Charging in a dusty plasma with a size distribution: a comparison of three models

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Adv. Space Res. Vol. 29, No. 9, pp. 1283-1288.2002 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0273-l 177102$22.00 + 0.00 PII: SO273-I 177(02)00067-4

CHARGING IN A DUSTY PLASMA WITH A SIZE DISTRIBUTION: A COMPARISON OF THREE MODELS L. Bringol-Barge and T. W. Hyde CASPER (Center for Astrophysics, Space Physics & Engineering Research) Baylor University, P. 0. Box 97316, Waco, TX 76798-7316 Tel:254- 71 O-251 1, [email protected]

ABSTRACT

There has recently been a renewed interest in the charging of dust grains in a plasma and its effect not only on dust dynamics in astrophysical and solar system environments but also in laboratory plasma processing and semiconductor manufacturing. The calculation of individual dust grain charges in a dust cloud with a size distribution is a challenging one, especially under sufficiently dense cloud conditions and moderately high plasma temperatures when secondary electron currents become significant. Both the capacitance of the dust cloud and the significant charging currents must be taken into account. Three models for calculating individual dust grain charges are compared. The first model considers the dust cloud to consist of isolated grains immersed in plasma. The second model uses a uniform potential for the dust cloud and thus finds the charges using an appropriate capacitance for the cloud. The third model assumes a nonuniform grain surface potential and uses a solution to Poisson’s equation to calculate the grain charges. The models are applied to a dense dusty plasma environment, appropriate for Saturn’s F-ring. The implications for coagulation in presolar environments are also discussed. 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved.

INTRODUCTION

There are many areas in which dusty plasma effects are observed in laboratory settings thus motivating computational modeling specific to this environment. The recent surge in interest in Coulomb lattice structures produced in dense laboratory dusty plasmas and predicted by computational models creates a need to better understand the fundamental charging process. Many dusty plasma models, such as the box-tree code (Richardson, 1993) require grain charge as input for the program, and the high grain densities studied necessitates an accurate calculation of grain charge. As grain size distributions are being considered more widely among researchers studying computational models of dusty plasmas (especially with close-packing of grains), a proper treatment of grain charging becomes crucial. Also, it has recently been discovered experimentally that dust grains can cause serious problems in the manufacturing of semiconductor material during plasma processing. In order to develop methods by which to remove such contaminants from the system, a proper understanding of grain charging becomes essential. As plasmas are generated in laboratory settings for the purpose of studying grain charging and effects, plasma sheaths exist on all cold boundaries with which the plasma is in contact. It has been shown (Chutov et al., 1998) that there is a strong mutual influence between dust grains and the sheath plasma, and again the determination of the grain charges is important in interpreting these laboratory measurements. Thus, the calculation of the charge residing on a single grain that exists as part of a dusty plasma ensemble is essential to understanding macroscopic effects in dusty plasmas. Considerable work has been done in calculating equilibrium potentials for grains existing in cloud formations with specified size distributions (Houpis and Whipple, 1987), and it is well-accepted that astrophysical processes in space seem to produce power-law distributions of dust grain sizes in clouds (Bums and Showalter, 1984). The elements that complicate the calculation of grain charge include intergrain effects as well as the nature of the charging currents involved. There has been extensive research into the conditions by which oppositely charged grains can exist in a dusty plasma (Horanyi and Goertz, 1990; Chow et al., 1990). The possibility of the existence of dust grains with opposite

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signs of charge in a cloud is important in a number of areas, especially as to their influence on coagulation rates in the protostellar environment. HorClnyiand Goertz (1990) found that one effective method for producing opposite charges on dust grains in a cloud is to introduce a temperature fluctuation. Such a change in temperature would cause the charging rates of the grains to vary significantly between the small grains and large grains due to the effect of the secondary-electron charging current. The model presented in this paper shows that opposite signs of grain charge are possible in a dusty plasma under quasi steady-state conditions, without the need for a plasma temperature fluctuation. As mentioned, any condition under which the coagulation of dust particles can be enhanced is of great interest, especially when studying the presolar nebula, protoplanetary disks and cometary environments (Goertz, 1989; Northrop, 1993). If the grains have the same charge sign, coagulation is decreased and can perhaps be halted altogether (Simpson et al., 1979). However, the existence of both positive and negative grain charges in nontransient plasma environments has not until now been demonstrated and is therefore of great interest to researchers studying grain accretion mechanisms. CHARGING MODELS Isolated Grain Model The simplest charging model that can be applied to a dusty plasma is one in which the grains are considered to be isolated from one another; that is, the influence on the charging of one grain by the other grains in the cloud is ignored. This model is satisfactory for a dust cloud in which the Debye spheres (spherical volume around the dust particles outside which the grain charge is effectively screened out by the plasma) of the grains do not overlap. This occurs when the Debye length (radius of the Debye sphere) is less than the intergrain distance in the cloud. This model is, however, still commonly used for denser dust clouds where this requirement is not met and is thus included in this paper for comparison purposes. The conditions for equilibrium in a dust cloud using the isolated grain model are as follows. The net current to the grain surface is zero. The currents considered in this work apply to a dusty plasma 1) removed from radiative sources. Thus only the primary ion, primary electron and secondary electron currents are considered. The forms for the currents to a single grain of radius a and surface potential U immersed in a plasma of species charge Z, species mass m, temperature kT, density n and potential V are well known and stated without proof (Havnes et al., 1987). Primarv Ion (i=i) and Primary Electron (i=e) Currents: 1-1

eZ.U klj

ex

’

zju

10

(1)

zju20

Secondarv Electron Current: J,, =el?z.na’-

z(kL)-2

exp(~).l;ES(E,a)exp(-~)dE, e

J,, = ene-m2.

U5 0

(24

c

U20

(2b)

where ~(E,Q) is the Jonker form for the secondary electron yield, adapted to a spherical grain geometry (Jonker, 1954; Chow et al., 1993) The strong size-dependent nature of the secondary-electron yield is shown in Bringol and Hyde (1997) with the double-maximum feature corresponding to the semiinfinite slab peak and the grainpenetration peak. The derivation of the secondary-electron current along with the definitions of the parameters can be found in Chow et al. (1993). The condition for a vanishing net current to the erain can be written as

where the summation is taken over all the currents (1): (2a) and (2b).

Charging

2)

in a Dusty Plasma: Comparison

of Three Models

1285

Charge is not depleted from the plasma. Therefore there is no charge conservation equation.

Plasma number density The simplest way of considering charging in a dusty plasma is to treat the plasma as being a non-depletable source. Thus, as the grains charge, no plasma particles are depleted, and the plasma potential Vremains constant at 0. This implies that ?z,= n,. (4) Using equations (3) and (4), the surface potential U is found for each grain size. The grain charge is subsequently found by using the isolated grain capacitance (Whipple, et al., 1985) Cl:

Q,(a) = C,(u)17 = u(l+ ka)U

(5)

where k is the reciprocal of the Debye length. Uniform Scale Potential Model The uniform scale potential model assumes a constant grain surface potential throughout the cloud. Since there can be currents which differentiate among grain sizes (such as the secondary-electron current), it is not generally expected that each grain would achieve the same surface potential. Thus, a scale potential cis introduced which is the surface potential achieved by a grain size between rminand r,,,, which satisfies the charging equations for the cloud. The conditions for equilibrium in a dust cloud using a scale potential are as follows. The net integrated current to the dust cloud with uniform scale potential c is zero: 1) ~I,(8,V)=O, n

(6)

Z”(U,V) = n-‘J.J&,LI,V)f(a)da

(7)

where I,, is defined to be the integrated current

with the size distribution function f(a) defined by ii = j-f (u)du = j,r

Cu-“da,

(8)

where i is the dust number density and C is a constant. Thus instead of considering the net current to each grain to be zero, the net current to the dust cloud is now taken to be zero within a scale length z. The scale length z is the representative size of a volume of the dust cloud, inside which a statistically significant amount of the dust representing the size distribution can be found. The plasma potential is allowed to vary in this model, thus causing the number densities of the electrons and ions to differ. A Maxwellian distribution for the plasma species is assumed, and the number densities can be expressed as (Havnes et al., 1990): nj = noj ex

ziev kl;

(9) . pi 1 The total charge is conserved in the dusty plasma. That is, the charges that are depleted from the plasma 2) are collected by the dust grains. If the dusty plasma is considered to be charge neutral in the absence of dust, the charge conservation equation can be written as

Qi + Qe + Qdc= V,(i&ni - me) + &jr;;

Q(4fW

- da = 0 ,

w

where Qi, Qc and Q~care the total charge contributed by the plasma ions, plasma electrons and the charges on the grains in the dust cloud, respectively. The volume occupied by the plasma in the dusty plasma is defined to be V, while the total volume occupied by the dusty plasma is V,,,,. The parameters rmi, and r,,,, are the minimum and maximum grain radii, respectively. The function Q(U) is the charge attained by a grain of radius a, and can be written in terms of the scale potential and the capacitance of a dust cloud with a size distribution and a uniform potential (the reader is referred to Houpis and Whipple (1987) for a derivation of the capacitance):

Q&O = ‘W>fl

(11)

L. Bringol-Barge

1286

and T. W. Hyde

Thus, the charge conservation equation can be writtenas Qi+Qe+Q&=

v,(~en,(V)-en(V))+~~~~S~-C,(~)f(~).da=O, ‘mi”

(W

The charging equations (6) and (12) can be solved for u and V, and the charges can subsequently be found using equation (11). Nonuniform

Potential Model

The nonuniform potential model allows the grain surface potential to vary from grain to grain throughout the cloud, as is expected for a dust cloud immersed in a moderate- to high-temperature plasma. It includes a method by which grain charge can be calculated given a nonunform surface potential following Bringol-Barge and Hyde (2002a). This is the charging model which makes the fewest assumptions and thus is the most genera1 when applied to various dusty plasma environments. The conditions for equilibrium in a dust cloud using a nonuniform potential are as follows. The absolute value of the net integrated current to the dust cloud with a nonuniform potential U(a) is a 1) minimum:

Ix

Z,(U(a), V) = minimum.

In

(13)

I

For a dust cloud with a specified size distribution, the notion of current conservation is much more difficult. In general, smaller grains can have a larger secondary-electron current than will grains of larger radii. Therefore streaming currents may exist between the grains that cause small fluctuations in grain charge. In this case, instead of requiring the current to each individual grain to vanish, we require it to be minimized within a given length scale z. The dusty plasma will not be in a strict equilibrium but rather in a “pseudo-equilibrium” defined by a state of minimum fluctuations in the grain charges. The total charge is conserved in the dusty plasma. Recasting equation (10) using the assumption of a 2) nonuniform surface potential U(a) , we present the condition for charge conservation (14) where

Q,,,(a) is the

charge on a grain of radius a, given the grain surface potential profile U(a)

in the dust

cloud. Equations (13) and (14) are used to,calculate the potentials and charges at the “pseudo-equilibrium.” complete derivation of this model is given in Bringol-Barge and Hyde (2002b). APPLICATION

A more

TO SATURN’S F RING

The results of the three charging models are compared below for two dense dusty plasma environments. The plasma and dust parameters at Saturn’s F ring, though not directly detected, can be inferred from magnetospheric models. A range of parameters are given by Goertz (1989), and we use dust and plasma number densities along with plasma temperatures that are close to these values for our examples. For Case A, we use a low hydrogen plasma temperature of 10 eV along with a dense dusty plasma condition of no = 1000 cmm3and <=I00 cm”. The power-law distribution parameter S is taken to be 1, while the minimum and maximum radii are assumed to be 0.01 micron and 1.0 micron, respectively. For Case B, all the parameters are the same except that the plasma temperature is 25 eV and S is 2. As can be seen in Figure 1, for the 10 eV plasma temperature environment, the isolated grain model predicts charges that are monotonically increasing toward negative charge as the grain size is increased. This is an expected result contributed by two factors: the decreased secondary-electron yield and increased charging cross-section for larger grains. The uniform potential model predicts very small negative values for grain charge. This can be understood in the following way. Since there is a higher proportion of small grains in the cloud, the scale potential will be skewed toward the actual potential achieved by the small grains, which is very small due to the high flux of secondary electrons for small grains. Thus the uniform potential would tend to agree with the nonuniform potential model only for the small grains. Examining the results for the nonuniform potential model, it is seen that all grains achieve a negative charge. The shape of the curve resembles that of the isolated grain model, although at more positive charge.

Charging

in a Dusty Plasma: Comparison

of Three Models

1287

In Figure 2, both the isolated grain model and the uniform potential model predict only positive charges for all grain sizes. Since the isolated grain model considers the charging of a single grain only, it is understood that at a plasma temperature of 25 eV, the secondary-electron yield is sufficiently high for all grain sizes to achieve positive charge. As well, the uniform potential model predicts a surface potential consistent with the smaller grains, and thus the grain charge increases with radius. The one feature present in the nonuniform potential model which is absent in the others is the charge attained by the larger grains in the cloud. When the potential is allowed to vary from grain to grain, it is seen that the larger grains do not achieve a positive charge, due to the lack of primary electrons with sufficient energy to cause large numbers of secondaries to be emitted from its surface. Thus, it is apparent that neither the uniform potential model nor the isolated grain model allows for the presence of both positively and negatively charged grains coexisting in the dusty plasma except under unusual conditions. The nonuniform potential model is the only one to predict the opposite charges that can be attained at “pseudo-equilibrium.” 1 .OOE-06

1 .OOE-05

1 .OOE-04

1 .OOE-06

1 .OOE-05

1 .OOE-04 1000 500 0 -500 -1000

1

kT=25eV

“1

-1500 -2000

Grain Radius (cm)

Fig. 1. Grain charges predicted by the three models for the dusty plasma parameters for Case A (see text). The curves represent the grain charges calculated using the a) isolated grain model, b) uniform potential model, and c) nonuniform potential model.

Grain Radius (an)

Fig. 2. Grain charges predicted by the three models for the dusty plasma parameters for Case B (see text). The curves are labeled in the same manner as in Figure I.

IMPLICATIONS OF MODEL RESULTS TO F RING GRAVITOELECTRODYNAMICS Studies of the magneto-gravitational resonance of charged grains in Saturn’s F ring with nearby satellites have been performed by Mendis et al. (1982). It was stated that the electron number density would have to become extremely low before the grain surface potential would achieve significant positive values on the sunlit side of the planet due to photoelectric charging. Thus, it was concluded that the grains in the F ring are negatively charged. It has also been pointed out that due to the high density of micron- and submicron-sized dust in the F ring, there is a low probability that a grain would attain even a single electronic charge (Griin, et al., 1984). However, it can be seen from the data generated by the nonuniform potential charging model that, even at moderate temperatures, opposite charging of grains is possible without the necessity of photoelectric charging. As a result, the coupling of grains to satellites in a magneto-gravitational resonance thus producing macroscopic effects (such as might contribute to waves, kinks and braids) is now a possibility under non-radiative, non-transient plasma environments. The inclusion of the photoelectric charging process in this model would only increase the positive charge attained by the smallest grains. IMPLICATIONS OF MODEL RESULTS TO COAGULATION There have been many mechanisms proposed to increase the coagulation rates of grains in presolar environments. The presence of grains immersed in a plasma in which there is a temperature fluctuation has been studied by Hon!myi and Goertz (1990), and they found that oppositely charged grains can exist with the inclusion of the secondary-electron current. In comparison with their charging study during temperature fluctuations there are interesting differences raised by the model given above. Hotinyi and Goertz (1990) assumed the Stemglass form for secondary-electron yield (for a planar geometry) while we consider the Jonker form adapted for a spherical grain geometry. Their results show that the larger grains (- 6 pm) attain a positive charge due to their larger charging cross-section while the smaller grains (- 1 pm) maintain a negative charge. We would expect the results to be similar for temperature fluctuations assuming the Jonker yield, since the sharp peak in the yield function is

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L. Bringol-Barge

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significant only for submicron grains. An interesting investigation would be into temperature fluctuations for a dust cloud with a size distribution ranging from 10 pm down into the submicron regime using the Jonker yield. One might expect that, under certain plasma conditions, a fluctuation in plasma temperature would cause both the largest grains and the smallest grains to maintain a positive charge while the grains on the order of 1 pm acquire a negative charge. Dusty plasma environments satisfying the above requirements could create an increase in the coagulation rate, as discussed earlier. It is clearly seen that there is a mechanism by which dust grains in a plasma may achieve opposite signs of charge without the necessity of a temperature fluctuation. Although we limited our discussion to the secondaryelectron current as a means for achieving opposite charges in a quasi steady-state dusty plasma, there are many other currents which have a size-dependence which causes the equilibrium potential to be differentiated among grain sizes. An interesting extension to this work would be to include such currents as the secondary-electron yield under ion bombardment, the photoemission current, and field emission current, all of which can enhance the possibility of positively charged grains. ACKNOWLEDGMENTS This work was partially funded through an NSF grant, a Department of Education Grant and the Baylor Research Committee. REFERENCES Bringol, L. A., and T. W. Hyde, Charging in a Dusty Plasma, A&. Space Res., 20, 1539-1542, 1997. Bringol-Barge, L., and T. W. Hyde, The Calculation of Grain Charge in a Dense Dusty Plasma with a Nonuniform Surface Potential, Adv. Space Res., this issue, 2002. Bringol-Barge, L., and T. W. Hyde, A Charging Model for a Dust Cloud with a Size Distribution and a Nonuniform Potential, Adv. Space Res., this issue, 2002. Burns, J. A., and M. R. Showalter, The Ethereal rings of Jupiter and Saturn, in Planetarv Rings, eds. R. Greenberg and A. Brahic, University of Arizona Press, Tuscan, AZ, 1984. Chow, V. W., D. A. Mendis, and M. Rosenberg, Role of Grain Size and Particle Velocity Distribution in Secondary Electron Emission in Space Plasmas, J. Geophys. Res., 98, 19065-l 9076, 1993. Chutov, Y. I., 0. Y. Kravchenko, and V. S. Yakovetsky, Non-linear Sheaths with Dust Particles, in Strongly Couoled Coulomb Svstems, eds. G. J. Kalman, J. M. Rommel and K. Blagoev, Plenum Press, New York, NY, 1998. Havnes, O., T. K. Aanesen, and F. Melandso, On Dust Charges and Plasma Potentials in a Dusty Plasma with a Dust Size Distribution, J. Geophys. Res., 95,6581-6585, 1990. Havnes, O., C. K. Goertz, G. E. Motftll, E. Griin, and W-H. Ip, Dust Charges, Cloud Potential, and Instabilities in a Dust Cloud Embedded in a Plasma, J. Geophys. Res., 92,228 l-2287,1987. Horanyi, M., and C.K. Goertz, Coagulation of Dust Particles in a Plasma, Asfrophys. J., 361, 155-161, 1990. Houpis, H., and E. Whipple, Electrostatic Charge on a Dust Size Distribution in a Plasma, J. Geophys. Res., 92, 12057- 12068, 1987. Goertz, C. K., Dusty Plasmas in the Solar System, Rev. Geophys., 27,271-292, 1989. Grtin, E., G. Morfill and D. A. Mendis, Dust-Magnetosphere Interactions, in Planetarv Rings, eds. R. .Greenberg and A. Brahic, University of Arizona Press, Tuscan, AZ, 1984. Jonker, J. H., On the Theory of Secondary Electron Emission, Phi&s Res. Rep., 7, l-20, 1952. Mendis, D. A., H. Houpis, and J. R. Hill, The Gravito-Electrodynamics of Charged Dust in Planetary Magnetospheres, J. Geophys. Res., 87,3449-3455, 1982. Northrop, T. G., Dusty Plasmas, Phys. Ser., 45,475-490, 1993. Richardson, D. C., Planetesimal Dynamics, Dissertation University of Cambridge, 1993. Simpson, J. C., S. Simons, and I. P. Williams, Thermal Coagulation of Charged Grains in Dense Clouds, Asfrophys. Space Sci., 61,65-80, 1979.