust kinetic temperature can significantly exceed not only their surface temperature (determined by the
gas temperature) but also the temperature of the electron component (~ 1-5 eV). Possible reasons for these
anomalously high dust kinetic temperatures include the space-time fluctuations of the dusty plasma parameters as
well as development of various plasma-dust instabilities [1-3].
In this paper, we consider the effect of electrostatic oscillations of a quasi-equilibrium plasma on the kinetic
temperature of an isolated dust particle. These oscillations appear as a consequence of the space separation of the
plasma charges because of their thermal motions and lead to the time fluctuations of the plasma electric field E(t),
with the mean-average < E(t)2> ∝Tpl, where Tpl is the plasma temperature. The main physics of the influence of the
electrostatic plasma oscillations on the kinetic energy of dust particles is that the electric field fluctuations induced
by the stochastic motions of the electron/ion plasma component lead, in their turn, to fluctuations of the electric
forces, ~ FE(t) = eZE(t), acting on the macroparticles. This causes extra chaotic movements (in addition to the dust
Brownian motion due to its collisions with the neutrals of gas) with the kinetic energy Tkd ∝ < Z2E(t)2> which is nonzero
even in the case when the stochastic charge fluctuations (because of the discrete character of charging plasma
currents) are ignored, Z(t)=const.
ELECTROSTATIC PLASMA OSCILLATIONS AND DUST KINETIC ENERGY
Consider electrostatic oscillations of an unmagnetized plasma due to its thermal fluctuations in the two limiting
cases: 1) for Langmuir oscillations of electrons on the background of stationary ions, and 2) for electrostatic
oscillations of the number density of cold single-charged ions on the adiabatic electron background (with the thermal
electron motions). The last case takes place when the electron temperature significantly exceed the ion temperature,
Te>>Ti. The field appearing because of the charge separation due to the thermal motion of plasma electrons and ions,
is determined by the Poisson equation.

NUMERICAL SIMULATIONS
Simulations were done for the conditions close to those of a typical laboratory experiment in the gasdischarges.
A macroparticle with the mass M, the radius R, and the fixed charge was confined in the center of the
cubic simulation box with the length 2L by the linear electric field with the gradient α=4πen, where n=Ne/(2L)3, and
Ne=Ni is the number of electrons/ions in the simulation cell. The macroparticle’s equation of motion was solved
taking into account the effect of plasma electrons and ions. For the electrons and ions, the Langevin equations were
solved taking into account the electric field of the dust charge as well as the plasma particle interactions.
We have considered argon with the pressure P=0.3-10 Torr. The effective collision frequencies vin (vеn)
were proportional to P and were taken as vin=8×106 s-1 (vеn=5.3×109 s-1) for P=1 Torr. Simulations were done for
Z=10-1000, R = 0.1-3 μm, Ti/Te = 0.01-0.1, Тi = 0.03 eV, νfr = νin/100 , and M=(100-10000)mi. The physical scale
of the simulation box was changed from ~1.5λDi to ~15λDi (where λDi is the ion Debye length) by varying the
number of plasma particles (n~109-1012 cm-3). The simulation was done until the macroparticle’s kinetic temperature
achieves a stationary value. The illustration of the time evolution of the heating of the macroparticle in the
simulation cell is presented in Fig. 1 for L/λDi≈15.
To determine the plasma eigenfrequency ω
р in the numerical simulation, we varied the values of Zα/M and νpn.
As a result, we have obtained that ω
р2≈ 8πe2n/(3mi). The temperature Tcal determined in the simulation run did not
depend on the dust radius R. The acquired temperature Tcal was dependent on the ratio β=L/λDi (see Fig. 2). With the
increase of β, the value of Tcal approached Td (6) with the parameters of the ion plasma component, i.e.,

Thus the dust particle in a plasma can acquire the additional stochastic kinetic energy Td. The value of Td is
determined by the fluctuations of the electric field caused by the thermal motion of plasma ions. If to consider the
conditions of typical laboratory experiments in a gas-discharge (in argon) and write the coefficient for the dust
friction in the free-molecular approach and the dust charge in the Orbit-Motion Limited (OML) approach, then the
acquired energy can be estimated as Tapp ≈ 0.5 (Те[eV])2Тi. Therefore, for Те ~2-5 eV and Тi ~0.03 eV, the dust
kinetic energy can achieve ~0.14-0.34 eV.
We note that for the conditions of a typical laboratory experiment with dusty plasma, collisions of plasma ions
with the neutral gas atoms can significantly decrease the dust charge as compared to that determined by OML [4].
Since T app ∝ Z2, this, in turn, can lead to significant decrease of the kinetic energy acquired (T app << Tr).
CONCLUSIONS
To conclude, we have presented here the analytic relations for the stochastic energy that is acquired by an
isolated solid macroparticle in a weakly-ionized plasma because of the thermal electrostatic plasma fluctuations. The
fluctuations can be related to the Langmuir plasma mode as well as to the electrostatic (cold) ion mode. The derived
expressions allowed us to estimate the minimum value of the kinetic temperature of the macroparticle in a quasiequilibrium
plasma for the conditions where there is no plasma-dust instabilities and no propagated plasma waves.
Numerical simulations have demonstrated that for the conditions of typical laboratory experiments in a gasdischarge
complex plasma (when, in particular, Ti/Te<<1), the most significant contribution to the energy acquired
by the macroparticle comes from the plasma fluctuations associated with the ion component. The kinetic
temperature of the macroparticle acquired by this channel, can significantly exceed the room temperature of the
background gas/plasma ions.
Although the simulations were done for an isolated particle, these results can be also useful for the analysis of
extended plasma-dust structures (consisting of many macroparticles) if the mean interparticle distance significantly
exceeds the ion Debye length. We note that this is the case for most of laboratory experiments.
ACKNOWLEDGMENTS
This work was partially supported by the Russian Foundation for Fundamental Research (No. 04-02-16362),
CRDF (No. RU-P2-2593-MO-04), the Program of the Presidium of RAS.
REFERENCES
1. V.V. Zhakhovskii, V.I. Molotkov, A.P. Nefedov, et al., JETP Lett., 66, 419 (1997).
2. O.S. Vaulina, A.A. Samarian, B. James, et al., JETP 96, 1037 (2003).
3. O.S. Vaulina, S.A. Khrapak, O.F. Petrov, et al., Phys. Rev. E 60, 5959 (1999).
4. A.V. Zobnin et. al. JETP 91, 483 (2000).

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