KINETIC SIMULATIONS OF ARGON DUSTY PLASMA AFTERGLOW INCLUDING METASTABLE ATOM KINETICS
A. L. Alexandrov* I. V. Schweigert, D. A. Ariskin
Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences
630090, Novosibirsk, Russia
Received July 16, 2012
The afterglow of a dusty plasma of rf discharge in argon is simulated by Particles In Cells-Monte Carlo Collisions (PIC-MCC) method. The experimental observation that heavy dust contamination of plasma leads to an anomalous increase in the electron density at the beginning of afterglow is explained by release of electrons from the dust surface. Under the assumption that the floating potential of particles is in equilibrium with plasma conditions, the fast cooling of electrons in afterglow plasma due to a rapid escape of hot electrons from the volume leads to a decrease in the magnitude of the floating potential and hence to a loss of charge by dust. The intensive desorption of electrons from nanoparticles is the origin of anomalous behavior of the electron density. At the next stage of afterglow, when the electrons become cool, the plasma decay is defined by ambipolar diffusion. The effect of metastable argon atoms is also considered. Additional ionization due to metastable atom collisions affects the electron temperature but does not change the behavior of the electron density qualitatively.
DOI: 10.7868/S0044451013040150
1. INTRODUCTION
The afterglow of dusty plasma is an interesting field of investigation duo to the complexity of various nonequilibrium processes that occur in an unsteady plasma containing charged nanoparticles after switching off the discharge fl 5]. In the experiment fl], the anomalous behavior of electron density was observed, namely, a fast increase in the electron density in the dusty plasma of the radio-frequency (rf) discharge immediately after switching off the discharge voltage. The release of electrons from charged dust particles was suggested as a possible explanation. Analytic approaches to the description of the processes in a dusty afterglow plasma [4, 5] are usually restricted to the case of weak dust contamination, when the dust space charge is small compared to the electron density (small Havnes parameter). The anomalous electron density behavior is observed in the opposite case of heavy dust contamination.
In [3], other possible mechanisms leading to the anomalous electron density increase were suggested, such as ion dust and metastable dust secondary elec-
E-mail: a-alex'flitam.nsc.ru
tron emission, and the ionization due to metastable argon atom collisions. The release of electrons from dust particles was not considered, because the authors of [3] stated that the dust discharging rate is defined by the work function of the nanoparticle material (4 5 oV for carbonaceous dust). On the other hand, it was suggested in [6, 7] that the electrons are kept on the nanoparticle surface by a self-induced short-range polarization potential and hence the electron binding energy is much less than 1 oV and the characteristic time of dust discharging is of the order of 1 //.s. This allows assuming that the electrons adsorbed on dust can be released quickly and the dust floating potential can vary rapidly enough to be in equilibrium with the local plasma environment even during the early afterglow.
A promising approach is to study the afterglow by-direct simulation of ion and electron motion, in particular, by the Particles In Cells Monte Carlo Collisions (PIC MCC) method, which is applicable for plasma with a large Havnes parameter. In this paper, the afterglow of rf discharge in heavy dusted plasma is considered using the PIC MCC method, with the simplifying assumption that the dust particle floating potential is in equilibrium with the plasma environment and at each instant is defined by the equality of electron and
ion currents to the particle surface. The kinetics of metastable argon atoms is also taken into consideration.
2. DUSTY DISCHARGE MODEL
The rf discharge and the afterglow plasma in argon are simulated by the conventional PIC MCC technique. The PIC MCC model of high-frequency discharge dusty plasma is based on the electron and ion kinetic equations, including their interaction with dust nanoparticles and the Poisson equation. In this model, the velocity distribution functions of electrons, fc(t,x,v), and ions, /.¡(t.x.v), in the approximation of three-dimensional velocity space and one-dimensional real space are obtained by numerically solving the Boltzmann equations
at
dx
II), <)v,
= Jr
II, (/..'•) = I h
(1)
dv.
and
dh at
an .E an
+ V«-r£ + —TT1 = Ji,
OX II), OV.j
>i,(!■■>■) = j fidvi,
(2)
where r,. <■,. ii, . n,. and id, . id, are respectively the electron and ion velocities, densities, and masses, E is the electric field, and Jc and J, are the collision integrals for electrons and ions, which include elastic and inelastic collisions with the background gas and negatively charged dust particles.
The electron and ion kinetics in the PIC MCC model is simulated with test particles with weight W", which is the number of electrons or ions represented by this test particle. The equations of motion are integrated for electrons and ions in the discharge electric field accounting for the collisions with neutral atoms and negatively charged dust. The probabilities of elastic and inelastic scattering events during an electron/ion time step are set by the cross sections depending on the test particle energy. If an ionization event occurs, the new electron and ion test particles with the same coordinates and weights are introduced. The test particle leaving the calculation area through the electrodes or captured by dust is excluded from the simulation. The interelectrode gap is covered with a calculation grid (one-dimensional in this case). The macroscopic characteristics of the plasma such as the
electron and ion average energies nc and n., are calculated by integrating the electron and ion distribution functions over the calculation grid nodes. The electric field in plasma is calculated at each time step from the Poisson equation
A<j> = 4tt6 [nc — n.j + Zd(x)nd(x)]
^ dx' ^
where nc and n, are obtained from the PIC MCC calculation. The boundary conditions are the sinusoidal applied voltage on one electrode and zero voltage on the other, grounded electrode. The space charge of the dust particles with the density nd(x) and the mean particle charge Zd(x), depending on the spatial coordinate, are included. The particle charge and density-are found self-consistently during the simulation, as described below.
The cross sections for electron argon reactions are taken from [8]. For argon ions, the resonant charge exchange is taken into account. The influence of dust in this model is included similarly to how this was done in the models in [9 11]. In the kinetic simulation, the cross sections of electron and ion capture by dust particles are included, which are taken from the Orbital Motion Limit (OML) theory:
fed —
7rr|(l + eUd/e 0,
> <
-cUd, -eUd,
(¿id = -eUd/£i),
where Ud = —eZd/rd is the (negative) floating potential of the dust particle surface gained in the discharge plasma, ec and e, are the kinetic energy of electrons and ions, rd is the radius of the dust particle, Zd is the particle charge number, and e is the elementary charge. The value of the floating potential Ud must provide equal electron and ion fluxes to the dust surface, thus making the particles charge steady. We used two methods of finding the Ud in the discharge simulation. The first was to control the number of ions and electrons collected by dust according to the mentioned collision cross sections and adjust the value of Ud to make them equal. The other method, more efficient for simulation of afterglow with rapidly changing plasma parameters, was to use analytic expressions for the electron and ion fluxes onto a dust particle, derived from the OML theory (see, e.g., [12]):
I, = -ii, fd<'r. ex11
Ii = 7nvrJeTi I 1 ■
UTc)
" m)
where nc and n, are local electron and ion densities, Tc and Tj are their temperatures, all taken from the PIC MCC simulation, k is the Boltzmann constant, and vtc and tare thermal velocities, taken as vT = s/sWJ ->n. Because the particle charge must be in equilibrium, both fluxes are equal and the equation I(: = I.j is solved by the Newton Raphson method to obtain the established I'd for each calculation grid cell. Also, during the afterglow simulation, the numbers of ions and electrons collected by dust are checked to verify the validity of Ud• This shows good applicability of such a method for calculating Ud-
The transport equation for the nanoparticlo density nd(x). including their diffusion and drift due to the electric field and ion drag force, is solved to obtain the equilibrium dust density profile:
(bid ~df
jL
dx
Dd
dtid dx
Vd(x)nd(x)
(5)
where Dd is the diffusivity of nanoparticlos and Vd(x) is their local average velocity, determined by the electrostatic force, ion drag force, and gas friction. The details of the dust transport model are described in [13,14].
The obtained space charge of nanoparticlos, Zdiid-is included in the Poisson equation to find the electric field in plasma.
In this paper, to take the effect of metastable states of argon atoms into account, the equation of metastable atoms kinetics is added to the described model.
3. METASTABLE ATOM KINETICS
Simultaneously with the discharge kinetics simulation, the balance equation for metastable argon atoms Ar* is solved. The model of metastable atom kinetics is simplified. Because argon atoms have two metastable levels with close excitation energies, they are regarded as one state with the energy offset 11.5 eV. The cross sections of the argon atom excitation to the metastable state were taken from experimental measurements [15].
We include the following main processes into the balance
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