Expansion of a dusty plasma into a vacuum: effects of charge nonneutrality

Planet. Space Sci., Vol. 45, No. 2, pp. 2514254, 1997 0 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0032-0633/97 $17.00+0.00

Pergamon

PII: S0032-0633(96)00080-3

Expansion of a dusty plasma into a vacuum: effects of charge nonneutrality Yasser El-Zein,’

Seungjun Yi’ and Karl E. Lonngren’p2

‘Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, IA 52242, U.S.A. *Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242, U.S.A. keceived 20 March 1996; revised 22 May 1996; accepted 22 May 1996

1. Introduction

Acertaining the properties of a quasi-charge-neutral

u’ust~~

plasma that consists of electrons, positive ions, and mass-

ive dust particles to which a significant number of electrons have been attached is of current interest. Due to its importance in the study of the space environment, such as asteroid zones, planetary rings, cometary tails as well as the lower ionosphere of the Earth and in plasma processing applications, there has been an extensive effort to gain an understanding of this plasma. The dust grains could be of micrometer or submicrometer size and can be considered to be very massive negatively charged particles in the background plasma. The negative charging of the dust particles is due to field emission, ultraviolet ray irradiation, plasma currents, electron attachment, etc. In addition to studying just the dynamics of the dust particles in the plasma such as their creation, trajectory, impact and fragmentation characteristics, there has been an effort to obtain an understanding of collective interaction of the dust particles with the ambient plasma. Recently, Rao et al. (1990) examined the long-wavelength low-frequency collective oscillations in a dusty plasma in which the dust particle dynamics is crucial

Correspondence to: K. E. Lonngren

rather than having modes that are just affected by the dust. They studied the collective motion of the negatively4 charged dust particles in a background of hot electrons and hot positive ions that are in thermodynamic equilibrium. The masses of the latter two species can be neglected with respect to the dust species. They found that a new type of sound wave that they called a dust acoustic wave could appear. In their analysis, they assumed that the motion of the dust particles would be described by nonlinear equations of continuity and motion. In their model, they assumed that both the electrons and the positive ions were in thermodynamic equilibrium and could be described with Boltzmann distributions. A self-consistent potential is determined from Poisson’s equation. Charge separation between the three species leads to possible dispersive effects. Since both nonlinearity and dispersion were included in the model, they demonstrated that dust acoustic solitons of either positive or negative electrostatic potential could propagate in the dusty plasma. Verheest (1992) examined certain intrinsic mathematical constraints found in the model. Linear dust acoustic waves have recently been detected in a laboratory experiment by Barkan et al. (1995) in a Q machine plasma. In this experiment, the frequency of oscillation of the dust acoustic wave was of the order of tens of Hertz in the same device that had been used to study the ion acoustic wave that had a frequency of oscillation that was of the order of several tens of kilo-Hertz. D’Angelo (1995) has shown that a low frequency oscillation cf~ 12Hz, 2x0.5 cm) that was detected in a dusty plasma that was performed in a discharge device by Chu et al. (1994) could be interpreted- in terms of the dust acoustic wave. In addition to studying the linear and nonlinear properties of waves in a dusty plasma, there is interest in studying the characteristics of the expansion of a plasma that contain dust grains into a vacuum. It has been shown that in cases where charge neutrality could be enforced, this nonlinear expansion could be described with a self-similar

Y. El-Zein et al.: Expansion of a dusty plasma into a vacuum

252 solution. In this case Poisson’s equation can be neglected with the potential being obtained from the Boltzmann assumption for the lighter particles. Solutions of the nonlinear set of fluid equations that describe the dusty plasma have been obtained (Lonngren, 1990 ; Luo and Yu, 1992 ; Yu and Luo, 1992 ; Yu and Bharuthram. 1994 ; Bharuthram and Rao, 1995 ; Rao and Bharuthram, 1995). The assumption of charge neutrality that is made in obtaining a self-similar solution does, however, impose restrictions on the charge separation that is allowed in the analysis. In this paper, we study the nonlinear expansion of a dusty plasma into a vacuum using a numerical technique. Our results will demonstrate that in addition to the dominant expansion of the dusty plasma into a vacuum that has been previously studied, there will be an acceleration of a small component of dust particles. The acceleration mechanism for the dust particles is due to the electric field set up by the electrons and positive ions that initially leave the dusty plasma before the more massive dust particles have a chance to move. The acceleration of positive ions in the expansion of a two-component plasma due to the departure of the electrons has been predicted by Crow et al. (1975) and observed in an experiment (Chan et al., 1984). The acceleration of positive and negative ions in a three-component plasma has also been noted in a recent simulation (El-Zein et al., 1995). In Section 2, we describe the model that is used to describe the dusty plasma. Section 3 reports the results of the simulation and contains the concluding comments.

2. Model We impose restrictions on our results by using a model for the dust grains that assumes the dust particles must have a uniform charge, mass and size and that they neither break up nor coalesce in the expansion. We use the following normalizations. The normalized space X, velocity U, time T,and potential 4 variables are

(1) The normalized densities are

electron

N,, positive

The dust particles are described linear equations of continuity

ion N,, and dust Nd

by the normalized

non-

aw,cr) stx=o and motion (4) The parameter

Z refers to the number

of electrons

that

reside on a dust particle. We chose Z = 5. In this simulation, the ratio (Z/M) is assumed to be constant. In equation (4) the dust particles have been assumed to be cold. The ion and electron number densities (N, and NJ are given by the Boltzmann distributions Ni=exp

(-$$J

N, = N,, exp(4) = [I- 4exp(4) where Ni,, N,,, normalized ion electron number temperature. In dust density N,,

(6)

T,and T, are the respective equilibrium number density, equilibrium normalized density, ion temperature and electron the simulation, we chose TJT,= 4.The is Nd = EZ.

(7)

Poisson’s equation is used to determine the electric potential 4 in terms of the densities of the three species that exist in the plasma in the region behind the front xF of the expanding dust particles

2 =[N,+N,-NJ. In the region x > xF, Poisson’s ~

equation

= [N,-Ni]

(8) becomes (9)

since there are no dust particles ahead of the front. The term (EZ) that appears in equations (6) and (7) is the normalized charge that is on the dust. It is restricted to the values (EZ)I 1 and it will be the parameter that we will use to emphasize the effects due to the dust concentration. In the simulations to be described below, we will choose two values for this parameter: (a) all electrons have been attached to the dust particles (EZ) = 1 ; (b) free electrons exist in the plasma along with those that have been attached to the dust particles (EZ) = 0.5. The initial plasma at T = 0 in the region - cc 5. As a check, different values of potential were assumed at this far boundary and no quantitative nor qualitative differences were observed. The momentum equation then advances the dust velocity in time using the new potential. The equation of continuity is then used to advance the dust density in time using the updated velocity. Given the new dust density, Poisson’s equation is again solved by relaxing the

253

Y. El-Zein et (11.:Expansion of a dusty plasma into a vacuum

0

distance

5

0

distance

5

Fig. 1. Expansion of a dusty plasma into a vacuum. The plasma contains no free electrons. The temporal interval between traces isAT=

Fig. 2. Expansion of a dusty plasma into a vacuum. The plasma contains free electrons. The temporal interval between traces is AT=5

potential and the free electron and positive ion density. This procedure is iterated, advancing the plasma quantities in time. In this simulation, the number of particles is not conserved. An infinite reservoir of all particles exists in the region X50. The Crank-Nicholson method was used to integrate the velocities and densities at every point. The values at this point were calculated from a fourthorder polynomial fit of the preceding variables. The code that we used was a modification of one that we had developed earlier (El-Zein et al.. 1995).

them in a self-similar analysis which requires the absence of any scale length in a problem. As noted above, the local acceleration of positive ions has been numerically and experimentally noted previously in a two-component plasma due to the departure of the Boltzmann electrons (Crow et al., 1975 : Chan et al., 1984). The acceleration of positive and negative ions in a threecomponent plasma was also noted in a recent simulation (El-Zein et al., 1995). The two species of ions were accelerated in opposite directions by the electric field created by the sudden departure of the Boltzmann electrons. The results of the simulation of a dusty plasma in which there are free electrons in addition to those that are attached to the dust particles are shown in Fig. 2. In this case, the parameter (~2) = 0.5. The global expansion characteristics of the three components is similar to those obtained in the recent study of the self-similar expansion of a dusty plasma (Bharuthram and Rao, 1995). For the results shown in their Fig. 2, they assumed that the free electron density was (EZ) = 0.1. In the simulation results that we are presenting, the free Boltzmann electrons will neutralize a fraction of the Boltzmann positive ions in the vacuum region X> 0. These are the charged particles that have been expelled before the massive dust particles have had a chance to move. This charge neutralization will reduce the electric field that is available to accelerate the dust particles. Since we are assuming that the plasma can be considered to be onedimensional, the accelerating electric field will be proportional to the net charge that has been expelled at T = O+.With our choice of (EZ) = 0.5, this reduction will be by a factor of two. This reduction is noted when we compare the simulation results in Fig. 1 where there are no free electrons and in Fig. 2 where free electrons are present. In conclusion, we note that dust particles will expand into a vacuum and the gross features of the expansion can be interpreted with a quasi-neutral dusty plasma fluid model. However, a small group of dust particles will be accelerated to a high velocity by the instantaneous expulsion of Boltzmann free electrons and positive ions. It would be interesting to extend this type of investigation to the case where the dust charge to mass ratio (Z/M)

3. Results and conclusions The results of the simulation of a dusty plasma in which all of the electrons are attached to the dust particles are shown in Fig. 1. In this case, the parameter (EZ) = 1 and there are no free electrons. This is the case that admits a self-similar solution for the expansion of a quasi-neutral plasma (Lonngren, 1990 ; Luo and Yu, 1992 ; Yu and Luo, 1992 ; Yu and Bharuthram. 1994; Bharuthram and Rao, 1995 ; Rao and Bharuthram, 1995). The density of the free positive ions expands into the background vacuum region at the initial time increment T = O+.During this time, the massive dust particles have not moved. As time increases, the dust density adjacent to the constant reservoir at XI 0 starts to expand into the vacuum region X>O. Positive ions from this reservoir will follow the expanding negatively charged dust particles. Close to the reservoir, it appears that the two species expand together as a quasi-neutral fluid that can and has been treated with the self-similar mathematics. We note a small group of negatively charged dust particles at the front of the expanding dust density that are not neutralized by the positive ion fluid. This group of ions is not predicted by the self-similar solution. The velocity of the fastest dust particles in this accelerated burst is greater than the dust acoustic velocity. We interpret these ions as being accelerated by the electric field in the region X> 0. This electric field is created by the positive ions that left the plasma at the time T = O+ before the heavy dust particles could move and now reside in the vacuum region at X+m. Since the size of this local dust density perturbation is finite, we should not expect to see

254 was not assumed distribution.

Y. El-Zein et al.: Expansion to be a constant

but

had

a realistic

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of a dusty plasma into a vacuum

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