Charging Of A Small Dust Grain In Collisional Plasmas
L.G. D’yachkov1, S.A. Khrapak2 and A.G. Khrapak1
1Institute for High Energy Densities, Moscow, Russia
2Max-Planck-Institut für Extraterrestrische Physik, Garching, Germany
Abstract. Analytical approximation for charging of a dust grain in dense low-ionized plasma is presented. The charging
fluxes, grain surface potential, and long range asymptote of electrostatic potential in the case a < li(e) << R under
assumption that ionization and recombination processes in the vicinity of the grain (r < R) can be neglected, are
estimated. Here a is the grain radius, li(e) is the ion (electron) mean free path, and R is the characteristic lengthscale of the
perturbed plasma region around the grain. The limiting cases li(e) >> a and li(e) << a are also considered.
Keywords: Dusty plasma, charging, grain floating potential, hydrodynamic approximation.
The grain floating potential and charge are important parameters of dusty plasmas. If the electron emission from
the grain surface is negligible, the grain potential and charge are determined by the balance of ion and electron
fluxes that the grain collects from the surrounding plasma. In low-pressure discharges usually applied for the dusty
plasma investigation, the ion li and electron le mean free paths exceed the characteristic lengthscale R of the
perturbed plasma region around the grain, and the orbit motion limited (OML) approximation can be used. In the
opposite limiting case li(e) << a, R, where a is the grain radius, the hydrodynamic approximation is applicable down
to the grain surface. Both the approximations are well known in the dusty plasma  and electric probe  theories.
In the present paper, we consider the intermediate case a < li(e) << R with distance between grains much larger
than R, so the grain can be regarded as a solitary one. Ionization and recombination in the vicinity of the grain are
assumed to be neglected, i.e. the plasma sources compensating for the loss of ions and electrons due to absorption on
the grain are far from its surface . This corresponds to a small dust grain in relatively dense low-ionized plasma.
We have obtained an approximate algebraic equation for the grain surface potential, expression for ion and electron
fluxes to the grain, and asymptotic behavior of the electrostatic potential at large distances from the grain.
EQUATION FOR THE GRAIN SURFACE POTENTIAL
With regard to spherical symmetry, after integration of the continuity equation for the ion (electron) density ni(e).
This study was supported by the Program “Thermal Physics and Mechanics of Intense Energy Actions” of the
Presidium of the Russian Academy of Sciences and the Russian Foundation for Basic Research through grant 05-02-
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