Frontiers in Dusty Plasmas : Proceedings of the Second International Conference on the Physics of Dusty Plasmas

Frontiers in Dusty Plasmas

This Page Intentionally Left Blank

Proceedings of the Second International Conference on the Physics of Dusty Plasmas ICPDP-99

Frontiers in Dusty Plasmas Hal > 1 and the inter-grain spacing is of the order of XD^ charged dust microspheres strongly interact with each other, and we have the possibility of forming Coulomb lattices in a strongly coupled dusty plasma. According to Ikezi[4], the critical value of F for the Coulomb crystallization of charged dust grains is about 170. Strongly coupled plasmas are also found in a highly evolved star, in a white dwarf, in planetary rings (narrow rings of Uranus, incomplete rings of Neptune, etc.), in the Jovian interior, in laser implosion experiments, as well as in colloidal systems. In this paper, we highlight the present status of collective processes in dusty plasmas. We focus our attention on waves, instabilities, and nonlinear structures in weakly and strongly coupled dusty plasmas. The consequence of the dusty plasma wave spectra with regard to the Coulomb crystallization is discussed. Furthermore,

RK. Shukla /Perspectives of collective processes in dusty plasmas

5

we enlighten the physics of soUtons, shocks, and vortices in dusty plasmas. It is remarkable that both the dust ion-acoustic shocks and vortices are observed in laboratory dusty plasma devices.

2

D u s t y p l a s m a waves

We consider a multi-component dusty plasma composed of electrons, singly charged positive ions, and extremely massive negatively charged dust grains, in a neutral background. The dust grain radius R is usually much smaller than the dusty plasma Debye radius A^. When the intergrain spacing d is much smaller than \D, the charged dust particulates can be treated as massive point particles similar to multiply charged negative (or positive) ions in a multi-species plasma. On the other hand, for d < \D the effect of neighboring particles can be significant, whereas for d > > A^) > > J? the grains are completely isolated from its neighbors. The dusty plasma quasineutrality condition for negatively charged dust grains is {rte/rii) + P - 1 = 0, where P = Zdrtd/rii. Generally, the spectra of dusty plasmas waves are obtained by Fourier analyzing the Vlasov, Poisson, and Maxwell equations, supplemented by the dust charging equation. In unmagnetized and weakly coupled dusty plasmas with Maxwellian particle distributions, the electrostatic wave spectra are obtained from the dispersion relation 1+

$Z Xa + Xqde + Xqdi = 0,

(1)

where the dielectric susceptibility is given by[14] X. = ^ [ l

+ ^.Z{^a)]

V2kvta

(2)

Here, k^a — ^pal'^ta^ ^pa the unperturbed plasma frequency of species a, Vta the thermal velocity, Z the standard plasma dispersion function of Pried and Conte, ^a = {^ — kUaQ + ii^an) IV^kvta^ and tx^o the unperturbed streaming velocity. The linear susceptibility Xqds{^ — ^^ i) arising from the dust charge fluctuation[15] produced by electrostatic perturbations is[16] Xqds =

—

/

VCTs [qdO^ V) dv,

(3)

where fso is the unperturbed distribution function of species 5, Vph = u/k the phase velocity, rj = e\Io\ UCTe)~^ + [CTi — eqdo)~ the charge relaxation rate originating from the variation in the effective collision cross-section due to dust charge fluctuations at the grain surface as experienced by the unperturbed particles, leo the equilibrium electron current reaching the grain surface, C the grain capacitance, qdo is the equilibrium grain charge, Te{Ti) the electron (ion) temperature, and (is^e.i the collision cross-section.

6

P.K. Shukla/Perspectives of collective processes in dusty plasmas

Let us first ignore the equilibrium drift, collisions, and the dust charge perturbation [viz. we set -u^o = 0, z/an = 0, and Xqde.qdi = 0 in Eq. (1)]. The dusty plasma wave spectra can then be deduced from 1 + Xe + Xi + Xd — 0. First, we consider the DIAW with Vtd.vu « uo/k « Vte so that Xe ~ V^^'^De? Xi ^ -^lil

(^^ - '^^'^^u) and Xd ~ -^Idl

( Eq. (5) yields u ^ kcda{'^—i^c/'^kcda)^

PK. Shukla/Perspectives of collective processes in dusty plasmas

1

which is the frequency of the damped DAW. Second, for uJ^ « u'^, i/„z/, we obtain LO ^ {k^c^J^n/^) (1 — ikcdaJ^c/^^^n)- Finally, the real and imaginary parts of k are U^

I

2

2 (a;2 + i/2) !_

2

2 c

1/2^

(u^ + uun) +u;^K

(6)

and " ^ " 2 ^ , ( 0 . 2 + ^2)'

^^^

where the wave vector, the wave and coUisional frequencies are normaUzed by tOpd/cda and cjpd, respectively. The properties of dust acoustic waves including dust-dust interactions and correlations have been investigated by Rosenberg and Kalman[19], Murillo[20], de Angelis and Shukla[21], and Kaw and Sen[22]. It is found that dust-dust interactions cause a spatial damping of the DAW. The effect of the plasma boundary[23] on the DIA and DA waves in weakly coupled dusty plasma has also been investigated. The boundary effect modifies the shielding term. For example, for the DIAW we have to replace k'^X^^ ^Y ^De(^^ + JI/RI), whereas for the DAW we must replace k^Xj^ by A|)(fc^+7^/i?o)- Here, 7^ is a root of zero order Bessel function JQ and RQ the radius of the cyhndrical waveguide. For jn » kRo^ the frequencies of the DIA and DAWs are ou ^ XDe^pi7n/Ro{^ + ^l^De/Riy^^ and u ^ \DUJpdlnRol{'^ + ll>^])IRlf'^^ respectively.

3

Dust lattice waves

The wave propagation in crystals is well studied in solid state physics. Here, we discuss the properties of the dust lattice wave (DLW) in a plasma crystal. The DLW arises due to the oscillation of charged microspheres under the action of a screened Coulomb potential which describes the interaction between neighboring dust particulates in the plasma crystal lattice. Using a linear chain model and considering only nearest-neighbor forces, we can write the equation of motion for the dust grains in a dusty plasma crystal as[24, 25] d T dv '^d-Q^ + rudi^dn-^ = OL (rn-1 " ^T^ + r^+i),

(8)

where r^ = A^/a, A^ is the deviation of the dust particles from its equilibrium position, a the mean lattice spacing, and the coupling constant is defined as a = (5^exp(—(^)[1 + (1 + 0^]/^^5 where we have assumed that negatively charged dust grains carry the same charge Qd, and ^ = a/A/). The constant Vdn = 2y/TxpgR?'Cs represents the dust-neutral collision frequency, where pg is the neutral gas density and Cs the thermal speed of the neutral gas atoms. The expression for Vdn holds as long as the mean free path for the gas molecules is much larger than the particle size. Making use of the Bloch condition, (8) can be written as[24, 25]

PK. Shukla/Perspectives of collective processes in dusty plasmas

^

+ i^an^

+ Pism\ka/2)ro

= 0,

(9)

where /3/ = Aa/ruda and k the wavenumber. The subscript 0 denotes the origin. The frequency of the DLW, which is deduced from (9), is given by uj'^ + ivdn^ - (5ism^{ka/2) = 0.

(10)

We note that the hnear dispersion relation (10), which describes the propagation and damping of the dispersive DLW, has been experimentally verified[26, 27]. The nonlinear propagation [24] of finite amplitude DLWs in a coUisional dusty plasma is governed by the modified Boussinesq equation ^

+ ^ . n ^ - % ^ ^ - t ; o ^ - Y ^ a — +

—

^

= 0,

(11)

where rjd represents a kinamtic viscosity, VQ = y/Pia is the phase velocity, u = dro/dz^ and 7z = {GQl/a"^) e x p ( - 0 ( l + ^ + ^V^ + ^V^)- For nonlinear DLW waves with phase velocity close to t;o) (H) ii^ ci moving frame (Z = Z — VQT^ t = T) can be written as the modified Korteweg-de Vries-Burgers equation

Equation (12) admits both rarefactive soliton as well as shocks. The rarefactive dust lattice soliton appear in a coUisionless dusty plasma with Vnd — 0 and ry^ = 0. On the other hand, in a coUisional plasma, (12) predicts monotonic and oscillatory dust lattice shocks. The latter can cause the melting of dust crystals due to the shock heating.

4

Instabilities of dusty plasma waves

The dusty plasma waves, as discussed in section 2, can be excited provided that there exist free energy sources. The latter include streaming beams of charged particles[28, 29, 30, 31], radio-frequency and laser beams, etc. In the following, we present two examples of linear instabilities that could be responsible for the excitation of dust acoustic waves. We first consider the kinetic instability of the DAWs in a coUisionless dusty plasma. Here, for Vtd « ^/k « Vte^vu and \u — kzUio\/k « Vu^ Eq. (1) reduces to 1

.

1

/7r\ V2 ijj - kujo _ (4d

Equation (13) admits an oscillatory instability of the DAW when Uio > (jOr/k » uji, where uOr = kX]:,ujpd/{l + A:^A|,)^/^ and the growth rate is given by

PK. Shukla/Perspectives of collective processes in dusty plasmas

UJI

1

\kUiQ-UJr\

, ..

Second, we focus on the ion-dust two-stream regime, where ^e < < 1? ^ind ^i, Cd » 1- For Ud « uj « Ue and Ui « |a;'|, where uj' = uj — kuio, (1) gives

where A = l + {l/k'^Xj)^).

In the absence of ion-dust colhsions, Eq. (15) for kuio

LO gives ujr '^ uji ^ (uJpiUJp^] /VA,

with a maximum growth rate at kuio ~

On the other hand, in a coUisional dusty plasma with uj « has the approximate solution[19] (1 + i) fu;^\'/'

1

»

ujpi/vA.

kUio ~ ujpi/y/A, i/j, (15)

..„.

which yields a dissipative instabihty.

5

Wakefield

In this section, we point out the importance of collective effects with regard to the generation of a wakefield in dusty plasmas. The wakefield, which is responsible for the attraction of charged dust grains of like polarity, arises due to the resonance interaction of a test dust charge which moves with a velocity close to the dustacoustic velocity Q . The wake potential of a test dust charge in the presence of the DAW in an unmagnetized plasma is found to be[32, 33] (t>w{p = 0,^t.t) = fcos{^t/L),

(17)

where ^t = \z — Vtt\, qt is the charge of the test particle L — Xjj [{vt — VQ)^ — c^] /Q is the lattice spacing, Vt is the test charge velocity, VQ is the equilibrium ion streaming velocity, and p and z are the radial and axial coordinates in a cylindrical geometry. For \vt — Vol ~ 30 c m / s , A^^ ~ 300/xm, and Q ~ 6 cm/s, we find L ~ 1 mm, which is in agreement with the observation[6]. The physics of the charged dust attraction is similar to the electron attraction in superconductors in which Coopers pairs are formed due to collective interactions involving phonons. In dusty plasmas, the latter are replaced by the DAWs, and negatively charged dust grains feel an attractive force in the negative part of the oscillatory potential (17), where the positive ions are focused. The wake-potential concept for charged dust attraction seems to be verified both in laboratory[34] and computer simulation[35].

10

6

RK. Shukla/Perspectives of collective processes in dusty plasmas

Nonlinear waves

The nonlinear waves in include solitons[l], shocks[36, 37], and vortices[38]. To the best of our knowledge, there are no observations of solitons in a dusty plasma. However, a recent laboratory experiment[37] has observed the compressive dust ionacoustic shock in a weakly coupled dusty plasma. The dynamics of that DIA shock is governed by

where $ = e(/>/Te, A^ = 5 + 3T, 5 = rtio/rieo, ^i = i^id/^pu Vi ^ ^iKnl^])e^ ^^d S = (3(5 — 1 + 12a)/5, Furthermore, (/) the electrostatic potential, Uid the ion-dust collision frequency, and A^ represents the effective mean-free-path. The time and space variables in (18) are in units of the ion plasma period {l/(jOpi) and the electron Debye radius X^e — '^tel^pe- An equation similar to (18) can also be obtained for the nonlinear dust acoustic waves. While the ion-dust drag is responsible for the monotonic dust ion-acoustic shock, the dust-acoustic shock can be formed even in the presence of dust charge perturbations[36]. It is suggested that future experiments should be designed to observe the dust lattice and dust acoustic shocks in an unmagnetized dusty plasma.

7

Discussion

In this paper, we have discussed the present status of waves, instabilities, and nonlinear structures that are observed in unmagnetized dusty plasmas. Both the laboratory experiments and computer simulations have conclusively verified the spectra of DIA and DA waves. The latter are excited by streaming ion beams in a dusty plasma. On the other hand, the DLW is excited by the radiation pressure of a modulated laser beam in rf discharges. Furthermore, we have discussed the physical mechanism for ion focusing and charged dust attraction in an oscillatory wakefield that is created by the motion of a test dust charge in the presence of DAWs in a weakly coupled dusty plasma. It may well turn out that the wakefield concept should still prevail in a strongly coupled dusty plasma in which the dielectric response function of the DAW is rather complex. However, the wakefield calculation has yet to be performed in a highly coUisional strongly coupled dusty plasma, taking into account dust-neutral and dust-dust interactions. Finally, we have also discussed the nonlinear properties of dust acoustic and dust lattice waves in a coUisional dusty plasma. The dynamics of experimentally observed dust ion acoustic shocks is governed by the KdV-Burgers equation. On the other hand, analytical and numerical models for vortical structures, which are observed in strongly coupled laboratory dusty plasmas (even under micro-gravity), have to be worked out. Vortices are formed in the presence of sheared plasma flows, the dust angular rotation, and the temperature and density gradients.

P.K. Shukla/Perspectives of collective processes in dusty plasmas

11

Acknowledgments: The author is grateful to Gregor MorfiU for useful discussions and support. This work was partially supported by the Max-Planck Institut ftir Extraterrestrische Physik, Garching.

References [1 N. N. Rao, P. K. Shukla, and M. Y. Yu, Planet. Space Sci. 38, 543 (1990). [2 J. Chu and Lin I, Phys. Rev. Lett.72, 4009 (1994). [3 H. Thomas, G. E. MorfiU, V. Demmel, J. Goree, G. Feuerbacher, and D. Molmann, Phys. Rev. Lett. 73, 652 (1994).

[4 H. Ikezi, Phys. Fluids 29, 1764 (1986). [5 P. K. Shukla and V. P. SiUn, Physica Scripta 45, 508, 1992; see also P. K. Shukla, Physica Scripta 45, 504 (1992).

[6 J. H. Chu, J. B. Du, and Lin I, J. Phys. D: Appl. Phys. 27, 296 (1994). [7 A. Barkan, R. L. MerUno, and N. D'Angelo, Phys. Plasmas 2, 3563 (1995).

[8] C. Thompson, A. Barkan, N. D'Angelo and R. L. Merlino, Phys. Plasmas 4, 2331 (1997).

[9] R. L. Merlino, A. Barkan, C. Thompson, and N. DAngelo, Phys. Plasmas 5, 1607 (1998).

[10 J. B. Pieper, and J. Goree, Phys. Rev. Lett. 77, 3137 (1996). [11 H. R. Prabhakara and V. L. Tanna, Phys. Plasmas 3, 3176 (1996). [12 A. Barkan, N. D'Angelo, and R. L. Merlino, Planet. Space Sci. 44, 239 (1996). [13 P. K. Shukla, Phys. Plasmas 1, 1362 (1994). [14 K. Miyamoto, in Plasma Physics for Nuclear Fusion (MIT Press, Cambridge, 1989).

[15 R. K. Varma, P. K. Shukla, and V. Krishan, Phys. Rev. E47, 3612 (1993). [16 P. K. Shukla, in Physics of Dusty Plasmas, Editors: P. K. Shukla, D. A. Mendis, and V. V. Chow (World Scientific, Singapore, 1996), pp. 107-121.

[17 P. K. Shukla, in Physics of Dusty Plasmas, Editors: M. Horanyi, S. Robertson, and B. Walch (AIP, New York, 1998), pp. 81-96.

[18 P. K. Shukla, G. T. Birk, and G. MorfiU, Physica Scripta 56, 299 (1997).

12

P.K. Shukla/Perspectives of collective processes in dusty plasmas

[19] M. Rosenberg and G. Kalman, Phys. Rev. E 56, 7166 (1997). [20] M. S. Murillo, Phys. Plasmas 5, 3116 (1998). [21] U. de Angelis and P. K. Shukla, Phys. Scripta 60, 69 (1999).

Lett.

A 244, 557 (1998); Physica

[22] P. K. Kaw and A. Sen, Phys. Plasmas 5, 3552 (1998). [23] P. K. Shukla and M. Rosenberg, Phys. Plasmas 6, 1038 (1999). [24] F. Melands0, Phys. Plasmas 3, 3890 (1996). [25] H. M. Thomas, R. J. Jokipii, G. E. Morfill, and M. Zuzic, in Strongly Coupled Coulomb Systems, Editors: G. Kalman et al. (Plenum Press, New York, 1998), pp. 187-192. [26] G. Morfill, G., H. Thomas, and M. Zuzic, in Advances in Dusty Plasmas, Editors: P. K. Shukla, D. A.Mendis, and T. Desai, 1997, World Scientific, Singapore, pp. 99-142. [27] A. Homann, A. Melzer, S. Peters, R. Madani and A. Piel, Phys. Lett. A 242, 173 (1998). [28] M. Rosenberg, Planet. Space Sci 41, 229 (1993). [29] M. Rosenberg, J. Vac. Sci. Technol. 14, 631 (1996). [30] D. Winske, S. P. Gary, M. E. Jones, M. Rosenberg, V. W. Chow, and D. A. Mendis, Geophys. Res. Lett. 22, 2069 (1995). [31] D. Winske and M. Rosenberg, IEEE

Trans. Plasma Sci. 26, 92 (1998).

[32] M. Nambu, S. V. Vladimirov, and P. K. Shukla, Phys. Lett. A 203, 40 (1995). [33] P. K. Shukla and N. N. Rao, Phys. Plasmas 3, 1770 (1996). [34] K. Takahashi, T. Oishi, K. Shimomai, Y. Hayashi, and S. Nishino, Phys. Rev. E58, 7805 (1998). [35] G. Lapenta, Phys. Plasmas 6, 1442 (1992). [36] F. Melands0 and P. K. Shukla, Planet. Space Sci. 43, 635 (1995). [37] Y. Nakamura, H. Bailung, and P. K. Shukla, Observation of ion-acoustic shocks in a dusty plasma, Phys. Rev. Lett. 83, No. 6, in press, (1999). [38] P. K. Shukla, M. Y. Yu, and R. Bharuthram, J. Geophys. Res. 96, 21343 (1991); M. SalimuUah and P. K. Shukla, Phys. Plasmas 5, 4502 (1998).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

13

Non-Ideal Effects in Dusty Plasmas Nagesha N. Rao^ Theoretical Physics Division Physical Research Laboratory Navrangpura, Ahmedabad-380009 INDIA

Abstract. Electrostatic waves and instabilities in weakly non-ideal magnetized dusty plasmas have been investigated by incorporating the van der Waals equation of state as well as the grain charging equation. For the homogeneous case, the propagation of linear dust-acoustic waves (DAWs) has been considered. It is found that the volume reduction coefficient enhances the DAW phase speed while the molecular attractive forces lead to a decrease in the speed. In the high temperature limit, there is a net increase in the DAW phase speed while near the critical point the phase speed is reduced. Inhomogeneous magnetized dusty plasmas support density as well as temperature gradient driven electrostatic drift waves in the low-frequency regime. On the other hand, when shear flows are present, dusty plasmas are susceptible to (parallel) Kelvin-Helmholtz (K-H) instabilities. In addition to the usual density gradient driven K-H (DGKH) instability, we point out the existence of a new type of temperature gradient driven K-H (TGKH) instability. At higher frequencies near the dust-gyro frequency, electrostatic dust cyclotron waves (EDCWs) can be driven unstable due to sheared flows. In aU cases, dust charge fluctuations lead to damping effects, thereby reducing the instability growth rates.

I.

INTRODUCTION

The presence of finite-sized, charged particulate matter in plasmas leads to a host of novel collective phenomena in plasmas. The so-called dusty plasmas contain grains in the s u b - or super-micron range and are highly charged, typically to about a fev^ thousand electrons. In the ultra-low frequency regime, dusty plasmas support new kinds of waves and instabilities arising due to the dust collective dynamics [1]. Examples are the Dust-Acoustic Waves (DAWs) [2], the Dust-Ion-Acoustic Waves (DIAWs) [3], and the Kelvin-Helmholtz (K-H) instabihties [4,5]. In laboratory dusty plasmas, such waves have been observed, and the frequency of DAWs is usually about a few hertz [6,7], while in space and astrophysical plasmas it could even be much smaller. ^) E-mail : [email protected]

14

N.N. Rao/Non-ideal effects in dusty plasmas

When the grains are in the super-micron range, dusty plasmas can exhibit n o n ideal behavior. First, large grains are highly charged and hence particle correlations as well as intergrain molecular forces come into play. Second, volume reduction contribution can be significant at higher gas densities, particularly for grains in the few tens of micron range. For example, for grains with a radius of about 50 microns, the volume reduction contribution is about 10% for dust gas density of about 10^ particles/cc. Third, grain charge can fluctuate because of the equilibrium charging processes as well as due to the wave induced perturbations in the electron and the ion currents flowing onto the grain surface. Research studies beginning in this decade have mostly concentrated on the ideal response of dusty plasmas, while parameter regimes wherein non-ideal contributions become signiflcant are within the scope of present day laboratory experiments on dusty plasmas. Recent analysis [8,9] of DAW propagation in non-ideal dusty plasmas has shown that the volume reduction enhances DAW phase speed while the molecular cohesive forces lead to a reduction in the speed. On the other hand, charge fluctuations are known to result in damping of waves [10,11], which otherwise would propagate as normal modes. In this paper, we present a systematic analysis of electrostatic waves and instabilities in a weakly non-ideal low-^ magnetized dusty plasma by including shear in the dust flow velocity as well as the charge fluctuation effects. Non-ideal effects are modelled through the van der Waals equation of state for the dust fluid. First, we consider the low-frequency regime and discuss electrostatic modes such as the DAWs, density and temperature driven drift waves and the electrostatic dust cyclotron waves (EDCWs). Next, we discuss the onset of Kelvin-Helmholtz (K-H) instabilities driven by shear flows. The existence of a new kind of K - H instability driven by dust temperature gradient is pointed out. Damping effects on the instability growth rates due to charge fluctuations have been computed.

II.

BASIC EQUATIONS

For low-frequency phenomena, we follow the dusty plasma model first suggested by Rao et al [2] wherein the electrons and the ions are assumed to behave like ideal massless thermal fluids in equilibrium at their respective temperatures, while the wave dynamics is governed by the heavier dust component. Accordingly, the electron and the ion number densities are given by the respective Boltzmann distributions rie = rieo exp {e(j)/KBT^),

n,- = riio exp {-e({)/KBTi),

(1)

while the dust component is governed by the fluid equations

(2)

{dnJdt) + V'{n,v,) = 0, n,m, [{dvjdt)

+ {v, . V)v,] = -n,q,

Vcf> + {n,qjc)[v,

X B,) - V p „

(3)

15

N.N. Rao/Non-ideal effects in dusty plasmas where the notations are standard [12]. The grain charge variable q^ is determined by the charge current balance equation

(4)

{dqJdt) + {v,-V)q, = h + Ii, where the average electron (/g) and ion (7^) currents are given by [13] /e = -TreR'ried

exp V Trme

—— , \f^B^eJ

7, = ireR riiJ V ^^i

1 V

— , f^B-^iJ

(5)

and ij; = qd/R is the dust surface potential relative to the plasma potential where R is the grain radius. The dust fluid is described by the van der Waals equation of state [14] {p, + Anl)

{l-Bn,)

= n,K,T„

(6)

where the gas constants A and B are given by A = dKaTc/Sric and B = l/Sn^; here, the subscript 'c' denotes the respective values at the critical point. Note that for A, 5 -^ 0, Eq. (6) reduces to the ideal gas law. Equations (l)-(6) are closed by the charge neutrality condition, qdn^ + eni — erie = 0, and thus constitute a complete system of governing equations to describe lowfrequency phenomena in non-ideal dusty plasmas. We now discuss the steady state equilibrium relevant to the (parallel) K - H configuration. Accordingly, there exists in equilibrium a transverse shear in the dust flow velocity component parallel to the magnetic field, BQ = BQZ. The equilibrium dust flow velocity is thus represented by, v^o = VyV + K^? where the components Vy and Vz are to be suitably evaluated consistent with Eqs. (2) and (3). The latter are identically satisfied provided the equilibrium quantities n^o? ^o-i Pao ^^d T4 are functions of X only, while V^ is a constant which is calculated from the i-component of Eq. (3) as, Vy = {c/Bo){d(l)o/dx) + {c/ndoqdoBo){dpdo/dx), Note that the right-hand —•

—*

side contains E x B a.s well as diamagnetic drifts. In equilibrium, the pressure gradients give rise to the dust diamagnetic drift defined by, V^ = yV^ where V^, = {dp^oIdx)j771^71^,0^1^0- In particular, when p^^ given by Eq. (6), we obtain V^ = ^{)^iVdn + ^2VdT) where Ai = 9(3 - T/)"^ - 9a7//4 and A2 = 3/(3 — r/) defined in terms of the non-ideahty parameters, a = T^Td and 7/ = rido/ric. Note that for an ideal dust fluid, a, 7/ —> 0, and hence Ai, A2 ^^ 1. Here, Vdn and VdT are, respectively, the characteristic drifts in the presence of density and temperature gradients, and are given by

In Eqs. (7), CDA = L)(^?+ ^l)]-'-

(13)

It should be noted that the damping rate vanishes for a tenuous dusty plasma (/-O).

(3)

Temperature Gradient Drift Wave (TGDW)

We now show the existence of a novel kind of drift wave driven purely by dust temperature gradient even when the density gradient is absent. For small kz^ dispersion relation (8) becomes u = -/3\2kyVdT

[Cs + A;>L(1 + l3CiC3)]~'.

(14)

For the case when charge fluctuations can be neglected, above equation yields LJ - -^A2a;^^ [1 + klpl^il

+ e ) ] " ' = uj"-,^,

(15)

The dispersion relation (15) is similar to (12), and represents temperature gradient driven drift wave (TGDW) [12] in a warm dusty plasma. For a tenuous plasma when the damping is weak, Eq. (14) yields real frequency as uJr ~ ^ T whereas the damping coefficient (7) is given by 7 « -fu,oJl

[1 + kliPpl,

where p^ = Cn/^do[cf. Eq. (13)].

+ epD] [{ul + Ul) {1 + klplil

+ e)}]"',

(16)

This is similar to the damping rate obtained for the DGDW

18

N.N. Rao/Non-ideal effects in dusty plasmas

(B) Dust gyro-frequency regime : u; ~ Qdo Consider first the case of homogeneous plasmas. For almost perpendicular propagation {ky ^ kl) without shear flows, Eq. (8) becomes

^

[uji + fuj2 - ^(^)

+

^y^DA

9^

9a/37]

(17)

(3 - IJY

which is a generahzed dispersion relation for the electrostatic dust cyclotron wave (EDCW) modified by dust charge fluctuations (the third term) as well as non-ideal contributions (the fourth term). For an ideal plasma with constant dust charge, we recover the usual dispersion relation for EDCWs [16,17], cJ^ = f]^o + ^I^DA0^ the other hand, for the non-ideal case with weak damping, we obtain

K =fi^o+ KCii^ + ^)'

2 , —2\-l

7 = -..fKci.^M

+ K)

(18)

Shear flow effects manifest themselves for finite, non-zero A:^. Accordingly, for homogeneous plasmas, Eq. (8) reduces to ^2 \

—2 7.2/-f2

(u;^ - klci.Oiu' - nl,) - uj^ci^c, = kyKni,cl,c,s,

(19)

which shows a coupling between DAW and EDCW. For the case when / -^ 0 so that charge fluctuation damping can be neglected, Eq. (19) has the roots LJ± given by u, —

mdo

2 L

(1 + k'pl^C)

± {(1 + Ppl.Cf

- Akyk.pl^K

-

S)}

1/2

(20)

where K = kjky, P = k^ + kl and C = (1 + ^ C i ) . The mode io = a;+ is the usual E D C W modified by the dust shear flow, while cJ == a;_ is a low-frequency mode which becomes unstable for S > K. This can be seen for the case when \K^ZPIAC{I^ - S)\ < (1 + k'^pljfif, which leads to 2^2

\-l

wl« ni, + k'ci.c + kyk,ci,c{s - K)(I + k'pl,c)-\ \-l

ui = -kyKCi,c{s - /c)(i + fcX.c)

(21)

(22)

Thus, a;_ is the purely growing zero-frequency unstable mode for S > K. The effects of inhomogeneities on the modes cJ^ and a;_ have been discussed elsewhere [12].

19

KN. Rao/Non-ideal effects in dusty plasmas

V.

KELVIN-HELMHOLTZ INSTABILITIES

Dusty plasmas are susceptible to (parallel) Kelvin-Helmholtz (K-H) type of lowfrequency (aJ ,E=<E>+JE where the brackets denote the statistical averaging on the ensemble of the dust particles since the fluctuations are related only with dust discreteness. For the average distributions (we omit the arguments r, f and take < E > = 0 for simplicity)the equations are :

^ { [h.i + Y^iUq))) ^^'(?) + Ti^lMSf'^M))^

(5)

where we have defined the plasma-dust collision frequency

(6)

t^o = —0.01, r^ = 0.2, ^i — 0.03 and 5^ — 0.1. It turns out that the shape of the sheath changes in the range of 0.025A^ < i < 0.033Ai:) by the presence of trapped ions and an electron beam. We illustrate the space charge density in the sheath for |Z^| = 10 and 5i = 1.8 in Fig.2. In the comparison of our case with that of Boltzmann ions, as is seen in Fig.2, the space charge density enhances due to the trapped ions and an electron beam. The structure of the electric field in the sheath is shown in Fig.3, where we used the same parameters as employed in Fig.2. We see that the electric field forms a triple-layer, which is newly found in this sheath. Positive ions are accelerated and decelerated frequently by this electric field, and are drastically accelerated near the electrode. Figure 4 shows temporalevolution of the grain-charge, which implies that dust grains take ~ 0.4to=~ 1.1/is to saturate its charge, whereas ^ 1.0to=~ 2.8/is in the case of Boltzmann ions, where Tg^lOeV and Ugo = lO^^m"^. It is because the space charge density greatly enhances due to trapped ions in the range of 0.025XD < ^ < 0.0d3XD when the grain-charge number \Zd\ — 10. This tendency is similarly true in the cases of the charge number |Z^|=20 and 30. It turns out that the time take to attain the equilibrium charge of the grain is shortened by the trapped ions and beam electrons.

Y'N. Nejoh /Multiple-sheath and the time-dependent grain charge

30

47

1.x 10'

1.x 10"



Trapped ions 0.0175

0.02 0.0225 0.025 0.0275

Boltzmann ions

0.03 0.0325

X/\D

0.02

0.022

0.024

0.026

0.028

0.03

X/XD

Figure 1 Spatial profile of the potential in the sheath with Zd = 10(solid line), 20(dotted line), and 30(dashed line) for (p,u = -100, n = 0.2, Td = 0.1 and (f)o = -0.01.

Figure 2 Spatial profile of the space charge density in the sheath for (f>^u = -100, Zd = 10, Si = 1,8, n = 0.2 and (/)o = - 0 . 0 1 .

"

'

"

"

•

'

"

'

120 100 80

Zd

/

60 40 20

// /

0 2

0.4

0.6

Trapped ions 0.02

0.0225

0.025

0.0275

0.03

0.0325

X/\D

Figure 3 Profile of the electric field as a function of the distance, for 0^ = -100, Zd = 10(solid line), 20(dotted line) and 30 (dashed line), Ti = 0.2, Td = 0.1 and 0o = -0.01.

Boltzmann ions Figure 4 Temporal-evolution of the dust grain-charge for (f)^, = —100, Zd = 10, Si = 1.8, Ti = 0.2, Td = 0.1 and 00 = -0.01.

48

Y.-N. Nejoh/Multiple-sheath and the time-dependent grain charge

4. Concluding discussion We understand that positive ions are trapped in the localized region by our calculation. If a secondary electron beam is only calculated numerically, we cannot find the modification of the sheath and the formation of the triple-electric field. The effect of trapped ions gives rise to the remarkable change for the profiles of the sheath potential, space charge density and electric field, i.e., the modification of the potential profile, an enhancement of the space charge density and the occurrence of the triple electric field, respectively. Therefore this fact imphes that our results may be applicable to the control of the sheath by modifying the potential of the electrode.

References l]G.M.Jellum, and D.B.Graves, Appl. Phys. Lett., 57, 2077, (1990). 2]L.Boufendi, A.Plain, J.Ph.Blondeau, A.Bouchoule, C.Laure and M.Toogood, Appl. Phys. Lett., 60, 169, (1992). 3]G.S.Selwyn, J.Heidenreich and K.L.Haller, J. Vac. Sci. Techn. A., 9, 2817, 1991). 4]R.N.Carlile and S.S.Geha, J. Appl. Phys. 6 1 , 4785, (1993). 5]P.D.Prewatt and J.E.Allen, Proc. Roy. S o c , A 3 4 8 , 435, (1976). 6]N.St.J.Breithwaite and J.E.Allen, J. Phys., D 2 1 , 1733, (1988). 7]H.Amemiya, B.M.Annaratone and J.E.Allen, J. Plasma Phys., 60, 81, (1998). 8]H.Yamaguti and Y.Nejoh, Phys. Plasmas, 6, 1047, (1999). 9]Y.Nejoh, Austr. J. Phys., 52, 37, (1999). lOjYa.L.Al'pert, Spact Plasma^ vol.2. Chapter 12. Cambridge University Press, Cambridge, (1990). [lljD.Bohm, Characteristics of Electrical Discharges in Magnetic Fields^ Chapter 3. McGraw-Hill, New York, 271, (1949).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

49

Surface Charge on a Spherical Dust H. Amemiya The Institute of Physical and Chemical Research Hirosawa, Wako, Saitama-Pref, 351-0198, Japan 1. Introduction The dust suspending in the plasma influences the charge balance due to the surface charge. The dust has been treated as a small probe to estimate the charge. 1"^) Recently, the electric charge on the dust has been estimated by applying the probe theory, where positive ions are assumed to follow the radial motion theory.4) When the dust density is not small, the absorption radius of each dust will overlap with each other and the radial motion theory will break down. When the dust radius is small, positive ions would follow the orbital motion theory. Although the pervious theories do not assume the existence of negative ions, this is necessary not only for a negative ion containing plasma but also for a plasma containing dust because the dust has usually a negative charge and behaves effectively as a negative ion. Therefore, the dust charge should be calculated inevitably by including negative ions. The objective of this paper is to evaluate the dust charge by applying the orbital motion theory of probe to a negative ion containing plasma. The result would be more consistent than the previous theories in that the dust charge is a function of the charge itself. 2. Model Assume that a plasma consists of positive ions with a density N+o, thermal electrons with the density of Neo and the electron temperature Te, negative ions, and dust particles. We define N^Q as the negative ion density and N^Q as the dust density, and Tn as the negative ion temperature and T^ as the dust temperature. The dust potential can be obtained from the current balance between positive and negative charged particles. If Te and Tn are known, the current I_ of negatively charged particles to the dust with a radius r^ at a potential -Vp against the space potential (V=0) is given by

I - = e N „ S ( ( l - a „ ) V ^ e . p < - ^ H a ^ exp(-^)|,

(,)

50

H. Amemiya/Surface charge on a spherical dust

where 8=4^^2, a^=^JZ^^^-^ QNdo/ZN+o=«+5Q, 5=Ndo/ZN+o' oc^N^^ZN+o' Q is the dust charge and e is the electronic charge. The first term denotes the electron current while the second term the negative ion current, but unless 1-a is very near to (Tj^m/TgMjj)^/^, I. is almost given by the electron current. The positive ion current 1+ to a spherical dust is coupled with Poisson's equation as^)

where Ti=eV/KTe, r|p=eVp/KTe, ^=rAD, ?^D= (eoKTe/Noe^)!/^, No=ZN+o^ P=T^/Te, and i=I+/lA.; lA.=eZN+Q47iA-D2(2KTg/M)l/2. Signs + and - stand for outer and inner sides of the absorption radius at which r|/p=4i/pl/2^2-l. ^ is given by 4>(Ti)=(l-a-6Q)exp(-Ti) + aexp(-Yn) + 5Qexp(-YdQTl), (3) where T=Tjj/Tg and Yd^Td/Tg. It should be noted that <E> is a function of Q and 5. The conditions for absorption radius and quasi-neutrality are given respectively as Ya=Xa-2 -1 , (4) X-2=4{(l+Y)l/2 - 4>} . (5) where Y ^ / p and X=^/(2il/^/pl/'^). The absorption point in the quasi-neutral region is given by the crossing points between eqs.(4) and (5) as (1 + Y)l/2 =2.

(6)

In the plasma where Y is very small, we have Y=l/[2+4P{l+a(Y-lH5Q(l-YdQ)}]X2.

(7)

The following four cases may be considered. Case I: rd(r|a) = (l+—)/2.

(10) (11)

P

To obtain the floating potential r|f, |LI should be balanced by I. given in the normalized form as

L,(,.„.8Q,y^exp(-„.>.

(12)

By assuming a capacitance in vacuum as 47C8Qr^, Q is determined as Q=47r8ordVf; Vf=KTeTif/e.

(13)

Introducing Q£)=47C8Q?i^(KTg/e), we can express Q=^i1fQj). This value is a minimum value due to a finite sheath thickness. Q^ may be defined as the Debye sphere charge. In the case IV (OML), ^^ corresponds to ^^j. Then

^ = 4^(1-^).

(14)

lo 4 p The dust potential r|f is determined from (l-a-6Q)^/^exp(-Tif) =V2p (1 + ^ ) . V Tim

ft

(15)

Besides eq.(13), the dust charge can be also calculated from dfx\ld'^ at the dust surface. The field must be continuous at ri=n-|a» ^=^a- Then, the surface should coincide with the absorption radius, i.e. — = - - ^ ^ = - — ( 1 + - ) ; r|=qF, ^=^d.

^^

I'

^

^

(16)

The surface charge Qp is given by QF=47trd28o— = 2 ^ / — (47reo^;iD).

ar

Then, the ratio Q/Qp becomes

QF/Q=2(Tif+P)/rif.

a^

(17) (18)

52

H. Amemiya/Surface charge on a spherical dust

3. Calculation of dust charge Calculation has been made for thin sheath and orbital motion limited cases under the condition that the dust radius is r^j^O.liim and the plasma ion species are Ar"*" and 02". First, the ion and electron currents were calculated for a tentative charge Q. Then, from the current balance, the floating potential, the surface field and a new charge were calculated. The process was repeated until Q coincided. Figure 1 shows Q vs P in both models for some values of 5 (T^ =leV, a=0). The case when 5 is small corresponds to "single dust case". As 5 is increased, Q departs from the value by the single dust model, i.e. Q decreases with 6. In the thin sheath model, Q decreases with P slightly up to about p=l, however, above P=l, Q falls with (3. This is because an increase of I^. due to the higher positive ion temperature causes a reduction of Vp. An abrupt fall of Q at p-^O.S for 6=2.5x10"^ is associated with an abrupt drop of the sheath edge potential by the dust behaving like a negative ion. Above 5=2.5xl0'3, some times no solution was found consistently. This means that it becomes difficult to sustain a stable plasma as the limit 6Q=1 is approached. Near this limit, all the electrons are absorbed by dust and Q becomes independent of the models. Such a "dust plasma" where the plasma consists of dust and positive ions (or positively charged dust) has been recently observed.^) In the OML model, Q is smaller than that for the thin sheath model by a factor of about 2 for small p and the variation against P is different. However, the charge Q F calculated from the surface field is larger than Q by eq.(18) especially at larger p. Q F VS P in the OML model is shown in Fig. 2 for some values of 6 (T^ =leV, a=0). At larger p, the region of "no solution" appears as 6 is larger. In such a region, a some sort of unstable plasma is predicted. Figure 3 shows Q vs 5 in both models for some values of a (T^ =5eV, p=0.1). It is seen that Q is decreased by negative ions and especially at a=0.9, a strong reduction of Q is seen. As 5 is increased, Q approaches asymptotically to the limit (l-a)/5 as shown by dotted curves. For a=0 in the thin sheath model, no solution was found before this limit is reached: it becomes difficult to sustain the plasma stably for 5>2xlO"3. In the presence of negative ions, however, it seems that negative ions can avoid the plasma destruction by dust even if the limit (l-a)/5 is approached. Figure 4 shows the dust charge Q vs a in both models for different values of 5 and T^ (P=0.01). The values of 5 are as shown labeled. 5=0 corresponds to the case of "single dust case" where the dust density does not affect the charge equilibrium. Some humps of Q appearing at (X=0.4-0.8 are due to an abrupt drop of the sheath edge potential caused by the dust and negative ions when a4-6Q reaches near 0.7.^) A reduction of the sheath

H. Amemiya/Surface

500

1

1—I—I I I I 11

T—r

I I i 11

1

1—I—I

53

charge on a spherical dust

2000

I I I1

-!

1

1—[ 1 1 I I I

Te=leV, a =0 Thin sheath

400

1

1

OML

IxlQ-^

1—I I I I I I

\

1

1—I M

M

Te=leV,a=0

1500

—ft

o

1000

1x10' 500 - • — » » »» • • -J

:

10-

I I I I 11

I

I

I

I I I I 11

10-'

-i

I

I I I I I

_i

10^

10"

r - r - r i i i j-

T

-

10^

•^ T

^ ^—•^^^

1—1' T"

10-^

10'

T

"*~^

1—

a=0.5

\ %

10-=

Te=5eV, p=0.1 . .......1 10-^

1

5x10-' i__j

,

' I 1111

I

10"

-

o

1 1 1 1 1 1 1 ^ p-n

1 1

Thin sheath Te=10eV

[Jr

a

•

1.

10-^

4

E OML • • ^ txhin sheath O D A h 1

10^

T-j-rTTT

*%/^Xa=0

a=0.9

I

10-'

'

I

I ' l l

10'

F

-

^

a

rTTj

,

I I 1111

--' no solution

Fig. 2. Surface chage Q^ determined by the surface field.

Fig. 1. Surface charge Q vs temperature ratio (3.

• r—' J

I

10-

• — • • »» • •.•

D

n

D-Q-

-g—.

^Q.

p

^

OML Te=

r 10'

—1

L..,X_I_

MI

1

1

10-^

Fig. 3. Surface charge Q vs dust density ratio 5.

I 1 1J. i "K

10--

10

0

Fig. 4. Surface charge Q vs negative ion density ratio a .

54

H. Amemiya/Surface charge on a spherical dust

edge potential causes an increase in the floating potential difference. For larger a above these humps, the current of negatively charged particles, eq.(l), also decreases with a and Q makes a smooth change with a. Q increases in proportion to T^ up to a^, a critical a, from which the effect of Q on the charge equilibrium becomes remarkable. As 5 is made larger, Q begins to decrease at a^, and as a approaches unity, Q drops steeply towards the limit (l-a)/5. In the thin sheath model, the ratio rjx^ remains almost near unity at p>0.3 while it increases in proportion to 1/p^ at p« 0 - 20 s

MD

(d)

^

".=• 0.063

(e)

^ ^ ^ * ^ ^ , J'.'Oia 1

' / '^ ^ . I

. s

^'/»005

«

s

»

-.

^

-»

*

•

At= 5000

MP

At= 2000

MD

^t= 2000

FIGURE 4. (a) and (b) Triagulated snap shots and trajectories showing the evolution of ±e (2,8,13) state from our experiment, (c) to (e) The trajectories from our MD simulation with increasing noise intensity- 7] o- The damping coefficient is kept at 0.15.

This strongly coupled system supports many interesting thermally induced collective excitations. Under the shell structure induced by the circular system symmetry, the collective inter-shell angular excitations predominate over other excitations, such as the intra-shell angular and radial motion, the radial vibration of the whole cluster relative to the confining center, radial breathing, etc.. Figure 4(a) and (b) give an experimental example of the time evolution of the N = 23 cluster. The elongated trajectories indicate the higher excitation energy for the radial motions. The sequential snap shots with triangulations show the relative particle and defect positions. From our simulation, the (2,8,13) cluster has three stable (meta-stable) configurations: a single 5-fold defect, a connected 5-7-5 defect pair, and a 5-7 defect pair with another 5-fold defect separated by one dust particle around the second shell. The center two particles prefer to line up with the two particles in the second shell. These three states are almost degenerated and the energy barriers among them are much smaller than the transition to other nearly degenerated states with different sets of Ni ,such as (1,8,14) and (3,8,12), in which the transition should be associated with the particle radial hopping. The experimental snap shots in Fig. 4 manifests that these three nearly degenerated states can be easily thermally accessed through the slight adjustment of the particle relative positions which causes the propagation, generation and annihilation of defects. Note that the state with defects at the center is unstable and seldom occurs. The triangularly packed center core is quite robust. The relative position of the second to the third shells is not that strongly locked and can be more easily excited. For the structure with magic packing such as the (2,8,14) state, the thermal defect can equally access to the entire cluster (20). These phenomena are also evidenced by our simulation results. Depending on the temperature and viscosity, the relative fluctuation amplitudes of the angular motions for

80

Lin I et al /Classical atom-like dust Coulomb clusters in plasma traps

different shells may change. The MD results in Fig. 4 further shows that at high temperature, the inter-shell hopping can also be excited.

FIGURE 5. Triangulated configurations (MD results) for the larger N clusters with tvpe IV and III interaction forms at zero temperature. The two states for N = 180 (type IV) have the same total energyup to the sixth digit. Figure. 5 shows a few simulated ground state configurations at larger A^ for the cases of Vjj = ln(l/ry ) and IMy obtained at zero temperature. For the former case, the defects mainly stays around the outer shells and leaves a uniform defect free core. However, for the latter case, the core part is relatively defect free only for A^ < 150. The former case has an almost uniform radial packing density distribution through the cluster but the latter has lower density as radius r increases (the mean lattice constant can increase about 30% from the center to the boundary) (11). Note that, Vy = In(l/r,y) is the interaction between two uniformly charged long wires, Vy = l/r^ is the interaction between two point charges on a plane. Va = rr is the field from a uniform neutralizing ion background. The packing density should be uniform for the first case in order to null the coarse grained space charge. For the second case, the leaking of the electric field along the axial direction weakens the radial field. Particles in the inner region are compressed to higher density to reach the force balance (10). The easier bending at larger A^ makes the defects move inwards from the outmost shell. In addition to accommodating the interface matching between the outer circular shells and the inner triangular cores, the defects have to move further inward for the latter case to accommodate the lattice bending due to the nonuniform packing. In our experiment, the higher rf power for supporting the larger A^ cluster also makes the background fluctuations larger than the cases at small A^. The fluctuations can cause lattice deformation and make the defect move inward. Our experimental results (Fig. 6) show that, unlike the large increase of the mean lattice constant Ur with r for the case of Vy = I fry, Qr is uniform in the inner region and only increases less than 10% toward the boundary. Namely, our larger A^ cluster is more similar to the case of long wire interaction. The contribution from the particles along the vertical chain and the slightly ion rich double layer boundary could be the possible causes (note that there could be about 15 particles along one vertical chain for the large cluster but less than 10 for the small clusters. Our MD simulation results show that in addition to the intrinsic defect around the cluster boundary for the case of Vy = ln(l/r;,), the thermal excitations cause

Lin I et al /Classical atom-like dust Coulomb clusters in plasma traps

81

the Spread of defects into the center part of the cluster (Fig. 7). N = 88 / V \y-.

Os

10s

^ .

0 - 10s

10-20 s

N = 179 •

K ' ' I ' ^ ' ' . V'.

v**;-*- i • ; . ^ ' t - f l ^ ^ •

• *^' ,^*^;--Vrt^'

...*-." M /

10s

0-IOs

10-20 s

N = 287

•::rn-^

•/.-.••''rSyti:.-.

Os

FIGURE 6.

"

10s

0-IOs

' " '

10-20S

The typical experimental results of the defect and particle trajectories at large .V.

(a) . V0.05

.••//.",• • V 0 ' ' 2

• • • • • ' • i ' •' •* '*>'.* ^'; -.' •' ^ ' ' ' - V - '•»

0.0001

0.0010

0.0100

0.1000

1.0000

FIGURE 7. (a) The trajectories and the corresponding snap shots of defect configurations showing the transition to the more disordered liquid state as noise intensity 7] o increases for the ideal 2D Coulomb interaction at A' =300 from our MD simulation, (b) The averaged power spectra of particle coordinate evolution (averaged over 100 particles in the core of the cluster).

Unlike the small A^ cases, the excitations become more isotropic and uniform as N reaches the level of a few tens because the system also has a larger inner triangular domain which provides space for the radial hopping. Figure 6 shows the typical vortex type collective excitations observed in our experiment. Only the particles along the outmost shell prefer angular excitation due to the circular shell structure. Similar to the

82

Lin I et al /Classical atom-like dust Coulomb clusters in plasma traps

previous observation in the very large N crystal (5,6), the cluster exhibits continuous excitation and relaxation of vortices with different sizes and life times, associated with the generation, propagation, interaction and annihilation of defects. The spatial and temporal correlation lengths of the excitations first increase with noise strength 7] o and then decrease as 77 ^ further increases to the liquid state (Fig. 7). The two peaks in the power spectra correspond to the center of mass vibration and the shortest wavelength acoustic modes. Increasing j] o causes the increase of the background floor and the slopes of the low frequency part. It corresponds to varying the temporal correlation length of the vortex type excitations which generates persistent diffusion (6). We also found that applying an axial magnetic field with a few tens of Gauss can cause rotation of the cluster along the ExB direction, with about 10"^ Hz frequency and no averaged angular velocity shear(9), where E is the radial (outward) space charge field in the sheath. Our rough estimate of the momentum transfer rates shows that the cluster is floating and drifling with the background neutral whose rotating speed is determined by the balance between the incoming angular momentum from ions and the viscous loss toward the surrounding walls. This research is supported by the National Science Council of the Republic of China under contract number NSC-88-2112-M008-008.

References [I] J.H. Chu and Lin I, Phys. Rev. Lett. 72, 4009 (1994). [2] Y. Hayashi and K. Tachibana, Jpn. J. Appl. Phys. 33, 804 (1994). [3] G.E. ^ Thomas, et al, Phys. Rev. Lett. 73, 652 (1994). [4] A. Melzer, T. Trottenberg, and A. Piel, Phys. Lett. A 191, 301 (1994). [5] Lin I, W.T. Juan, and C.H. Chiang, Science, 272, 1626 (1996). [6] C.H. Chiang and Lin I, Phys. Rev. Lett, 77, 647 (1996). W.T. Juan and Lin I, Phys. Rev. Lett, 80, 3073 (1998). [7] W.T. Juan, Z.H. Huang, J.W. Hsu, Y.J. Lai and Lin I, Phys. Rev. E, R6947 (1998) [8] J.M. Liu, W.T. Juan, J.W. Hsu, Z.H. Huang, and Lin 1, Plasma. Phys. Control Fusion, 41, A47 (1999) [9] W.T. Juan, J.W. Hsu, Z.H. Huang, Y.J. Lai, and Lin I, Chinese J. Phys. 37, 184 (1999). [10] Y.J. Lai and Lin I, to be published. [II] J.J. Thomson, Phil. Mag. S. 6. 7, 39, 236 (1904). [12] H. Ikezi, Phys. Rev. Lett. 42, 1688 (1979). P. Leiderer, W. Ebner and V.B. Shikin, Surface Science, 113,405(1982). [13] Nanostructure Physics and Fabrication, edited by M.A. Reed and W.P. Kirk (Academic, Boston, 1989). [14] D. Reefman, H.B. Brom, Physica 183C, 212 (1991). [15] W.I. Glaberson and K.W. Schwartz, Phys. Today, 40, 54 (1987). [16] D.Z. Jin and D.H.E. Dubin, Phys. Rev. Letts, 80, 4434 (1998) [17] V.M.Bedanov and P.M. Peeters, Phys. Rev. B. 49. 2667 (1994). [18] V.A. Schweigert and P.M. Peeters, Phys. Rev. B. 51, 7700 (1995). [19] A. A. Koulakov and B.I. Shklovskii. Phys. Rev. B. 55. 9223 (1997) [20] J.W. Hsu, Master Thesis. National Central Umversit\' (1998).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

83

Crystallography and Statics of Coulomb Crystals Yasuaki Hayashi and Akito Sawai Department of Electronics and Information Science Kyoto Institute of Technology Matsugasaki, Kyoto, Japan 606-8585

A b s t r a c t . Three-dimensional and two-dimensional Coulomb crystals have been formed in plasmas by growing carbon fine particles, of which size has been controlled using the Mie-scattering ellipsometry. Lattice constants of the crystals were determined from CCD images of top view and side view taken at the same time and at the same position. Smaller particles of L4 M m in diameter formed the structure of a three-dimensional Coulomb crystal, which was face-centered orthorohmbic. It is suggested that the ratio of the lattice constants of the crystal is decided so that Coulomb energy takes a minimum value under the condition of constant particle density in horizontal layers. Larger particles of 5 M m in diameter formed a two-dimensional crystal structure, simple hexagonal one. The result of the dependence of lattice constants with particle density for the crystal indicates that the force acting on particles in the vertical direction is repulsive for smaller value of the vertical lattice constant while attractive for larger one. The vertical directional forces are mainly external ones for smaller particles, however a directional attractive force between particles plays an important role for the formation of Coulomb crystal for larger particles.

INTRODUCTION Since Coulomb crystals in fine particle plasmas were experimentally discovered several years ago [1-3], many types of the structure have been reported to be observed. They are simple hexagonal, body-centered cubic (bcc), face-centered cubic (fee) and so on. Because structures of Coulomb crystals reflect forces acting on particles, the quantitative or qualitative analysis of the forces should be possible from the analysis of the structures. In order to determine the three-dimensional structure and lattice constants of t h e crystals exactly, top and side views have been taken with two CCD video cameras at the same time and at the same position.

84

Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals

EXPERIMENTAL A schematic drawing of an experimental system for the formation of Coulomb crystals is illustrated in Fig.l. Coulomb crystals were formed by the growth of monodisperse spherical carbon fine particles in methane/argon R F (13.56 MHz) plasmas. As in the previous experiments [1,4-6], the diameter of growing carbon fine particles was monitored by the Mie-scattering elhpsometry [7,8]. For the elhpsometry measurement and the observation of particle arrangement, argon-ion laser hght of 488 n m in wavelength was irradiated upon particles, which were suspended above an R F electrode and trapped in a potential bucket formed by a ring of 3 cm in inner diameter on the electrode. Three-dimensional structures of particle arrangement were observed by two CCD video cameras from the top and the side at the same time and the same position.

C( 2 D (Tc

Ar+ Laser *"""

CCD (Side)

Mie-Scattering Ellipsometry

-*iHii ilPl

(*-^

*^ 1

FIGURE 1. Schematic drawing of the experimental system.

RESULTS AND DISCUSSION Three-dimensional Coulomb crystal First we controlled the diameter of fine particles to 1.4 /x m by decreasing R F power from 5 W to 1 W with the use of the Mie-scattering ellipsometry. The

Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals

85

arrangement of the fine particles is shown in Fig. 2.

SOD am F I G U R E 2. Top and side views of formed three-dimensional Coulomb crystal.

TOP VIEW

SIDE VIEW

F I G U R E 3. Explantion drawing of three-dimensional crystal structure.

Laser Hght beam was directed to illuminate fine particles in a few lowest layers

86

Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals

and in several vertical layers. Thus, in the top view, fine particles in the lowest layer are indicated by bright spots and those in the second lowest layer by rather dimmed spots. By the comparison with the result of the two-dimensional MonteCarlo simulation, the particle arrangement was supported to be in the solid phase [9]. Some particles in the lowest layer are seen to be piled up by fine particles in another layer. From the correspondence to the side view, it is found that they are in the third lowest layer and the structure of the arrangement is face-centered or body-centered as shown in Fig.3. The following relationships hold among the lattice constants a, 6, and c: a = 6 = c for fee and 2^/^ a = 6 = c for bcc. The average lattice constants and the standard deviations of the formed crystal in the experiment were evaluated for 11 images during 10 seconds to be a = 106 ± 2.6 At m, 6 = 159 ± 4.1 n m, and c = 162 ± 2.4 A6 m (see Fig.4). Since the relationship among these three lattice constants is 2^/^ a F I G U R E 6. Top and side views of formed two-dimensional Coulomb crystal.

Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals

89

is seen in the figure that the increase of the lattice constant c with the decrease of the density is larger than the lattice constant aiov N^ 7 X 10^ cm~^ or c < 220 n m, while smaller for iV < 7 X 10^ cm~^ or c > 220 IM m. The repulsive shielded Coulomb force isotropically acts on paricles in the plain parralel to an electrode, although the resultant of the Coulomb force and directional forces does in the vertical direction. Therefore the difference in the tendency of the change of lattice constant between a and c should be caused by such directional forces. The experimental result indicates that the force acting on particles in the vertical direction is repulsive for c < 220 M m while attractive for c > 220 /i m. Gravity, ion drag force, or electrostatic force is directional in the vertical direction, however they does not produce an attractive force between particles. If negatively charged particles and positive ions surrounding them are polarized in the vertical direction, distant particles attract each other while close particles repulse it in the direction. Such polarization may arise from electrostatic field near plasma-sheath boundary or Wakefield generated by positive ion flow toward an electrode [13]. CONCLUSION Coulomb crystals were formed by growing carbon fine particles in plasmas and lattice constants were determined from CCD images. Smaller particles formed the structure of a three-dimensional Coulomb crystal, while larger particles did that of a two-dimensinal one. The three-dimensional crystal structure was strictly facecentered orthorohmbic. It is suggested that the ratio of the lattice constants of the crystal is decided so that Coulomb energy takes a minimum value under the 1

^600 =1. ^^^ ^-» C

£(0

1

0

J

0

a

J

8

§400 O o

1—1—r-i

Q

(0 (0

0)

1

0

-

-"200 -

•

•

• :

•\

,

C •• —

10*

J

1

1—•-•

•

1

10"

.-3»

Particle Density (cm ) F I G U R E 7. Change of lattice constants a and c of the crystal with fine particle density.

90

Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals

condition of constant particle density in horizontal layers. T h e structure of the twodimensional Coulomb crystal was simple hexagonal. The result of the dependence of lattice constants with particle density for the crystal indicates that the force acting on particles in the vertical direction is repulsive for smaller c while attractive for larger c. T h e resultant of the isotropic shielded-Coulomb force and directional forces act on fine particles in the vertical direction. The directional forces are mainly external ones for smaller particles, however a directional attractive force between particles plays an important role for the formation of a Coulomb crystal for larger particles.

REFERENCES 1. Hayashi, Y., and Tachibana, K., Jpn. J. Appl. Phys. 33, L804 (1994). 2. Lin, I, and Chu, J., H., Phys. Rev. Lett. 72, 4009 (1994). 3. Thomas, H., Morfill, G., E., Demmel, V., Gorre, J., Feuerbacher, B., and Mohlmann, D., Phys. Rev. Lett. 73, 652 (1994). 4. Hayashi, Y., and Tachibana, K., J. Vac. Sci. Technol. A 14, 506 (1996). 5. Hayashi, Y., Takahashi, K., and Tachibana, K., Advances in Dusty Plasmas , Singapore: World Scientific, 1997, pp.153-162. 6. Hayashi, Y., and Takahashi, K., Jpn. J. Appl. Phys. 36, 4976 (1997). 7. Hayashi, Y., and Tachibana, K., Jpn. J. Appl. Phys. 3 3 , L476 (1994). 8. Hayashi, Y., and Tachibana, K., Jpn. J. Appl. Phys. 33, 4208 (1994). 9. Hayashi, Y., submitted . 10. Totsuji, H., Kishimoto, Y., and Totsuji, C , Phys. Rev. Lett. 78, 3113 (1997). 11. Totsuji, H., Kishimoto, Y., and Totsuji, C., Jpn. J. Appl. Phys. 36, 4980 (1997). 12. Hammerberg, J., E., Holian, B., L., Lapenta, G., Murillo, M., S., Shajiahan, W., R., and Winske, D., Strongly Coupled Coulomb System,s , New York: Plenum Press, 1998, pp.237-240. 13. Melandso, F., and Goree, J., Phys. Rev. E 52, 5312 (1995).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

91

Monolayer Plasma Crystals: Experiments and Simulations J. Goree, D. Samsonov, Z. W. Ma, A. Bhattacharjee Dept. of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52246

H. M. Thomas, U. Konopka, G. E. Morfill Max-Planck Institute for Extraterrestrial Physics, 85740 Garching, Germany Abstract. Experiments and simulations are reported for two kinds of monolayer plasma crystals. In a monolayer with a small number of particles, ranging from 1-19, we measured the microscopic structure of very small crystals. The crystals have concentric rings. For certain numbers of particles, corresponding to 'closed-shells' in the outermost ring, the configurations are the same as for hard spheres. Yukawa molecular-dynamics simulations accurately reproduce our experimentals results. By adding far more particles (-10,000) and operating at higher power, we discovered Mach cones. These are V-shaped shocks created by supersonic objects. They were detected in a two-dimensional Coulomb crystal. Most particles were arranged in a monolayer, with a hexagonal lattice in a horizontal plane. Beneath the lattice plane, a sphere moved faster than the lattice sound speed. The resulting Mach cones were double, first compressional then rarefactive, due to the strongly-coupled crystalline state. Molecular dynamics simulations using a Yukawa potential also show multiple Mach cones.

INTRODUCTION Many experimenters have reported Coulomb crystals in dusty plasmas (1-13). In this paper, we report experiments and simulations with monolayer Coulomb crystals. These v/crc carried out by suspending microspheres in a horizontal electrode sheath. Two experiments were performed. In the first experiment, we prepared crystals with small numbers of particles, ranging from 1-19 particles. These configurations are arranged in concentric shells. These arrangements can be categorized, as a function of particle number, as a "periodic table" of small plasma crystals. In the second experiment, which was carried out with -10,000 microspheres in the monolayer, we observed Mach cones. These are V-shaped shock waves produced by supersonic particles. Mach cones are familiar in the field of gasdynamics (14) where they are created, for example, by supersonic aircraft. Less commonly, Mach cones also occur in solid-state matter. For example, surface waves along a borehole, in seismographic testing, propagate faster than the sound speed in solid rock, thereby creating Mach cones in the rock (15). The existence of Mach cones in dusty plasmas was predicted theoretically by Havnes et al (16). Moreover, they predicted that Mach cones are produced in the dust

92

J. Goree et al /Monolayer plasma crystals: experiments and simulations

of Saturn's rings by boulders moving in Keplerian orbits. Dust moves at a different speed, nearly co-rotational with the planet. In the case of Saturn's rings, the dust is probably weakly-coupled, and the relevant sound wave would be the Dust Acoustic Wave (DAW).

PERIODIC TABLE EXPERIMENT In the experiment, we used a modified GEC reference cell with a capacitivelycoupled lower electrode. By filling the chamber to a low pressure of 55 mTorr, and applying a 166 V peak-to-peak radio-frequency voltage at 13.56 MHz, we produced a Krypton glow discharge. The dc self bias was -55 V, and the Debye length measured by a Langmuir probe and using the ABR method, is estimated as 370 |im. We used microspheres of diameter 8.9 ± 0 . 1 |im and density 1.51 g/cm^. By dropping these through a 1-mm opening in an upper electrode, we were able to introduce only one sphere at a time. After it came to equilibrium, we imaged it, using a 512 X 512 resolution digital camera equipped with a Nikon micro lens. Then we added a second particle, and so on. The particle charge was measured as 2 = -12,300 e, using a variation of the resonance method of Melzer and Trottenberg (4,5). Our variation avoids Langmuirprobe measurements of ion density, which can have uncertainties of a factor of 3 or more due to various factors including placing the probe at a location other than the particle height. Instead, we use measurement of the dc self bias, dc plasma potential, and particle levitation height, which can be measured more accurately. Using the assumptions that the dc electric potential in the sheath varies quadratically with height and that the particle height is determined by a force balance with gravity but not ion drag, we are able to compute the particle charge with a smal random error. Particle x-y coordinates were identified from the camera images, and then plotted, as in Fig. 1. We applied Delaunay triangulation to show the bond configuration. For comparison, we also show results from a Yukawa molecular dynamics simulation, which is described later. The simulation parameters were Q = -15,300 e, Xj) = 370 |Lim, m = 5.57 x 10"^" g, and the curvature of the confining potential was parameterized by /: = 1.15 x 10'^ g/s^. We also show results for a hard-sphere experiment, which was carried out simply by dropping metal balls into a spherical bowl. We found that plasma crystals at the smallest size are predictable. Their positions are accurately modeled by a Yukawa simulation. We believe that the Yukawa potential is suitable because the particles lie in a two-dimensional plane that is perpendicular to the ion flow. In a three-dimensional crystal, it is known that ion focusing leads to strongly non-isotropic potentials that cannot be modeled accurately with a Yukawa potential (17). Certain numbers of particles have multiple multiple stable equilibria. One might call these "isotopes." These were most easily identified in the numerical simulations, which we repeated for 100 different random initial seedings of particle positions.

J. Goree et al /Monolayer plasma crystals: experiments and simulations

soft disk triangulation (experiment)

soft disk

•

experiment

O

simulation

93

tiard disk

©

©

(•K ^1 • )

©

closed shell

© ®©

4

® ©©

^

o

P®

© ©o

ers)

03^1 85%

o®© ©

^ 98% closed shell

m^

©I®

4 nnnn

4 mm

FIGURE 1. Particle configurations from the dusty plasma experiment are shown with dots in the left column, and in the Delaunay triangulation in the center column, (continued next page)

94

J, Goree et al /Monolayer plasma crystals: experiments and simulations

FIGURE 1. (continued) Simulation results are shown as open circles in the left column. In some panels, a percentage is indicated, showing the fraction of simulation runs that yielded the equilibrium (isotope) shown; other runs resulted in different isotopes.Hard disk experiment results are shown in the right column. These results are the beginning of a series for particles 1-19. The complete results will be presented elsewhere.

Comparing the dusty plasma results to the hard-sphere analog, we find that the plasma crystal tends to arrange with an azimuthally-symmetric outer ring, whereas the hard spheres arrange in a perfect triangular pattern even if the outer ring is incomplete. For certain numbers corresponding to complete shells, the two are the same. This occurred for 3, 7, 12, and 19 particles. Further details of the these experiments and simulations will be reported elsewhere.

SIMULATION We performed molecular-dynamics simulations of the experiments. Particles were constrained to move in a horizontal plane. The particle equation of motion m d r / d f = - (2 V (jf) - / d r I dt was integrated for N particles. The electric potential (p consisted of a parabolic potential to model the radial confinement from the plasma, plus a Yukawa inter-particle repulsion, (p- -kr^l 2 - 2 (2 / r^) exp (-n / Xjj). Here r is the distance from the central axis, r/ is the distance to particle /, and the sum is over all other particles. The parameter k determines the curvature of the bowl-shaped confining potential. The particles were loaded with random initial positions, and then their motion was followed by integrating their equations of motion simultaneously, using a simple leapfrog integrator. Their motion eventually ceased, as the excess kinetic energy was dissipated by drag, / d r / d f, leaving the particles in a stable equilibrium.

MACH CONE EXPERIMENT Our experiments were carried out in a strongly-coupled plasma, with a particle separation that was smaller than the Debye length. Under these conditions, the sound wave is the Dust Lattice Wave (DLW), which is different from the DAW that propagates under other conditions. The results we report here are peculiar to the crystalline strongly-coupled state. We expect that certain Mach cone features, which we will identify, will be different in a weakly-coupled dusty plasma. We used a larger 230 mm diameter electrode, without a glass insert in the upper ring electrode. Krypton gas was used at the low pressure of 0.05 mBar. Higher pressures result in a damping rate too high to observe mach cones. Approximately 10^ microspheres were shaken into the plasma above the electrode. The rf input power was much higher than in the periodic table experiment. We operated at 50 W, yielding a self-bias of -245 V on the lower electrode. This bias levitated the negatively-charged

J. Goree et al /Monolayer plasma crystals: experiments and simulations

95

particles 6.5 mm above the lower electrode. In the radial direction a gentle ambipolar electric field trapped the particles in a disk approximately 40 mm in diameter. This disk, which we term the "lattice layer," was a two-dimensional lattice, with a particle spacing A = 256 |im. The Debye length was smaller, Xj)- 124 |Lim, with an accuracy of a factor of two. There was very little particle motion. Mach cones in the lattice layer were produced by charged particles moving in a second incomplete layer, 200 \im beneath the lattice layer. This lower layer was populated by less than 10 particles. Unlike the particles in the lattice layer, they moved rapidly in the horizontal direction. Presumably they were accelerated by a horizontal electric field in the sheath, which we cannot explain. Sometimes they traversed the entire disk in a nearly straight line. They may have been single spheres or agglomerates of two or three. Imaging the cones with the digital camera and a horizontal laser sheet, we observed the Mach cones in the lattice layer. These cones are easily seen in the moving video, but more difficult to identify in a still image. The video rate was 50 frames/sec. To process the data to produce a still image of a Mach cone, we carried out the following computational process. Images were analyzed to identify the x-y coordinates of the particles. Particles were threaded from one image to the next, and their velocity was computed as the change in position, divided by the 0.02 sec frame interval. Then, by computing the magnitude of the velocity vector, we produced a map of the particle speed. To reduce noise, we averaged this over nine consecutive frames, which we displaced spatially so that the position of the fast particle coincided. This yielded the image shown in Fig. 2.

2 mm FIGURE 2. Map of particle speed in the lattice layer. Darker grays con-espond to faster particles. A supersonic particle moving at 4 cm/sec moved toward the lower right, producing the Mach cones shown here.

96

1 Goree et al /Monolayer plasma crystals: experiments and simulations

Some peculiar features to note are the multiple cones and the rounded vertices. Analyzing the particle motion, we determined that the first cone is compressional, with particles displaced forward, while the second cone is rarefactive, with particles moving in the opposite direction. We attribute the presence of multiple cones to the strong coupling in the crystalline state. In our experiment, the particles were arranged in a crystalline lattice, and they were deformed elastically as the fast particle passed by. Unlike a gas atom, an atom in a crystal has a memory of its original position. When the crystal is deformed elastically, the atoms are restored toward this position. In our experiment, the particles over-shoot this equilibrium position, and oscillate about it. The oscillation is damped by the gas drag. The rounded vertices are probably due to the finite size of the Debye sphere surrounding the fast particle. In gasdynamics, it is well known that a finite-size supersonic object, such as a sphere, creates a U-shaped Mach cone, in contrast to a needle-shaped object, which produces a V-shaped cone. We carried out molecular dynamics simulations of these experiments. This was done using 2,500 particles, which we allowed to settle into equilibrium positions. We then injected an additional particle, which we constrained to move on a horizontal plane 200 microns below the lattice layer. These simulations also revealed multiple Mach cones. The simplicity of the physics in the simulation demonstrates the simple physical nature of the Mach cones. As predicted by Havnes (16), Mach cones can be used as a diagnostic of the dusty plasma. In our case, we measured the particle charge, 2, from the Mach angle // = sin"l (1 / M), where M = v / c is the Mach number of an object moving at speed v through a medium with an acoustic speed c. Doing this requires a model for the acoustic speed. In our case, with a strongly-coupled two-dimensional suspension and particles separated by a distance large compared to the Debye length, it is valid to use the DLW dispersion relation of Roman et al (9). In the long-wavelength limit, X > 2nA, the DLW has only weak dispersion. Thus, the acoustic speed is a constant, as required for Mach cones. It happens to vary linearly with the particle charge Q. By measuring the Mach angle and the speed of the fast particle from the video, the particle separation from a correlation function analysis of a still image, and the Debye length from a Langmuir probe, we find a particle charge. In our case, Mach angle \i = 33.5° ± 3° for the first second cone yields find 13,000 ..,., I f t ^ f 1 !• i ^ i i f I f i

wssm

, 0. The calculations where performed up to the edge of the structure and the conditions for structure to exist were found from boundary conditions at the surface of the structure. It was found that the structure is regulated by one parameter Nd = {47T/:i)J{P/z)r'^dr. By increasing it one finds its maximum possible value when the structure can exist. It is of order of 1. The structures were investigated in broad range of parameters for the case where the ion-neutral collisions do not play any role and for the case where they are important. Fig.'s 4 illustrate the case of large ion-neutral collision rate; the ion density has a minimum inside the center of the structure and, increases to its periphery and than decreases to n = 1. The drift velocity has maximum close to the surface of the structures. The theory as applied to problems of proplanetary condensations in dust-molecular space clouds. The spherical structures in space are new astrophysical objects- "dust star", their maximum mass can be of the order of planetary mass.

F I GURE 4 Parameter in spherical dust structures a)a:„ = oo h)xn = 1.

DUST VOIDS Dust voids observed in recent experiments have sharp boundaries (^'^). They explanation can be based on same set of nonlinear equations assuming the presence of ionization in the central region. The ionization creates the drag force which moves the dust from central region crating a dust void. The theory of dust voids was formulated both in the limit where the ion-neutral collisions do not play important

104

VN. Tsytovich/Nonlinear dust equilibria in space and laboratory plasmas

role (/^) and in the case where they are important ^^ The corresponding cases are named the collision less voids and collision dominated voids, in analogy with collision less and collision dominated plasma sheaths. But for both types of voids the ion-dust collisions are very important and create the drag force which is the force responsible for the void appearance. The void boundaries are sharp. The voids appear if the ionization rate exceeds critical value. For collision less voids it is illustrated on Fig.5a where the dependence of the void size on the ionization rate Ijxi is shown. The theory of moving voids was created which takes into account the neutral-dust friction force. The phase diagram of the moving void shows the presence of one stationary void position for given ionization rate (Fig.5b). The hart beating voids or breathing voids are explained by an change in equilibrium ionization rate after void expansion or contraction. The jump of the parameter P at the void surface is minimum at the equilibrium position corresponding to the stationary void.

—

'

r-

V

1

1

a1

1

0,05

"

"1

•

r

1

0.01 i

1

—

i , , .-

1

,

\

F I GURE 5 a) void size iis function of l/x^ bjpha^e diagram of a moving void. The collision dominated voids are described by two parameters Xi and Xn^ The quasi-neutral void corresponds to x^ > dijy/r = ri^, the mean free path in ion-neutral collisions much larger than the electron Debye radius. The size of the quasi-neutral void appears to be proportional to Xn with numerical coefficient increasing with AV The Fig.6 shows the parameters of the quasi-neutral void for A^- = 0.1. The quasi-neutral voids for fixed Xi exist for Xn > Xn^rnin- The non quasi-neutral voids exist for ^e < Xny/2Xi = y/2xiX^, The global stability of ID structure including the voids for perturbations along the direction of the parameter distribution in the structure was proved numerically in computations of moving boundaries between the dust side and the void side of the structure. The structures can be shown to be unstable for creation of perturbations in the perpendicular direction which is related to appearance of dust convection (^). There exist a critical parameter related both with dust neutral collisions and ion-neutral collisions which determines the threshold of convection instability.

VN. Tsytovich/Nonlinear dust equilibria in space and laboratory plasmas

X

105

CUM

1.&

»^2

i

o. O.OM

t

,gi»P'--' S^mliis*--

•^

F I G U R E 6 Parameters of quasi-neutral coUisional void

APPENDIX The system of nonlinear equation used is: dm^Vd

1

— = —Vd - £• + nzuadr] dt z

T-

+ 2u . —u + —X- ] =E\dt dr ndrj dn dnu ^ I dt dr xi

1 One ^ _ g ^ Tie or

{A\)

1 + auu)

zodruP

(.42)

x„ ^P ^^ P^r dt z or z

(\

(A3)

106

V.N. Tsytovich/Nonlinear dust equilibria in space and laboratory plasmas

OjTp

The dependence of the drag coefficient adr and the dust capture coefficient ac on the ion drift velocity and the ratio t = r/z is given by the expressions

erf{u) 8ti3

—^

t{-l + 4u^ + 4tt'') + 2t{-l + 2u^) + 4ln T-)] +

'- \t{til + 2u^) + 2 - 4/n ; a,: = -^^^{1 L \aJi Su

+ t + ZuH)

The growth rate F of structurization instability for k = 0: '°

2v^(zo(^o + T + 1)(1 - Po) + Po(^o + r))

^l—-l 40r (A5)

^''''^

REFERENCES {') Tsytovich,V.N., Physics Uspekhi 4.0, 53 (1997). (^) Tsytovich,V.N.,Ya.K.,Khodataev, Bingham,R., and Resendes, D.,Comments on Plasma Pkys. and Contr. Fusion 17, 287 (1996). (^) Tsytovich,V.N.,Khodataev, Ya.K., Bingham,R., Morfill, G., and Winter, J., Comments on Plasma Pkys. and Contr. Ftision 18, 345 (1998). C) Tsytovich,V.N.,Khodataev,Ya.K.,Bingham,R.,and Tarakanov,V.A.,Jounia/Plasma Physics, 5, 32 (1998). i^) Morfill G.,and Thomas, H., J. Vac, Sci. TechnoL, A14, 490 (1996) (®) Bouchoule,A.,Morfill,G., and Tsytovich, V.}^.,Comments on Plasma Phys. and Contr. Fusion 21, 52 (1999). C) Morfill, G., and Tsytovich,V.N.,Pftys«C5 of Plasmas (submitted) (1999). (**) Goree, J.,and Samsonov, S., Phys. Rev, £" (accepted), (1999). n Meltzer, M.,Piel, A. et all in press, (1998). 0°) Goree, J.,Morfill,G.,Tsytovich,V.N.,and Vladimirov,S.V., Phis. Rev. E (accepted) (1999). (11) Goree, J.,Morfill,G.,Tsytovich,V.N.,and Vladimirov,S.V., Phis. Rev. E (submitted) (1999).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

107

Dust Particle Structures in Low-Temperature Plasmas Anatoly P.Nefedov High Energy Density Research Center, Russian Academy of Sciences, Izhorskaya 13/19, 127412, Moscow, Russia

Abstract. Dust particle ordering is discussed in various types of low-temperature plasma, such as thermal plasmas at atmospheric pressure, DC glow discharges, inductively-coupled plasma, UVand radioactivity induced plasmas. The investigations of UV-induced dusty plasma were made under microgravity conditions. Experimental data are presented. Properties of ordered structures are discussed and the conditions of formation considered.

INTRODUCTION A plasma containing macroscopic particles or grains (often referred to as a dusty plasma) has the feature that a particle introduced into such a plasma or produced in it by, say, condensation may be charged by an electron or ion flux or by photo-, thermoor secondary electron emission. Electron emission from a grain surface produces a positive charge, increases the electron density in the gas phase and hence enhances its electrical conductivity. Capturing electrons makes dust grains charged negatively, producing the reverse effect, a reduction in the electron density (1,2). The distinguishing feature of a dusty plasma is that, owing to the relatively large size of a grain (from hundredths of a micron to tens of microns), its charge Zp may assume extremely large values of up to 10-10 elementary charges. Therefore the (Z/-dependent) Coulomb interaction energy of the particles may, on average, be much greater than the thermal energy, which makes the plasma strongly non-ideal. Equilibrium calculations show that under certain conditions a strong intergrain correlation leads to a gas-liquid- solid phase transition and makes the grains arrange themselves into spatially ordered structures similar to those existing in liquids and solids. The electron and ion gases remain ideal analogous to those in a Debye plasma. Crystallike structures that form in plasmas from low-thermal-energy, strongly electrostatically coupled charged particles have been analyzed theoretically by Ikezi (3) and have come to be known as Coulomb or plasma crystals. In later studies, the Coulomb crystals of dust particles were found in a weakly ionized plasma in a radio-frequency (RF) discharge at low pressure (4, 5).

108

A.P. Nefedov /Dust particle structures in low-temperature plasmas

In parallel with the study of the properties of plasma crystals under RF-discharge conditions, in recent years attempts to obtain extended, essentially three-dimensional ordered structures in the bulk of a quasineutral plasma have been made, and structure formation processes for various charging mechanisms, particularly secondary emission and photoemission, have been investigated. An example is the observation of macroscopically ordered structures in a bulk thermal plasma under quasineutrality conditions at atmospheric pressure and temperature of about 1700 K (6, 7). The plasma studied was homogeneous, relatively large in size (its volume of 30 cm""^ corresponding to a particle number of the order o

7

'^

10 and particle density 10 cm"), and free of external electric and magnetic fields. Owing to the large plasma volume and the availability of reliable plasma diagnostics techniques, various types of gas and particle measurements have been performed, plasma state parameters obtained and also comparisons with numerical simulation results has been made. Here we present our studies of the formation of a macroscopic ordered structure in strongly coupled plasmas over a wide range of plasma pressures and temperatures. The plasma was investigated under conditions of low pressure DC gas discharge, inductively-coupled plasma and under thermal plasma conditions. The plasma was also formed from the positively charged dust grains in the presence of a flux of ultraviolet (UV) photons and radioactivity radiation. The investigations of UVinduced dusty plasma were made under microgravity conditions. PLASMA CRYSTALS AND LIQUIDS IN THE DC GLOW DISCHARGE A glow discharge is a non-isothermal room-temperature low-pressure plasma. Experiments were made in neon over a pressure range of 0.1 to 1.0 Torr and an electron and ion density range of 10^ to 10^^ cm""^. The electron temperature ranged between 20000 and 50000 K, and the ion and atom temperature between 300 and 400 K. Under these conditions, strong fields of the stratified discharge kept particles within the volume of the plasma. Observations showed that under certain conditions in such a plasma, quasicrytalline structure forms in the positive column far away from the electrodes, in the ionization instability region (stratum) with a relatively high degree of quasineutrality (-'0.01%) (8). These structures are essentially three-dimensional and their vertical dimension may be as great as several tens of centimeters, with an intergrain spacing ranging from 300 to 400 |im. The grain charge may reach unusually high values (10^ e), which makes it possible to keep the grains in the relatively weak stratum field of order 10 V/cm (to be compared with about 100 V/cm in the near-electrode field in an RF-discharge). By varying the discharge parameters (pressure and current), dust cloud shapes in the range from nearly spherical to cylindrical can be obtained.

A.P. Nefedou/Dust particle structures in low-temperature plasmas

109

Four types of dust spherical particles were dispersed into the column which allowed us to vary their diameters in the wide range from 1.9 |Lim to 63 |am, masses from 5 10'^^ g to 10'^ g and their charges in the range from 10"^ up to 5 10^ e. In the presence of standing or weakly vibrating strata in the column, the particles appeared (usually) as ellipsoidal clouds in the center of their glowing regions. Usually, there are several dust clouds located in neighboring strata, several tens of centimeters away from the tube electrodes. The cloud diameter was 5 to 10 mm for glass microspheres and increased to 20 mm for AI2O3 particles. Note the ordered structure and nearly equal spacing of the grains. In the vertical plane the grains were seen to arrange in chains. In the elliptic case the dust grains arranged themselves in 10-20 (glass spheres) and more (AI2O3) planar layers. The interlayer separation ranged between 250 and 400 i^m and the intergrain spacing in the horizontal plane between 350 and 600 |Lim, corresponding to the particle densities rip^-lO^AO^ cm'^. Clearly, the observed particle pattern is quasicrystalline and essentially three-dimensional in nature. Experimental data show that a stratified discharge has regions of strong and weak electric field alternating periodically along the tube axis. In the region of a strong (10 V/cm) longitudinal electric field in a stratum, a potential well arises in the vertical plane as a result of the balance of the electric and gravitational fields. A similar potential well in the horizontal plane is formed by the high (30 V) floating potential on the walls of the tube. The conclusion to be drawn from these facts is that the dust grains are kept - indeed trapped - by a strong electric field. By varying the discharge parameters (pressure and current) it is possible to change the size of the potential well and hence the shape of the dust cloud. For instance, reducing the discharge current and pressure causes two neighboring elliptic clouds to gradually transform into a cylindric shape with a vertical dimension of several tens of centimeters. Varying the current may violate particle ordering thus causing the quasicrystal to 'melt'. In the case of small melamine formaldehyde particles, an action of the ion drag force beginning to play here an important role may lead to an origin of a convective movement of dust particles. As a result, the dusty plasma structures with a simultaneous existence of crystal, liquid and gaseous phases are observed. GRAIN ORDERED STRUCTURES IN AN ALUMINIZED PROPELLANT FLAME The formation of the particle ordered structure in a solid propellant flame was studied. In our experiment we use aluminized solid propellant. The flame is formed

110

A.P Nefedov /Dust particle structures in low-temperature plasmas

rjf

b) FIGURE 1. Video image (a) and pair correlation function g(r) (b) of the dust particle cloud in the boundary sheath. The bar corresponds to 100 jim.

by ignition of a propellant tablet ('-10 mm diameter and -^30 mm height ) with the aid of an electrical heater. The spectral measurements revealed that a plasma spray of particles contains sodium and potassium atoms with a low ionization potential. As a result, the basic constituents of the plasma studied are charged alumina (AI2O3) particles, Na^ and K^ ions, and electrons. As a diagnostic base to support our experiments, we measure basic plasma parameters such as the alkali atom and ion number densities, plasma temperature, and the diameter and number density of the grains. The plasma diagnosis has been taken for different heights h above a propellant tablet surface. The measurements were performed at a temperature of 1900-2250 K. Microscopic structure observations were made by illuminating a horizontal or vertical plane with a sheet of Ar"^ laser light, with a thickness of 30 |Lim and a breadth of 10 mm. It is adjustable to various heights. Scattered light was viewed at 70^ to the horizontal plane through a transparent wall of glass tube. To observe particles in the vertical plane, receiving optics was positioned at an angle of 90° to the vertical. Individual particles were observed with a charge-couple device (CCD) video camera fitted with a macro objective. A 58-mm macro objective with extension tubes provided magnification from x30 to 140. Our investigations show that propellant flame includes three zones: combustion region (/z-lO mm, rg'-2000 K, ^2p -600

-800 -15001000-500 0 5001000

a) 2000

horizontal position (/^m)

2500

E

2000

o

1000

3. C 1500

'i?

(0

o

500

Q. 75 c o N

0 -500

-1000 -1500

14

16

18

time (s)

22

FIGURE 4. Behavior of the two-particle system during transition into the separated state at 200 Pa. In (a) the particle trajectories from the side and in (b) the horizontal position as a function of time are shown. That means that for alignment the attractive force on the lower particle must exceed the repulsion by a factor of almost 3 under the conditions of this experiment. This instability is not the same as that responsible for the phase transition in plasma crystals. The melting instability is due to a collective interaction of many dust particles. That instability sets in as a direct consequence of reduced pressure since the energy transferred into the plasma crystal cannot be dissipated by dust-neutral

A. Melzer et al. /Attractive and repulsive forces in dust molecules

121

£.V\3\J

£ 3

1000

c

o

0|

*^

"w o •1000 a 75 c o

•2000

,N

O 3000

x: .

laser on

a) 0

2

4

6

8 10 12 14 16 18 20

time (s)

3500

4

6

8

time (s)

F I G U R E 5. Horizontal position of the two particles at 120 Pa as a function of time when (a) the upper particle and (b) the lower particle is pushed by the laser beam. The shaded areas indicate the time when the laser beam was switched on.

collisions. Here, the instability is only due to the strength of the attraction and is only indirectly connected to the discharge pressure, as shown below. In the experiment the strength of the attractive force can be determined from following situation (see Fig. l b ) : when the upper particle is pushed by the laser, both particles move in the same direction. The attractive force on the lower particle (e— \-)QiQ2{^i — X2)/{4:7Teod^) is then balanced by the neutral drag m2(^2^2- Taking

122

A. Melzer et al /Attractive and repulsive forces in dust molecules

the values of X2 = 1.27 m m / s and {xi — X2) = 300 //m from the laser push in Fig. 5a {t = 15.4 s to 18.8 s) the drag force is determined as Fdrag = 6.87 • 10"^^ N and the repulsive force as 1.16 • 10"^^ N yielding e = 5.9. This value is decisively larger than the critical value of ec, thus showing that the observed aligned situation is in the stable region predicted by the model. The breakup of alignment at higher pressure can be understood by recalling how the attractive positive ion cloud arises: The streaming ions in the sheath are deflected below the dust particles by ion-dust Coulomb collisions. Ion-neutral charge exchange collision play an important role as a scattering process that tends to destroy this ion cloud [10]. With increasing discharge pressure the ion mean free path for these collisions is reduced and the ions are deflected closer to the dust particle, thus reducing the attraction provided by the ion cloud. When the attraction parameter e drops below its critical value the aligned situation becomes unstable and the interaction forces become net repulsive thus establishing the separated state. In conclusion, the two-particle system is a very useful tool for the quantitative determination of the interaction forces between dust particles. By laser manipulation of a two-particle force probe we have shown the existence of net attractive and asymmetric forces leading to the formation of dust molecules. With increasing pressure a transition to a net repulsion is found. This behavior has been explained in terms of our previous plasma crystal models.

REFERENCES 1. Chu J.H. and I Lin Phys. Rev. Lett. 72 4009 (1994) 2. Thomas H., Morfill G.E., Demmel V., Goree J., Feuerbacher B. and Mohlmann D. Phys. Rev. Lett. 73 652 (1994) 3. Melzer A., Trottenberg T. and Piel A. Phys. Lett. A 191 301 (1994) 4. Trottenberg T., Melzer A. and Piel A. Plasma Sources Sci. Technol. 4 450 (1995) 5. Melzer A., Homann A. and Piel A. Phys. Rev. E 53 2757 (1996) 6. Thomas H. and Morfill G.E. Nature 379 806 (1996) 7. Vladimirov S. V. and Nambu M. Phys. Rev. E 52 2172 (1995) 8. Melands0 F. and Goree J. Phys. Rev. E 52 5312 (1995) 9. Melzer A., Schweigert V.A., Schweigert I.V., Homann A., Peters S. and Piel A. Phys. Rev. E 54 46 (1996) 10. Schweigert V.A., Schweigert I. V., Melzer A., Homann A. and Piel A. Phys. Rev. E 54 4155 (1996) 11. Schweigert V. A., Schweigert I. V., Melzer A., Homann A. and Piel A. Phys. Rev. Lett. 80 5345 (1998) 12. Schweigert I. V., Schweigert V. A., Bedanov V. M., Melzer A., Homann A. and Piel A. JETP 87 905 (1998) 13. Takahashi K., Oishi T., Shimomai K., Hayashi Y. and Nishino S. Phys. Rev. E 58 7805 (1998) 14. Homann A., Melzer A., Madani R. and Piel A. Phys. Lett. A 242 173 (1998) 15. Homann A., Melzer A. and Piel A. Phys. Rev. E 59 3835 (1999)

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

123

Self-Organization in Dusty Plasmas S. Benkadda^ V.N. Tsytovich^ S.I. Popel^ and S.V. Vladimirov^ ^ Equipe Dynamique des Systemes Complexes, CNRS- Universite de Provence, Centre de St Jerome, Case 321, 13397, Marseille Cedex 20, France ^ General Physics Institute, Vavilova 38, Moscow 117942, Russia, ^ Institute for Dyn. of Geosph., Leninsky pr, 38, hid. 6, 117979 Moscow, Russia ^ School of Physics, The University of Sydney, NSW 2006, Australia

A b s t r a c t . Dusty plasma systems as open and highly dissipative media have an inherent tendancy to self-organization resulting in formation of dissipative structures. Two types of self-organized structures are analyzed, the dustplasma sheaths and dust nonlinear drift vortices. It is shown that all properties of the dust-plasma sheath depend on only one parameter - the Mach number of the ion flow, which have allowed zones and can be less than unity, in contrast to the features of the plasma sheath without influence of dust. It is also shown that high rate of dissipation in dusty plasma increases the growth of drift vortices.

INTRODUCTION Self-organization processes in dusty plasma are expected to be very important since the latter is an open sytem with a high rate of dissipation. The latter is produced by dust particles which absorb plasma particles. Dust plasma systems can not survive in absence of either external sources of electrons and ions or plasma particle fluxes from the regions where there is no dust. The plasma particle fluxes recombine on the dust particles as well as the energy fluxes are absorbed by the dust particles. Thus external sources of energy and particles are necessary to exist to compensate the absorption in the plasma particles and absorption of the energy on dust particles [1]. The presence of a high rate of disssipation provides rapid developement of self-organization processes and formation of long lived dissipative structures. These structures should be investigated flrst as single isolated structures to uderstand they properties which will be alterated in presence of other structures in the system. We consider here two types of such dissipative structures : the

124

S. Benkadda et al /Self-organization in dusty plasmas

dust-plasma sheath and the dust-plasma drift vortices. This choice is related to different applications such as plasma etching, dust crystals and turbulence in edge tokamak plasma and ionospheric plasmas. Nonlinear effects are very important for all these structures. These nonlinearities are very different from those in absence of dust and are related to large dust charges, as well as to the interaction of plasma particles with dust particles. The presence of high dissipation rates make these structures unique. We show that the dissipation lowers the possible dependence of the structures on parameters and often the dissipative structures depend only on very few parameters. The dustplasma sheaths depend only on the Mach number of the incident ion flux. Drift vortices in dusty plasmas become much more unstable than in absence of dust, then they evolve to the marginal stable state depending on few parameters. This tendency is natural since the other possible nonlinear states dead out due to the high rate of dissipation and only the structures for which the balance between dissipation and excitation occurs finally survive.

DUST-PLASMA SHEATHrA NEW DISSIPATIVE SELF-ORGANIZED STRUCTURE The presence of dust near the wall of a low-temperature plasma leads to formation of a dissipative structure confining dust and plasma particles in a selfsimilar way and creating the specific dust-plasma layer. This equilibrium structure is formed self-consitently with the fields of the dust particles. The problem of plasma-wall boundary is known in physics since the pioneering works of Langmuir in the 20s. The specific feature of the plasma sheath [2], which is formed near the conducting wall, is the existence of flows of plasma particles towards the wall, the latter being negatively charged because of the difference of electron and ion masses and temperatures. In the presence of a relatively high energy flux, the impurity contaminants, or dust^ consisted of macro-scaled (comparing with sizes of electrons and ions) grains can be ejected from the wall. Dust strongly influences all parameters of the sheath, in particular, the electric field distributions and the ion flow velocities. The electric fields of dust particles contribute to the distribution of the electric potential in the sheath and influence the ion flow; at the same time, the grain fields themselves strongly depend on the flow. In the absence of dust, there are two regions in the near-wall plasma [2]: the plasma sheath itself, where the main drop of the electric field potential occurs, and the pre-sheath, where the potential drop is small and ionization as well as ion acceleration occur to result in the plasma recombination on the wall. In the presence of dust, there are three distinguished layers: the plasma boundary pre-sheath layer (containing no dust), the dust cloud (there is the main drop of the electric field potential), and the wall-plasma layer (containing no dust). In contrast to the plasma sheath in the absence of dust, the dust-plasma sheath can support only spe-

125

S. Benkadda et ah /Self-organization in dusty plasmas

cific Mach numbers (which can now be less than unity) of the ions flowing toward the dust cloud and the wall, thus creating regions of allowed ion stream velocities. These results are important for dust-plasma experiments, plasma processing and fusion (formation of dust layers in edge plasmas). Consider Boltzmann distributed electrons rie — noexp(e(/p/Te), UQ is the unperturbed electron density in the region where quasineutrality holds (i.e., in the presheath), (/? is the electric field potential (which is practically zero in the pre-sheath), e is the electron charge, and Tg is the electron temperature in the energy units. The dissipation on dusts for electrons is negligible as compared to pressure and electric forces. The forces acting on the dust particles are due to the ion drag and the electric field —ZdeE^ where Zd is the dust charge in units of the electron charge e. For simplicity, we do not account for other forces such as gravity, thermophoretic force, etc., and suppose the electron and ion temperatures being constant in the sheath region. The ion drag force due to collisions of ions with dust can be written as ^dr = f^i^iUid = aPzuadr/TeXj)-^ where m^-, t;^-, and Uid are the ion mass, speed, and the effective collision frequency with the dust grains. Here, we have introduced the dimesionless quantities: the parameter P = UdZd/no^ where Ud is the dust number density in the sheath, the normalized dust charge z — Zdc'^/aTe^ where a is the dust grain size, the normalized ion speed u — v^|\/2vTi'> where VTI = {Ti/miY^'^ is the ion thermal velocity (with Ti being the ion temperature), X^i = {Ti/Anrioe'^y^'^ is the ion Debye length, and defined the drag coeflBcient adr- The force balance equation for dust is then P{-E + zunadr) = 0, (1) where E = eXj^-E/aTe is the dimensionless electric field (this is the field in which ions get the energy Tg at the distance Xj^Ja)^ and n = rii/no is the normalized ion density. Thus E = zuna^r when P 7^ 0. Equation (1) describes the balance of the electric field and ion drag forces; the important point here is that the electric field E is the self-consistent field which includes fields created by charged dust grains. Note that in our approximation (zero dust temperature) there can exist a step change of P from zero to a finite value. Since Zd is determined by the parameters of plasma particles, its value is continuous, and therefore rid can have a jump. To clarify that, generalize the force balance equation (1) for the nonzero dust pressure d fP\

P f

r.

2

\

X = xa/Xj)- is the dimensionless space variable and Td = Tde'^/aT^ characterizes the dust temperature Td. We see that if this parameter is small, then the boundary of the dust cloud is sharp. Eq. (1) is derived for r^f -> 0 when there are two solutions, P = 0 in the dust-free region, and P > 0 in the region occupied by dust. The balance of the electric and pressure forces acting on plasma electrons is given in dimensionless form by £' = —dne/riedx. Thus we have from (1) — —— = -zunadr^

(3)

126

S. Benkadda et al /Self-organization in dusty plasmas

The drag coefficient adr is a function of u and r jz^ where r = Ti/Te^ and includes the charging collisions as well as the Coulomb scattering collisions. In the limit T/Z ei^i{ps/Ln)' where k^ is the wave vector component parallel to the magnetic field, z>e is the frequency describing the momentum transfer from plasma particles to dust particles [8]. The coefficients a and /? describe the infiuence of charging and Coulomb collisions: 0>i{ps/Ln)' where v^.i are the collision frequencies in the continuity equations for electrons and ions (see [1]). To describe the excitation of the drift vortices we have to find the unstable solutions of the linearized set of the modified Hasegawa-Wakatani equations. The linearized set written in Fourier-components is

{-ioj + a + Ce)n - azC, + {iky - Ce)'il^ = 0, {-iuj + /9)C = a{l + P)n. These equations allow us to obtain the dispersion relationship for ui = w / a :

' \

{1 + P)a

a{l + P)[-iL0 + l + {cJa)-{z{l

+ P))/{-iu

+ (3/a)]

S. Benkadda et al /Self-organization in dusty plasmas

133

The dispersion equation has been solved for different limiting cases. The unstable solutions have been found. For example, for Cg ^ a (the case when the effect of non-adiabaticity is much weaker than that of dust charging) and P denotes the statistical average. In MD simulation, the time average over a sufficiently long time period is used to evaluate < > after the system reaches a thermodynamical equilibrium and periodic rescaling of particle velocities is no longer employed. It is also know that the self-diffusion coefficient D may be evaluated from the velocity autocorrelation function Z{t) through

D = ll^Z{t)dt.

(4)

The velocity autocorrelation function Z{t) is defined by

Z(0 = (v(t).v(0)), where v(i) is the velocity of a particle at time t. We have evaluated self-diffusion coefficient D from MD simulations using both Eqns. (3) and (4).

138

S. Hamaguchi, H. Ohta/Dynamical properties of strongly coupled dusty plasmas

r

r

r

,—

r

0.14

r-

1

1—

A B

o + + G

,

0.12

~

0.1

-

0.08

Q

6 H

• \

G

o

-

s* * ®

0.06

6

9

n

•j

#*

0.02

-\ H

9

0.04

+

L

L

i_

5

T*

6

10

Figure 2: The normalized self-diffusion coefficients D* as a function of the normalized temperatures T* for various values oi K {OA < K < 5). A(o) and B(+) indicate D* values evaluated from Eqns. (3) and (4). Figure 2 plots the dimensionless self-diffusion coefficient D* = ^/SD/CJE^^^ for various K, values (0.1 < AC < 5), where u^ is the Einstein frequency for the fee lattice of Yukawa systems defined by CJ-

Here (f) is the Yukawa potential given by Eq. (1), and particles are assumed to be at the fee structure sites. The Einstein frequency UJE denotes the harmonic oscillation frequency of a particle around its equilibrium site (an fee site in our case) when all other particles are located at their equilibrium sites. The Einstein frequency UJE depends on K, and OUE —^ ^p (plasma frequency) as K —> 0, i.e., in the limit of the classical one-component plasma (OCP), which is a system of mobile charges immersed in a strictly uniform neutralizing background..^"^^ The abscissa of Fig. 2 is the Yukawa system temperature T normalized by the melting temperature (i.e., fluid-solid phase transition temperature) T^, i.e., T* — TjTm ~ r ^ / r . As shown here, when the system is in a fluid state close to

139

S. Hamaguchi, H. Ohta /Dynamical properties of strongly coupled dusty plasmas the melting point (T* < 10), the normahzed self-diffusion coeJB&cient D* almost hnearly depends on the normalized system temperature T* and hardly depends on n. Therefore we may scale

Z ) * - a ( r - l ) ^ + 7,

(5)

with a c^ 0.01, /? — 1, and 7 c^ 0.003 for the range 1 < T* < 10, all coefficients being independent of K. The scaling of Eq. (5) is consistent with earlier simulation results for OCP by Hansen et alJ and those for Yukawa systems with limited range of parameters (5 < ^c < 16, 0.5 < T* < 2) by Robbins et al}^ However, our simulation also shows t h a t the dependence of those coefficients on K (that we ignored above), especially that of /?, becomes pronounced for large T* ( > 50).

IV,

Summary and discussion

We have obtained the self-diffusion coefficients for Yukawa systems in a wide range of the thermodynamic parameters. When the system temperature is not too high above the melting point (T* < 10), the normalized self-diffusion coefficient D* is almost independent of K and its dependence on the normalized temperature T* is given by Eq. (5). The independence of D* on K, may be accounted in the following manner. When the system is in a fluid state close to the melting point, motion of particles may be seen as oscillation (at lea^t for a short duration of time) about their equilibrium sites and particle diffusion results from hopping motion of such an oscillating particle from one equilibrium site to another. This self-diffusion process may be characterized by the diffusion coefficient given by D = C A r ^ / A t , where C is a constant, A r is the oscillation amplitude, and A i = u^^ is a typical time scale. The Lindemann criterion^^ states that fluidsolid phase transition occurs when the ratio Ar/a is nearly constant regardless of the interparticle potentials of the system, we may consider systems of the same T* have approximately the same ratio JR = Ar/a^ regardless of /^ as long as T* is relatively small. Therefore D* oaD/uji^o^ — CB? is independent on n for a given T*. However, for larger T* ( > 50), our simulations show that the the K dependence of the parameters for Eq. (5) is more pronounced and cannot be ignored.

Acknowledgements The authors thank R. T. Farouki for useful discussion on MD numerical methods.

140

S. Hamaguchi, H. Ohta/Dynamical properties of strongly coupled dusty plasmas

References ^S. Hamaguchi and R. T. Farouki, J. Chem. Phys. 101, 9876 (1994). ^S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, Phys. Rev. E 56, 4671 (1997). ^S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, J. Chem. Phys. 105, 7641 (1996). ^R. T. Farouki and S. Hamaguchi, J. Chem. Phys. 101, 9885 (1994). 5R. T . Farouki and S. Hamaguchi, J. Comp. Phys. 115, 276 (1994). ^S. G. Brush, H. L. Sahlin, and E. Teller, J. Chem. Phys. 45, 2102 (1966). ^J. -P. Hansen, I. R. McDonald, and E. L. Pollock, Phys. Rev. A 11,1025 (1975). *M. Baus and J.-P. Hansen, Phys. Rep. 59, 1 (1980). ^Strongly Coupled Plasma Physics, (F. J. Rogers and H. E. DeWitt, eds.). Plenum Press, New York (1986). i"R. T. Farouki and S. Hamaguchi, Phys. Rev. E 47, 4330 (1993). " M . O. Robbins, K. Kremer, and G. S. Grest, J. Chem. Phys. 88, 3286 (1988). ^^F. A. Lindemann, Z. Phys. 11, 609 (1910).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

141

Structures and Structural Transitions in Strongly-Coupled Yukiiwa Dusty Plasmas and Mixtures Hiroo Totsuji,* Chieko Totsuji, Kenji Tsuruta, Kenichi Kamon, Tokunari Kishimoto, and Takashi Sasabe (*[email protected]) Faculty of Engineering^ Okayama University, Tsushimanaka 3-1-1, Okayama 700-8530, Japan

Abstract. Regarded as Yukav^a systems in external confining potentials, structures and transitions in dusty plasmas have been analyzed theoretically and by numerical simulations. The one-dimensional confinement in three dimensions and the radial confinement in two dimensions are considered. The results of simulations have been reproduced to a good accuracy in both cases. It is shov^n that the inclusion of the correlation energy or the effect of strong coupling is of essential importance in the theory. A crossover from the surface freezing of Coulomb system to surface melting of systems of short-ranged interactions is observed. INTRODUCTION Physics of dusty plasmas, assemblies of macroscopic particles immersed in plasmas, is closely related to important practical problems in plasma processes of semiconductor manufacturing and also to subjects of basic statistical physics. Direct observations of structures such as crystals and their transitions have inspired us to ask what determines these structures. The purpose this paper is to answer this question through numerical simulations and theoretical analyses. We simplify the system as far as possible and try to find essential factors in the formation of structures and their transitions. The main message is that the contribution of the correlation (cohesive) energy in dusty plasma is of essential importance in these phenomena and we can reproduce them by taking the correlation energy into account properly. SIMPLE MODEL OF DUSTY PLASMA Let us consider the case where our dusty plasma is formed above a wide horizontal plane electrode as shown in Fig.l [1]. Dust particles are levitated by the electric field in the vertical gravitational field in the direction of ~z. We adopt the ion matrix sheath model and, for simplicity, assume that the density of charges in the sheath (except for those of dust particles) is given by erish, e being the elementary charge, and is nearly constant in the domain of interest.

142

H. Totsuji et ah /Structures and structural transitions in strongly-coupled Yukawa plasmas

The potentials for a dust particle of mass m and charge —q are written as mgz + 271 qerishZ^' In the domain z < 0, dust particles are in the potential well: (l>ext{z < 0) = mgz + 27rqenshz'^ = (l>ext{zo) + 2nqensh{z ~ z^f,

(1)

where ZQ = —{g/^T^^'^shYj^/Q) < 0. In the domain z > 0, we have (j)ext{z > 0) == mgz. In our model, the external potential (t>ext cannot keep dust particles afloat in the domain of neutral plasma. We regard dust particles as interacting via the isotropic repulsive Yukawa potential (5^/r)exp(—r/A), where ~q is the (negative) charge on a dust particle. It has been pointed out that there exists an anisotropic interaction coming from the ion flow in the sheath We may, however, expect to have the cases where the isotropic part of interaction potential plays the central role to determine the overall structure in z-direction, even if the configurations in the a:y-plane relative to adjacent layers are aff'ected by the anisotropic part. ROLE OF CORRELATION ENERGY IN STRUCTURES OF DUSTY PLASMA UNDER ONE-DIMENSIONAL CONFINEMENT In the case of one-dimensional geometry discussed above, we have analyzed the structure of the Yukawa system confined by the potential v^xti^) — {}./2)kz'^ by molecular dynamics simulations and theoretical approaches [2-6]. In this case, k — ATrqerish' At low temperatures, dust particles form layers conformal with the confining potential and the number of layers are determined by the parameters ^ = a/A, and T] = {TI^I'^/2)[{l/2)ka^/[q^/a)], a = {TTNS)-^^^ being the mean distance determined by the surface density Ns. The main results of simulations are expressed as the phase diagram shown in Fig.2. This phase diagram has been reproduced by our theoretical calculation and it is shown that the correlation (cohesive) energy of Yukawa particles makes an essential contribution in constructing the phase diagram [5]. TWO-DIMENSIONAL YUKAWA SYSTEM We now consider the case where dust particles are confined in a plane near the boundary between the plasma bulk and the sheath and they are also confined laterally by an electrode surrounding dust particles. We denote the coordinates as r = (R-^z)^ R being the xy components. The electrostatic potential of the surrounding electrode may be approximately expressed in the form {1/2)KR\

(2)

We thus have a two-dimensional Yukawa system confined by a parabolic potential. At low temperatures, this system is characterized by a single parameter

a = q^lK\\

(3)

H. Totsuji et al /Structures and structural transitions in strongly-coupled Yukawa plasmas

143

Structures at Low temperatures (Simulation) We have performed molecular dynamics simulations at constant temperatures on this system. Some results for the structures at local minimum of the total energy at low temperatures are shown in Fig. 3. For smaller systems, the star-like structure becomes global minimum for N==6 and larger. With the increase of the number of particles, these structures gradually change into triangular lattice in the central part and surrounding circular structures. The surface number density p{R) is shown as a function of radius in Fig.4. In the case with relatively large number of particles, the distribution is characterized by this function. In the case of unscreened Coulomb interaction, two-dimensional clusters have been simulated [8,9]. Structures of dust clusters have recenly been observed experimetally [10]. Structures at Low temperatures (Theory): Role of Correlation Energy In the case of relatively large number of particles on a plane, we may describe the distribution of particles by the isotropic surface density p(R) — p{R) and p{R) = 0 for Rm < R. When the Yukawa particles are distributed uniformly on a plane z = 0 with the surface density po, the interaction energy per unit area is thus given by irq^Xp^. Based on this, we estimate the interaction energy Uint neglecting the edge effect. Together with the external potential, we have

Uint - / dRWMRf.

Ue,t = I dIi^KR''p{R).

(4)

We find p{R) and Km which minimize the value of Uint + Uext- The results are AV(^) = ^

{{Rm/Xy - (R/X?) ,

{ ^ y

- SaN.

(5)

When compared with results of simulations, this result underestimates the surface density, as shown in Fig.4 or particle distributions are more compact in reality. In this calculation, the correlation (cohesive) energy between particles has been neglected. Since the correlation energy is negative, particles can be distributed more closely when the correlation energy is taken into account. The correlation energy (per unit area) of the two-dimensional Yukawa lattice of the surface density po is expressed by a function ecoh{^/^pl ) as q'^pl Ccohi'^/^Po ) [6]. This expression provides us with approximate values of the cohesive energy of two-dimensional Yukawa system at low temperatures. When we take the cohesive energy between particles into account within the local approximation, we have finally the results are plotted in Fig.4. We observe that theoretical results for the density and the maximum radius are greatly improved and the results of simulations are almost reproduced. We here emphasize again that the contribution of the correlation energy is of essential importance for this improvement.

144

H. Totsuji et al /Structures and structural transitions in strongly-coupled Yukawa plasmas

Surface Melting vs. Surface Freezing We have analyzed the melting of Yukawa system confined in the onedimensional geometry and have shown that with the increase of the temperature, the intra-layer melting and inter-layer melting occur in this order [7]. We here observe the melting of the laterally confined two-dimensional Yukawa system. The results of numerical simulations are shown in Fig.5 as orbits of particles in a certain duration of time. We note the crossover from the surface melting to core melting with the increase of the parameter a. When a 1, separate two-dimensional Yukawa systems, each being composed of one species, when ry 1.5), the assumption of the long-wavelength limit is not correct any more. Instead, a more detailed calculation taking into account the ^-dependence of the collision integrals has to be performed, which is also feasible.

158

A. Wierling et al /Dynamical structure factor of dusty plasmas including collisions

IV

CONCLUSIONS

Based on a many particle approach the dynamical structure factor for a dusty plasma has been calculated taking into account collective effects as well as collisions. This was achieved using a consistent perturbation expansion based on a generalized linear response theory. The heart of our treatment is the systematic determination of the force-force correlation function which in turn leads to expressions beyond the relaxation time approximation. Making use of the structure factor the dispersion relation for collective modes in dusty plasmas can be derived. In particular, we study the dispersion for dust acoustic waves and obtain Eq. (16) as a result, taking into account dust-ion, ionneutral and dust-neutral collisions. In particular, for dust-neutral collisions the result of Pieper and Goree was re-derived. Since the proposed treatment is valid in the entire {k^uS) plane, improved dispersion relations can be obtained be higher order expansion beyond the static or long-wavelength limit. In this way, diffusion and/or the dispersion of the plasmon can be included.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

F. Melaiids0, Phys. Plasmas 3, 3890 (1996) A. Homann, A. Melzer, S. Peters, A. Piel, Phys. Rev. E 56, 7138 (1997) J.B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1997) A. Baxkan, R.L. Merlino, and N. D'Angelo, Phys. Plasmas 2, 3563 (1995) R.L. Merlino, A. Barkan, C. Thompson, and N. D'Angelo, Phys. Plasmas 5, 1607 (1998) N.N. Rao, P.K. Shukla, and M.Y. Yu, Planet. Space Sci, 38, 543 (1990) X. Wang and A. Bhattacharjee, Phys. Plasmas 4, 3759 (1997) P.K. Kaw and A. Sen, Phys. Plasmas 5, 3552 (1998) M. Rosenberg and G. Kalman, Phys. Rev. E 56, 7166 (1997) M.S. Murillo, Phys. Plasmas 5, 3116 (1998) A.V. Ivlev, D. Samsonov, J. Goree, G. MorfiU, V.E. Fortov, Phys. Plasmas 6, 741 (1999) G. Ropke, Phys. Rev. E 57, 4673 (1998), G. Ropke and A.Wierling, Phys. Rev. E 57, 7075 (1998) D.N. Zubarev, V. Morozov, and G. Ropke, Statistical Mechanics of Nonequilibrium Processes (Wiley-VCH, Berlin, 1997) L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics (Addison-Wesley, Redwood, 1989) M.J. Baines, LP. Williams, and A.S. Asebiomo, Mon. Not. R. Astron. Soc. 130, 63 (1965) T.K. Aslaksen, O. Havnes, J. Plasma Physics 51, 271 (1994) R. Redmer, Phys. Reports 282, 35 (1997)

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

159

Melting of the defect dust crystal in a rf discharge I.V Schweigert\ V.A Schweigert^, A Melzer\ and A. PieP ^Institute of Semiconductor Physics, Novosibirsk, Russia, [email protected] ^Institute Theoretical and Applied Mechanics, Novosibirsk, Russia ^ Institut fur Experimentelle und Angewandt Physik, Christian-Albrechts-Universitat, 24098 Kiel, Germany Introduction Recently a lot of experimental and theoretical studies are devoted to phenomena of the Wigner crystallization of dust particles in the sheath of a rf discharge. The experimental observations of the dust crystal indicate that a Debye-Huckel model of isotropic interparticle interaction breaks down in moving plasma. The dust crystal in the sheath exhibits a specific crystalline structure in which the negatively charged particles are arranged in vertical chains, whereas in the horizontal plane the particles form the usual hexagonal lattices. This is not the most energetically favorite structure for the Coulomb (screened) isotropic potentials. The ion flux flowing through the sheath is focused by negatively charged particles that leads to formation of areas of enhanced ion density behind upperstream particles These areas provide the attractive force for lower particles and explain the vertical alignment of particles. The phenomena of particle attraction in ion flux was considered in [1] within a coUisionless approach and in [2], using a fluid approximation The trajectories of ions flowing through the sheath containing the bilayer crystal were calculated in [3] by the Monte-Carlo technique. It was established that the inter-particle interaction is anysotropic due to asymmetric screening of particles into ion flux. Model of the dust crystal in the sheath In order to study the dynamics of the dust crystal we developed the semi-empirical model with asymmetric inter-particle interaction [3]. In this model the inhomogeneous ion density around a particle is replaced by the isotropic ion distribution characterized by the screening length and an effective positive charge placed below a particle. The value of effective positive charge Zc and its coordinates for experimental conditions of Ref [4] were found in [3] in non-self consistent Monte-Carlo calculations. Late selfconsistent calculations of the dust bilayer crystal in the sheath were carried out with using 3D PIC MCC method (Three Dimensional Particle In Cell Monte Carlo Collisions) [5] It should be noted that 3D PIC MCC algorithm allows to obtain all the main parameters of the plasma-crystal system: a particle charge, the ion distribution, the inter-layer and the inter-particle distances. Besides, these calculations are very time consuming and it is not possible to apply this technique to dynamics problems such as melting or wave propagation. The important point of our semi-empirical model is that positive charges Zc are rigidly connected to their parent upperstream particles. In Fig. 1 the ion distribution is shown for the case when particles are in equilibrium position and

/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge

160

when the lower particle is shifted, ( see Ref. [5]). The ion distribution fast relaxes 2 0 0

2 0 0

H 150 H 100

H 50

3000

3200

3400

3600

3000

3200

3400

3600

z, miorons o

-3

Fig. 1 Ion density distribution (in units of 10 cm") for shifts 5x of the lower layer relative to upper one along the x axis: (a) 5x=0, (b) 5x=130 \im\ z=0 - cathode location. around a parent particle while its moving. It is apparent from Fig. 1 that the ion distribution around the upperstream particle does not depend on lowerstream particle position. We calculated forces acting on the particles into ion flux as function of their shifting and found coordinates and a value of positive charges, which fits well the attractive force acting on a particle in the horizontal plane. Fig. 2. 100

1o J 1

0,1

-0,08

0,1

0,2

0,3

X / e

0,4

Fig. 2 Restoring force acting on the particle in ion flux as fiinction of particle shifting for different ion free paths: 50|im(l), 100|im (2) 200|im(3).

0,10

0,15

0,20

0,01

Fig. 3 Particle kinetic energy in the upper layer (triangles), in the lower layer (squares) and in experiment (circles). The vertical dashed lines shows the instability (from linear analysis) and melting threshold.

Thus, the model dust crystal consists in the hexagonal lattices of negatively charged

/. V Schweigert et al. /Melting of the defect dust crystal in a rf discharge

161

particles and lattices of effective positive charges between them. The effective charges are rigidly connected to upperstream particles. The particles and effective charges move in the horizontal plane ( the experiment [4]). Melting transition. Linear analysis. The experimental observations [4,6] showed that the melting transition of dust crystals in the sheath is a non-ordinary process. The decrease of gas pressure causes rise of the particle kinetic energy Ek, even though the surrounding gas remains of the room temperature. What initiates the particle heating? The mechanism of particle kinetic energy growth was explained in [3] The linear analysis of particle motion equations has demonstrated that the system is unstable relative to short wave oscillations at gas friction less than some critical value. We have obtained the eigen vectors and eigen values of the dynamics matrix which are complex. It was established that with decreasing gas friction the unstable modes arise in the crystal that leads to substantial increase of Ek The sheath of a rf discharge containing the dust crystal is an open system and additional input of energy is provided by ion flux. The simulation input parameters were taken from the experiment [4]. The dust particle radius is R= 4.7|j-m corresponding to a dust mass M = 6.73-10"^"^ kg. The dust charge Z=16000 elementary charges, the interparticle distances a=450|j,m and the interlayer distances d=0.8a. The corresponding dust plasma frequency for the experimental conditions is (Op = (Z e /so M a ) =110 s" . The transition from the solid crystal structure to the gas-like state in the experiment is observed by reducing the pressure from 120Pa down to 40Pa corresponding to an Epstein dust-neutral friction coefficient n = 32s"^ down to lOs'V The critical friction of instability was found to be «=0.1575a)p. Applying the fluid approach, in [7] it was found also that the particle temperature rises at some critical gas viscosity. Melting transition. Molecular dynamics calculations. To understand the crystal melting in details we have performed the molecular dynamics simulations of particle behavior with decreasing pressure [8]. The asymmetry of interparticle interaction was described on the base of our model above. The simulation parameters also correspond to the experimental data [4]. We found that the particle heating is a two step process. In the crystalline state at higher gas friction the particles perform harmonic oscillations about equilibrium sites. Reducing gas pressure, we reach the first critical point of friction n=nins and the particle energy Ek increases quickly by orders of magnitude. Fig. 3. The crystal evolves to the state with developed oscillations, but the crystalline structure survives. Further with decreasing pressure at second critical point of friction n= «^^/the crystal transits to the isotropic liquid. Fig. 3. The melting is accompanied with a jump of the energy Ek- It was surprising that the parameter r = Z^/akfiT, (where Z is the particle charge and a is the inter-particle distance) for the 'hot' crystalline state was essentially smaller than the critical melting F of a single layer crystal with the same screened inter-particle interaction. The vertically aligned crystal in ion flux turned out to be more stable relative to melting than a single layer crystal. The particle velocity distribution function (PVDF) displays features of particle motion in the different regimes In crystalline regime PVDF is Maxwellian, Fig. 4(a).

162

/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge

0 , 0 0

0,0-1

eo

0 , 0 ^

0 , 0 0

n/^' = 0 . 1 e s

0 , 0 ^

0 , 0 s

0,1:2

i^/w„=0. 1 e i

2 3

# Ul

-40

1 O

2 0 (si)

Cb)

\

.^Wp=0. 1 Si

^/w

= 0 . 1

^ S

i -^

Cci>

Co)

0.0

O, 1

0 , 0

0,2

0 , 1

Fig. 4 Particle velocity distribution function at various frictions. The dashed curves are a Maxwellian distribution (a,d) and the distribution of harmonic oscillator (b). 2 0 0

i/w.. = 0 . 1

STe 1 SO

«/wp=o.:^i

Ni o 2 0

CQ>

i:k/w_=0. 1 : 2 s

Ck^) It/W_=0. 1

_..^

fthhmAtA

1

Cci>

Co) •v^V.,v^

1 .0

0 . 0

O.S

1

.0

>3VX'VX.''

Fig. 5 Spectrum of particle velocity autocorrelation function at various friction constants. The dashed curve (a) refers to a single layer crystal.

/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge

163

At n=nins which correlates with arising instability PVDF corresponds to the harmonic particle oscillations, Fig. 4(b). At lower gas friction the distribution becomes broader. Fig. 4(c), and eventually after melting PVDF looks like Maxwellian, Fig. 4(d). We calculated the particle velocity autocorrelation ftinction and, applying the Fourier transformation, derived the spectra of excited phonons at different gas pressures. At the crystalline state the dust crystal has usual spectrum for 2D crystals. Fig. 5(a). At «=««,* the spectrum has sharp peak shape. Fig. 5(b). A few modes with frequencies about the dust crystal frequency (Dp are excited. This is the reason of enhanced stability of the dust crystal, since the high frequency modes increase only particle energy ER. whereas the long wave oscillations cause the crystal to melt. In the liquid state the structure exhibits the reach phonon spectrum including zero mode. Fig. 5(d). We have explained heating of the dust crystal in the sheath by developing self-excited oscillations arising with decreasing pressure. Note, that the calculated Ek agrees with the measured one within an order of magnitude. However the experiment displays a more complex picture of crystal melting With decreasing pressure first some 'hot' spots are created and then streamline motion develops around heated parts of the crystal. Melting of the dust cluster with defects. In the case of simulation of the perfect crystal the transition to the 'hot' crystalline regime is accompanied with quick rise of Ek within a narrow interval of the pressure. In contrast to calculated data, in the experiment the particle temperature enlarges smoothly. We supposed that defects which are always present in the crystal even far from critical temperature are responsible for continuous melting. Two kinds of defects are considered. The first type of defects is point defects (vacancies and interstitials) and dislocations which are typical for the 2D systems and destroy the translational and the angular order. The second type of defects is additional particles surrounding the dust crystal. We study the melting transition of the defect bilayer crystal with asymmetric inter-particle interaction, using the Langevin molecular dynamics simulations. The crystal consists of 996 particles and in the crystal phase the particles and the positive charges are arranged into two parallel almost hexagonal lattices. In our earlier works [3,8] we have considered an infinite in the horizontal plane bilayer crystal. We took the fragment of the crystal and used the periodic boundary conditions. Here we take a finite cluster in order to create point defects and dislocations. The main reason of defect formation in the crystal phase is the presence of the external boundary of the system. The particle motion is described by the following equation —-i-= — F -n—- + —Fj dtM ' dt M ^ M

WU(r).

where ri==(xi^H-yi^)^^, Xi, yi are the transverse coordinates of the ith particle, M the mass of a particle, Fi is the electrical force, n the friction constant and F\ is the Langevin force which refers to the room temperature of gas. The electrical force includes interparticle repulsion and attraction between particles and positive charges. U is the external confining potential in the horizontal plane. Note, that each particle does not interact

/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge

164

with its downstream positive charge which mimics the asymmetric ion distribution around a particle, but interacts with all other particles and positive charges. The electrical force acting on the particle can be written as

F,=Z'Y^ ^

'^\e^'^

r. - r.. I"'

(1 + k

:, I)

ZZ^

^!-~'"\e-' r-^^A^l^k\r-r„\) 13

r - r^

where \/k is the screening length, j,n denote the summation over particle layers and layers of charges, respectively, j = 2, n = 2 for the crystal with point and extended defects, j = 4, n = 4 in the case of additional particle defects. The coordinates of charges ^-^/: where dc the distance between positive charge and the lower layer in the vertical direction and k = 2/a [9]. In this simulation, the positive charge is taken to be Zc= 0.5Z with a vertical distance dc = 0.6a [3]. Vacancies, interstiteals and dislocations are formed during slow numerical cooling of the system from the high temperature liquid state. Initially the particles are placed randomly within area rrrr..v.ivi.r

\ #1

-10 -10

-5

0

10

K>\ o

4i

1

J

•

\m

o

•

L *^ V '

*

'

'

*

•

^1 A

•

•

J

•

•

\

o

o^

^0,1

5

o oi

•

1

•

•

'

'

l i l t . . I . 1 . _L l_ 1

0,10 0,15 0,20 0,25 n/w Fig. 6 Particle trajectories in the crystal with point defects and dislocations (I-type defects) at «=0.0721a)p. (Top view)

]]

0,30

Fig.7 Kinetic energy of particles as ftinction of friction for the crystals with I-type defects (solid circles), with extra particles (open circles), and the experiment data (squares).

/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge

165

It contains uncorrelated dislocations with Burgers vector equal to 2. The crystal has also a lot of defects near the external boundary, since the hexagonal structure can be fitted in a circle only with the certain number of defects. The second layer is supposed to be of the same structure. First at given gas friction we allows the system to achieve quasi-equilibrium state during 2-10"^ ^6-10*^ MD steps in which the particle temperature stays constant while calculation. The mehing scenario of the bilayer crystal with first type defects turns out similar to a non-defect crystal melting described above. The change of ER with decreasing pressure is shown in Fig. 7 (solid circles). At gas friction n < «ins=0.105(Dp the system exhibits the crystalline structure, Fig. 7. Note, that even at friction n ^ Wins, the particle trajectories do not display some marks of occurrence of dislocations. Fig. 6. At fi = Wins an instability begins to develop. Within friction interval nins< n

'

:

-

'

*

•

'

•

•

•

-

•

•

-

'

•

•

•

•

•

•

'

* % '^ *

*^ . ^ -•. ' . ^ . ' . •. ' . •* - - . - '

Fig. 8 Particle trajectories in the crystal with extra particles for (a) n=0.15 ©p, Ek=10eV, (b) «=0.14 Qp, Ek=12eV, (c) «=0,13 ©p, Ek-14 5eV, (d) «=0.12 ©p, Ek=17eV.

166

I.V Schweigert et al /Melting of the defect dust crystal in a rf discharge

transition takes place. As followed from our results the occurrence of points defects and dislocations does not make the substantial contribution to the particle heating. Let us consider melting of the bilayer crystal with extra particles defects. The additional particles are placed randomly above and below with distance a from the bilayer crystal (added to the crystal shown in Fig. 6). The number of these particles equals 45 or 5% of the total number of particles. Our calculation showed that the heating evolution of the bilayer crystal with extra particles has interesting features. At sufficiently high friction the particles perform harmonic oscillations near their equilibrium sites with small amplitude, and, however, the positions of additional particles are clear seen. In Fig. 8(a) the particle trajectories of the upperstream layer of the bilayer crystal is plotted. The enhanced particle motion areas point out the locations of the additional particle placed above the bilayer crystal. At lower friction the bilayer crystal exhibits fragments of streamline motion and also crystalline regions. The crystal begins to melt locally and liquid fragments appear first under additional upper particles. Fig. 8(b-d).The occurrence of additional particles modifies the heating dynamics and leads to a visible increase of ER, Fig. 7 (open circles). The instability relative to short wave oscillations appeares also at higher gas friction. Now two step melting of the bilayer crystal with the asymmetric inter-particle interaction is smeared out by the local heating and it is difficult to identify the melting point. The excitation spectra of the bilayer crystal with extra particles have principal features. In the crystalline state the spectrum has a second sharp peak about © = 0.6(Dp. which corresponds to high-frequency phonons excited by the additional particles placed above the crystal. In conclusion we have developed the new model of asymmetric inter-particle interaction in the sheath of a rf discharge. Using this model linear analytic analysis and MD simulations of the dust crystal melting were carried out. It was shown that an increase of the dust particle kinetic energy with decreasing gas pressure is explained by instability related to excitation of short wave oscillations. The coexistence of liquid and crystalline fragments in the bilayer crystal observed in the experiments is explained by influence of additional particles situated above the crystal, whereas point defects and dislocations do not practically affect the particle temperature. [1] S.V. VladimirovandM. Nambu, Phys. Rev.E 52,2172(1995). [2] F. Melandso and J. Goree, Phys. Rev.E 52, 5312 (1995). [3] V.A. Schweigert, I.V. Schweigert, A.Melzer, A. Homann, and A.Piel, Phys. Rev.E 54, 4155 (1996); A. Melzer, V.A. Schweigert, I.V. Schweigert, A.Homann, and A.Piel, Phys. Rev.E 54, 46(1996). [4] A. Melzer, A. Homann, and A. Piel, Phys. Rev.E 53, 2757 (1996). [5] V.A. Schweigert, V. Bedanov, I.V. Schweigert, A.Melzer, and A.Piel, JETP 88, 905 (1999). [6] H. Thomas and G.E. MorfiU, Nature 379, 806(1996). [7] F. Melandso and J. Goree, J.Vac. Sci. Technol.A 14, 511 (1996). [8] V.A. Schweigert, I.V. Schweigert, A.Melzer, A. Homann, and A.Piel, Phys. Rev. Lett. 80, 5345 (1998); I.V. Schweigert, V.A. Schweigert, A.Melzer, and A.Piel, JETP 87,905(1998). [9] A. Homann et al., Phys. Rev.E 56, 7138 (1997); A. Homann, A. Melzer, R. Madani, and A. Piel, Phys. Lett.A 242, 173 (1998).

Part III. Industrial Applications

This Page Intentionally Left Blank

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

l69

On the powder formation in industrial reactive RF plasmas Ch. HoUenstein, Ch. Deschenaux, D. Magni, F. Grangeon, A. Affolter, A.A Howling P. Fayet Centre de Recherches en Physique des Plasmas Ecole Polytechnique Federale de Lausanne CH-1015 Lausanne Switzerland Tetra Pak (Suisse) SA 1680 Romont Switzerland Abstract. Particle formation in reactive plasmas plays an important role in industrial plasma processes. In the present study, inia-ared absorption spectroscopy, emission spectroscopy and mass spectrometry have been simultaneously applied to dusty organosilicon and hydrocarbon plasmas. Powder formation and its dependence on important process parameters in methane, acetylene, ethylene and m helium/oxygen diluted hexamethyldisiloxane (HMDSO) gas mixtures have been investigated. The influence of the chemical structure of the monomer gas on the neutral and charged plasma species and on possible polymerization reactions leading to powder precursors is discussed. In addition, cavity ring down has been applied to these different dusty plasmas in order to investigate in detail the powder precursors, their origin and then- consequences.

INTRODUCTION Powder or particles are found in deposition and in etching plasmas. However, the origin of these particles might be of quite different nature. Powder formation in plasmas used for film deposition, in particular (diluted) silane RF plasmas, have been intensively investigated during the last few years (1). The different stages of the powder formation such as the nature of the powder precursors, the agglomeration and accretion phases have been identified and investigated experimentally and theoretically. Besides this very prominent reactive plasma, many other plasma processes employed in industry show powder formation. In most cases, the powder formation leads to problems either in the quality of the film or to process interruptions due to prolonged maintenance and cleaning of the reactor. Hexamethyldisiloxane (HMDSO)/helium/oxygen plasmas, for example, are used for the deposition of silicon dioxide as a permeation barrier in the packing industry. Nano- to micrometer-sized Si02 particles are also formed in these plasmas (2). Another important category of plasma where strong powder formation can be observed are hydrocarbon plasmas. These plasmas find wide applications in plasma polymerization and in the production of hard carbon coatings. Various diagnostics have been applied to investigate the powder formation. These include in-situ particle diagnostics and diagnostics covering the plasma composition and parameters (3). In the present study, mass spectrometry, infrared absorption

170

Ch. Hollenstein et al/On the powder formation in industrial reactive rf plasmas

spectroscopy and emission spectroscopy have been applied to these plasmas in order to elucidate the origin and the different phases of the powder development and powder composition as a function of various external parameters and monomer types. From the point of view of the production of nanoparticle-seeded coatings (4), diagnostic methods measuring nanometer-sized particle and their density are needed. Besides exotic methods such as the laser explosion technique, no in-situ methods are available to measure the size and density of the (sub-) nanometer sized proto-particles. Such a diagnostic would also be of great interest to understand the transition from the large clusters to the proto-particles. In this paper we report on the cavity ring down method applied to dusty plasmas (5) in order to obtain new information on the early phases of the powder development.

EXPERIMENTS The experiments were performed in two different reactors with different substrate sizes. The Balzers KAI 1 reactor allows deposition on 35 x 45 cm substrates (6), whereas the substrate size was limited to 10 x 10 cm in the second reactor. For these investigations, both reactors were operated at an excitation frequency of 13.56 MHz and at approximately the same RF power density. For monomers silane, hexamethylsiloxane (HMDSO), methane, ethylene, aceteylene and oxygen were used. The influence of different diluting gases such as argon and helium on the powder formation was also investigated. Infrared absorption spectroscopy of the dusty plasma was performed by means of a commercial FTIR instrument and is described in more detail elsewhere (2). In addition to the infrared absorption measurements, optical emission spectroscopy and quadrupole mass spectrometry were applied at the same time. A Balzers PPM 422 plasma monitor with a mass range up to 512 amu was used with the probe head positioned adjacent to the electrode gap at 1 cm from the plasma boundary. Rayleigh-Mie scattering, using a He-Ne laser or an Argon ion laser as light source, was utilized to qualitatively visualize the powder within the discharge volume. The ring down cavity consists of two high reflectivity dielectric mirrors (>99.99%) in vacuum. To avoid instabilities of the confocal arrangement, the distance between the mirror was 120 cm. A pulsed dye laser tuned at 566 nm, operated at 10 Hz (4 ns pulse length) is introduced into the back face of the entrance mirror.

RESULTS AND DISCUSSION Silane discharges High mass anions up to 1760 amu containing at least 60 Si-atoms have been detected in silane plasmas (7). From various investigations it was concluded that these negative ions are the most probable polymerization pathway for powder precursors in pure RF silane plasma at low or moderate power densities. The regular nature of the mass distribution in the mass spectra suggests that a simple statistical approach might be sufficient to explain the form of the mass spectra. A simulation of the mass spectrum

Ch. Hollenstein et al/On the powder formation in industrial reactive rf plasmas

171

by means of random bond theory shows good agreement with the measured mass spectrum (8). As in the case of the pure silane discharge, the oxygen diluted silane plasma also shows considerable powder formation. Mass spectrometry has also been applied to the SiH4/02 plasmas (9) and also in this case very high mass anions were detected, whereas neutrals and positive ions remain limited to moderately high masses. For high masses the rich variety of Si-H-0 radical chemistry completely changes the regular structured spectrum found for the pure silane plasma case. Higher abundances of some mass values in the anion spectra may indicate clusters preferentially formed within the plasma. The masses of the preferential clusters coincide with hydrosilasesquioxanes(Si2nH2n03n n=3-6) which consist of a polyhedral cage made up of Si-O-Si linkages. A simple iterative model (9) has been adopted to explain the cation spectra in the SiH4/02 plasmas. Similarly to the anions for the pure silane plasma case, a statistical model including isotope effects reproduces the overall features of the observed cation spectra. However the complexity of the anion spectra in this condition implies that additional reaction mechanisms must also be included in the model.

HMDSO discharges In-situ infrared absorption spectroscopy was used to investigate the chemistry in diluted HMDSO plasmas. The admixing gases investigated were oxygen, helium and argon respectively. For the dilution with oxygen the infrared spectra reveal the presence of the oxidation steps of the methyl-groups: formaldehyde (COH2), formic acid (CO2H), CO and CO2 and with water as combustion by-product. The presence of methane (CH4) and acetylene (C2H2) found in the dilutions with argon, helium and oxygen indicate also the presence of hydrocarbon chemistry in these plasmas. As in the case of silane/oxygen plasma, silicon oxide (SiOx), particles were also formed in argon diluted HMDSO plasmas. This is in contrast to helium and oxygen diluted plasmas where no visible particles have been observed under similar conditions. Metastables produced in helium are supposed to induce a Penning dissociation breaking the polymerization precursors and thereby inhibiting the formation of particles. From the infrared absorption spectra, the HMDSO depletion was calculated based on three characteristic absorption bands for the Si-O-Si, Si-CHs and CH3 bonding within the HMDSO molecule. It was found that the depletion calculated from the different absorption bands gives new insight into the fragmentation of the HMDSO molecule and its dependence on the type of the diluting gas. In the case of helium and oxygen diluted HMDSO plasmas, the three absorption bands lead to similar depletion values as a fiinction of the dilution. However considerable differences are found in the case of the argon diluted HMDSO plasma. The depletion calculated from the Si-O-Si absorption band is 100% at high dilutions whereas the other two absorption band give about 30-40% less. We suppose that for the case of the argon diluted HMDSO plasma the molecule is fragmented into small radicals still containing Si-CHs bondings and

172

Ch. Hollenstein et al./On the powder formation in industrial reactive rf plasmas

CH3. This partial fragmentation might also be responsible for further polymerization which in turn can lead to the powder formation as observed in these plasmas. In the case of the oxygen and helium dilution a high degree of fragmentation of the HMD SO molecule is reponsible for reducing the powder formation.

Hydrocarbon discharges The combination of neutral mass spectrometry and FTIR absorption spectroscopy clearly shows that the production of acetylene is one of the most important processes in all three hydrocarbon plasmas investigated. In all experiments, we observe formation of acetylenic compounds, which appear to be very well-favoured products for the polymerisation reactions in the plasma. In the cases of methane and ethylene, the production of acetylene under the present discharge conditions is most probably not sufficient to influence strongly the chemistry of the anion and cation formation. In the acetylene plasmas, the triple C=C bonding remains mainly intact, imposing the characteristic even carbon number spectra of the ionic components, whereas double bonds from the ethylene are cracked within the plasma and play an insignificant role in the ongoing plasma chemistry. The main bonding type of the solid particles within the different plasmas is the sp bonding. Only limited evidence for the presence of some sp bonding in the powder could be found for the case of the acetylene plasma. This fact might be also of interest for astrophysical topics. Powder formation in the sp bond-containing acetylene is found to be strong whereas single and double bonds in accordance with previous reports in the literature show less powder formation. This might be partially explained by the production of hydrogen, observed in each case, which may prevent or delay the formation of powder. The limited production rate of acetylene might be a second reason for restricted powder formation. In the anion spectra of all three gases the absence of any CHx' groups has to be mentioned. This absence might indicate that C2Hx' rather than CHx" anions are produced by electron attachment. The observed formation of larger anion clusters might proceed by recombining C2Hx' anions with CH4 or most probably with C2HX. The different tendency to powder formation of the gases investigated might therefore be related to the different reaction rates for the production of the neutral C2HX (in particular C2H2) and their corresponding attachment rates. It might therefore be concluded that the formation of acetylene and subsequent electron attachment to this molecule leads to larger anions, which by analogy with the silane plasma could end in powder formation. In-situ powder diagnostics by cavity ring down technique The cavity ring down technique has been applied to investigate the powder formation in argon diluted silane plasmas, diluted HMDSO plasmas, and in pure methane plasmas. The change in the ring down time of the cavity is due to absorptions of

Ch. Hollenstein et al./On the powder formation in industrial reactive rf plasmas

173

various origins. Besides photodetachment of negative ions, and line absorption due to electronic excitation, the main losses of the laser beam are due to absorption within the nanoparticles and scattering. The extinction due to small particles is w^ell described by Rayleigh scattering theory and depends on R (R particle radii) whereas absorption depends on R"^ (volume fraction) and on the refractive index. In general absorption is the dominating effect for very small particles such as in the case of silicon- and carboncontaining particles. However the Si02 particles must be treated as non absorbing due to their very small imaginary part of the index of refraction in the visible. The time development of the powder formation in (diluted) silane, HMDSO and methane plasmas has been measured. In each case the scattered intensity at 135° and the single pass extinction of the Ar-ion laser beam was simultaneously monitored. The appearance of any scattered intensity indicates the presence of powder particles in the range of about 40-50 nm.

1000

E

c

time (s)

FIGURE 1. Powder formation in pure methane plasmas. Particle size and density for cleaned reactor (circles), after two (squares) and four (triangle) time developments of the powder formation.

Fig. 1 shows the formation of particles in a pure methane plasma. In a cleaned plasma reactor the powder formation takes a few hundred seconds. In a contaminated reactor, powder appearance is much faster, since powder formation might be triggered by particles emitted from the particle-contaminated electrodes. It can be concluded that contamination may strongly influence the powder formation. Therefore care must be taken in order to avoid artefacts due to this effect. The particle size and particle number density in this case were determined by assuming that absorption is dominant for very small particles and that the volume fraction is constant during the development. The particle size can be estimated from the ratio of the extinction due to absorption measured early in the particle development, to the measured absorbance (proportional to R ) and from this finally the number density can be obtained. For the case of Si02 particles this estimation cannot be applied due to the non-absorbing character of the particles. In this case only the behaviour of NR^ can be given. For

174

Ch. Hollenstein et al/On the powder formation in industrial reactive rf plasmas

larger particles, scattering is the dominating process. This leads to an increased extinction and therefore large particles even at low density might strongly influence the absorbance and dominate over the contributions from small particulates. In Fig.l the presence of a large particles is observed late in the time development. These large particles arrive near the plasma sheath due to rearrangement of the powder as was visually observed.

time

(s)

FIGURE 2. Addition of oxygen to an argon diluted HMDSO plasma (15sccm Ar, 2 seem HMDSO) a) Extinetion eoeffieient from eavity ring down method, b) extinetion eoefficient determined from Ar ion laser beam (single pass).

Fig 2 shows the absorbance in an argon-diluted HMDSO plasma. In these plasmas, first indications of powder formation appear after about 50 seconds. However, on adding 0.7 seem respectively 2 seem, of oxygen, very fast powder formation is observed. Recently the cavity ring down technique has been applied to determine the negative ion density in oxygen plasmas using photodetachment (5). These investigations show that the electronegativity of the oxygen results in a large fraction of negative charge carriers being negative ions. Therefore also in a dusty electronegative plasma many negative charges must be negative ions. This leads to a reduction of the particle charging and its consequences on the coagulation (larger particles) and powder dynamics (larger neutral forces and smaller electrostatic forces). It was found that Si02 particles in these plasmas are in the range of 400-500 nm, whereas powder particles in other reactive plasmas such as silane plasmas are typically only around 100 to 200 nm

Ch. Hollenstein et al./On the powder formation in industrial reactive rf plasmas

175

(10); this is probably a consequence of reduced charging of the particles in these plasmas. In the case of powder formation in very low power argon diluted silane plasmas, periodically varying absorbances have been observed (5). These oscillations are characteristic of powder creation/growth/elimination cycles in the reactor as already shown by other diagnostics. However, at low silane dilutions the ring-down signal is rapidly lost. The absorbance presented in fig. 3 follows an exponential law.

1 0- 4

5 0

F I G U R E 3 . Extinction coefficient in a argon diluted silane plasma (500 seem argon, 15.4 silane, lOW and 0.2 T).

Contrary to Si02, amorphous silicon shows considerable absorption at the dye laser wavelength and the beam extinction is rather given by the absorption than by scattering losses. The absorbance signal can therefore be interpreted for particle sizes up to about 40nm (about Mie scattering onset) as the total volume fraction of the powder in the plasma. An exponential increase of the volume fraction can be explained by a simple theory including coagulation and the growth of particulates and where the volume of the particulates remains unaffected by the coagulation (11). Up till now most of the theories applied to the coagulation in the plasma (1,12,13) assume constant volume fraction. In order to interpret the measurements, more sophisticated theories including simultaneous nucleation, condensation and coagulation in the plasma should be applied. A model including the simplest particle forming system by gas to particulate conversion (14) allows to describe the dynamic behaviour of such a system. In addition the influence of the negative ions on the charging of the particles needs clarification since charging infiuences not only the coagulation and the expected size, but also the governing forces on the particles and therefore the powder dynamics. For the powder formation and also for powder processing important information on changes in the polydispersity due to the coagulation might be obtained. Further investigation is under way.

176

Ch. Hollenstein et al/On the powder formation in industrial reactive rf plasmas

CONCLUSION Infrared absorption spectroscopy, mass spectrometry and emission spectroscopy have been applied to investigate the powder formation in different hydrocarbon and organosiUcon RF plasmas. The plasma composition and the powder formation depends on the chemistry of the monomer. Noble gas dilution can strongly influence the fragmentation of the monomer and with this the tendency to powder formation. The presence of negative ions such as negative oxygen ions will strongly affect the particle charging and in consequence the powder dynamics. The cavity ring down technique is a powerful method to investigate the early phases of the powder genesis.

REFERENCES 1. A. Bouchoule ed., "Dusty Plasmas between Science and Technology," (Wiley, 1999) 2. C. Courteille, D. Magni, A.A. Howling, V. Nosenko and Ch. Hollenstein, "Infra-red absorption spectroscopy of Si02 deposition plasmas", 40th Annual Technical Conference Proceedings of the Society of Vacuum Coaters , 304-3308 (1997). 3. Ch. Hollenstein, Plasma and Polymers, to be published 4. J. Dutta, H. Hofinann, C. Hollenstein and H. Hofineister, "Plasma-produced silicon nanoparticle growth and crystalization processes," in Nanoparticles and Nanostructural films, edited by Fendler (Wiley-VCH, Weinheim, 1998). 5. F. Grangeon, C. Monard, J.-L. Dorier, A.A Howling, Ch. Hollenstein, D. Romanini and N. Sadeghi, Plasma Sources Sci. TechnoL, to be published 6. L. Sansonnens, A.A. Howlmg and Ch. Hollenstein, Mat.Res.Soc.Symp.Proc. 507, 541-546 (1998). 7. A.A. Howling, C. Courteille, J.-L. Dorier, L. Sansonnens and Ch. Hollenstein, Pure & Appl. Chem. 68 (5), 1017 (1996). 8. Ch. Hollenstein, W.Schwarzenbach, A.A. Howling, C.Courteille, J.-L. Dorier and L. Sansonnens, J. Vac. Sci. Technol. A 14, 535-539 (1996). 9. Ch. Hollenstein, A.A. Howling, C. Courteille, J.-L. Dorier, L. Sansonnens, D. Magni and H. Muller, Mat.Res.Soc.Symp. Proc. 507, 547-557 (1998). 10. C. Courteille, D. Magni, C. Deschenaux, A.A. Howling, Ch. HoUenstem and P. Fayet, "Gas phase and particle diagnostics of HMDSO plasma by infrared absorption spectroscopy," 41st Annual Technical Conference Proceedings 1998 Society of the Vacuum Coaters, 327-332 (1998). 11. M.M.R Williams and S.K. Loyalka, Aerosol Science Theory and Practice (Pergamon Press, 1991). 12. C. Courteille, Ch. Hollenstein, J.-L. Dorier, P. Gay, W. Schwarzenbach, A. A. Howling, E. Bertran, G. Viera, R. Martins and A. Macarico, J. Appl. Phys. 80 (4), 2069-2078 (1996). 13. U. Kortshagen and U. Bhandarkar Phys. Rev. E submitted (1999). 14. S. E. Pratsinis, J. Colloid Interface Sci. 124 (2), 416 - 427 (1988).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

177

Trapping and Processing of Dust Particles in a Low-Pressure Discharge E. Stoffels, W.W. Stoffels, G.H.P.M. Swinkels, G.M.W. Kroesen, Department of Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. [email protected]. tue. nl

Abstract. Formation, behaviour and modification of dust particles in a low-pressure plasma are discussed. Coulomb interactions of negatively charged particles together with other forces in the plasma result in efficient trapping. A single particle can be kept motionless in the plasma, and fully controlled by the experimentalist. We are able to accurately determine the particle properties by angle-resolved Mie scattering, and modify them by etching or deposition in the plasma. This research is related to the industrial demand for particles with specially tailored properties. We show that a low-pressure discharge is well suitable for production or surface modification of special dust particles.

INTRODUCTION Dusty plasmas are nowadays a vast research field. Numerous aspects of the formation, interactions and consequences of dust particles in plasmas have been investigated, both from the fundamental and industrial side. The fundamental research covers the problems of charging and dynamics of dust particles in plasmas. In particular, formation of ordered structures called Coulomb crystals has been investigated, and a variety of wave phenomena in dusty plasmas have been described (1). Moreover, dust particles in the interstellar space attract much attention in relation to the rings of Saturn. The applied side emerged in the late eighties, and triggered dynamic development of the dusty plasma research (2). Its origin is the semiconductor processing industry, where sub-micrometer particles are formed in situ in the processing reactor. This is a common phenomenon in plasma enhanced chemical vapour deposition (PE-CVD) reactors, e.g. during deposition of amorphous silicon in the fabrication of solar cells. In deposition plasmas particle formation and co-deposition on the substrate can lead to layer defects. In the reactive ion etching (RIE) of semiconductor elements the purity of the surface is the major issue; there the consequences of particle formation are much more severe. Dust particles, when deposited on the processed surface, are the major cause for device failure. The major aim has been to minimise particle

178

E. Stoffels et al /Trapping and processing of dust particles in a low-pressure discharge

contamination, either by avoiding particle formation or by preventing their deposition. Many advanced diagnostic techniques for in situ contamination control have been developed to meet the increasing demands from the industry. Progressive miniaturisation of semiconductor elements implies that even nanometer size particles can be dangerous contaminants, and their presence must be well diagnosed. At the moment, detection of nanometer size particles is possible by means of inelastic laserparticle interactions (2). An immense research effort has resulted in the elucidation of in situ particle formation mechanisms in deposition plasmas, as well as the mechanisms of surface sputtering and flaking in etching plasmas. The powerful experimental diagnostics, the understanding of particle origin and the knowledge of various forces and interactions of particles in the discharge allow now a good control of reactor contamination. A dust particle has become a predictable object, whose behaviour and properties can be manipulated by the researcher. In recent years new trends have emerged, in relation to potentially interesting properties of plasma-produced particles. The knowledge acquired during the combat with dust contamination is now being used to explore the novel applications of dusty discharges. Two trends can be observed. First, particles formed in situ or injected externally, are processed in the discharge in order to adjust their properties (3). Applications include coating of particles with active layers for various purposes and separating large particle conglomerates in a discharge in order to obtain homogeneously coated grains, etc. Fabrication of catalysts and pigments, or improvement of toners for copying machines are just a few examples. For such applications it is essential to gain a good control of the particle behaviour in the discharge, and to understand plasma-particle interactions, like charging, etching and deposition processes on the particle surface. Previous studies were mainly concerned with systems containing many particles, e.g. with characterisation of particle clouds formed in the plasma. However, it is difficult to study surface modification processes when large amounts of not well-defined particles are involved. Alternatively, some investigators injected well-defined particles into the discharge in order to study plasma-particle interactions. Particle dynamics, force balance and formation of Coulomb crystals were intensively studied (4). Here we shall briefly describe particle charging and trapping phenomena, before introducing the particle processing. In order to be able to study the mechanisms of plasma processing on a clear model system, we propose a single particle experiment. A well-defined particle is introduced into the plasma and pinpointed in a specially constructed potential well. We can accurately determine the particle properties and influence them in a controlled way by means of plasma processing. In particular, angle-resolved Mie scattering is employed to monitor the particle radius and the particle size is adjusted by means of etching in a low-power radio-frequency oxygen discharge. The second major trend in dust particle applications is the production of hybrid materials by particle sintering or particle enclosure in a solid material. Small particles formed in the discharge can be embedded in a layer on the surface, in order to improve the layer quality. For example, the quality of amorphous silicon solar cells

E. Stoffels et al/Trapping and processing of dust particles in a low-pressure discharge

179

deposited in a silane plasma can be significantly improved by co-deposition of nanometer size silicon particles formed in the same discharge (5). One of the recent applications of hybrid coatings is the production of hard layers for the protection of mechanical tools. In most cases, lubricating the contact surface is needed to reduce friction in order to improve the mechanical yield of the system and reduce the wear of the parts in contact. It has been shown that coatings with an addition of lubricants like M0S2, combine wear resistance with a significantly lowered friction (6). Addition of M0S2 poses a technical problem, because of its poor adhesion and a bad oxidation resistance. To circumvent this problem, a coating where M0S2 in the form of nanometer particles is immersed has been proposed. However, production of nanometer particles and their co-deposition within a layer is not a straightforward task. In the second part of this paper we will describe the recently developed method of plasma-enhanced production of M0S2 particles and their handling in a PE-CVD reactor.

PARTICLE TRAPPING The characteristic feature of an object immersed in the plasma is the negative electrical charge it acquires. In the steady state, the fluxes of positive ions and electrons from the plasma towards the solid surface must be equal. As electrons have a higher mobility than positive ions, an appropriate negative surface charge and electric field will be established to accelerate the positive ions and repel the electrons. Such effects occur for bounding surfaces and floating objects in the plasma. The potential difference between the particle and the plasma (Vp) is a function of the electron temperature (Tg). In low-pressure discharges Vp is usually two to three times kTe/e, i.e. about 10 eV for typical electron temperatures of about 3 eV. The potential difference determines the energy of positive ions, reaching the particle surface as well as the number of elementary charges on the particle (Z). The latter can be estimated from the classical expression Ze = 47isoa Vp, where 47rsoa denotes the capacitance of a sphere with a radius a. This simple expression for the particle charge yields about 3 elementary charges per nanometer radius of a particle: Z ^^ 3-10^ a. A particle in the plasma is subject to various forces, the most important being the electrostatic force, gravitation, neutral and ion drag and thermophoretic forces. Coulomb repulsion between the negative charged particle and the electrodes provides trapping of the particle in the plasma glow. This in combination with the other forces allows sustaining the particle in a fixed position. Previous experiments have shown that one can easily control the particle position by influencing the force balance. Coulomb and ion drag forces are tuned by varying parameters like plasma power and pressure or changing the electrode geometry, thermophoretic force is influenced by introducing temperature gradients, and gas dynamical forces are determined by the gas flow pattern (4,7). The ability of controlling the particle position is essential in a study of particle processing in the discharge.

180

E. Stoffels et ah /Trapping and processing of dust particles in a low-pressure discharge

ETCHING OF A SINGLE PARTICLE Below we describe trapping and etching of a single particle in a low-pressure oxygen discharge (8). A schematic overview of the experiment is shown in Figure 1. We use commercial melamine-formaldehyde particles with a diameter of about 12 |Lim and a refractive index of 1.68. For the particle injection, we use a particle-filled sieve, which is mounted on a manipulator arm. The measurements are performed in a GEC reference cell: a parallel-plate capacitively coupled 13.56 MHz radio-frequency reactor. An aluminium ring (diameter: 2 cm, thickness: 1.5 mm) is placed on the lower powered electrode in order to create a potential trap, so the particles shed from the sieve are immediately "caught" at the glow-sheath edge above the metal ring. In this way it is possible to prepare a Coulomb crystal, but also to "freeze" a single particle in a fixed position. Angle-resolved Mie scattering is applied to monitor the particle size and refractive index as a function of time during plasma processing. The light source is an argon ion laser, linearly polarised in the direction perpendicular to the detection plane (see Fig. 1). The scattered light is collected by an optical fibre, passed through an interference filter and fed to a photo-multiplier. The optical fibre is mounted on a moveable stage. This allows continuous scanning of the detection angle in the range of 1 to 15 degrees for forward scattering, with an angular resolution of 0.06 degree. The angle-resolved scattering intensities are fitted to a numerical model for Mie scattering using the particle radius and refractive index as tuning parameters. A typical angle-resolved measurement and its fit to the data are shown in Figure 2. As can be seen, there is a good agreement between the measured data and the fit. The fitted particle radius (a = 5.90 |im) and refractive index (n = 1.68) agree with the size determination using SEM pictures and the data provided by the particle manufacturer. The particle size can be determined with accuracy better than 1% by the angleresolved scattering technique. In order to demonstrate a good control of the particle radius and the sensitivity of the diagnostics, and to study surface modification of particles, we use a low power oxygen discharge to etch the organic polymer, of which the particles are made. There is a large difference between the standard etching of a substrate placed on the electrode (RIE) and the etching of a free floating object in a plasma. In the sheath of the powered electrode a high potential in the order of 1 kV accelerates positive ions towards the surface. Sputtering and etching of the material is performed by highenergy ions, reaching the electrode surface. In contrast, the potential difference between the plasma and a floating particle is much lower, in the order of 10 V. Therefore, plasma chemical effects due to low-energy ions and radicals are expected to be most important for microscopic particle etching. In order to check the influence of physical ion sputtering, we monitored time changes in the angle-resolved scattering signals of single particles trapped in argon. In argon, where no chemical effects can occur, no remarkable changes in particle properties were observed even after hours of plasma operation. In an oxygen plasma substantial variations of the scattering signal were recorded already after a few minutes of processing.

E. Stoffels et al /Trapping and processing of dust particles in a low-pressure discharge

detection plane

181

argon ion laser

moving detector

FIGURE 1. A scheme of the experimental setup. A single particle is trapped in the plasma above the rf electrode and irradiated by an Ar ion laser. The angle-resolved scattered light intensity is collected by an optical fiber mounted on a moving stage and fed to a photomultiplier.

angle (d^)

100

^

FIGURE 2. Angle-resolved Mie scattering data of a single melamine-formaldehyde particle (squares), trapped in a radio-frequency oxygen plasma. The data are fitted by a theoretical scattering curve for a particle of 5.90 [am radius and 1.68 refractive index. FIGURE 3. Time-dependent angle-resolved scattering intensity of a single melamine formaldehyde particle treated in a 0.2 mbar oxygen plasma. The particle is processed for 30 minutes in a 1.5 W plasma, then for 30 minutes in a 5 W plasma and finally for 60 minutes in a 7 W plasma. The etch rate increases with increasing plasma power from 0.06 to 0.13 nm/s.

Oxygen is a well-known etching gas suitable for etching of organic polymer materials. Ion and radical chemistry in radio-frequency oxygen discharges was extensively studied in the past (9). Free oxygen radicals and oxygen ions provide an ideal composition of active species to etch organic polymer particles trapped in the discharge. The etching process is determined by various plasma parameters. Tuning of these parameters provides a means to obtain a good control of the etch rate and consequently of the particle radius. In Figure 3 the time evolution of the angleresolved Mie scattering signal is shown for a particle treated in a 0.2 mbar oxygen discharge with a varying plasma power. The characteristic Mie fringes are clearly visible, and it is evident that the fringes shift towards higher angles as the etching

182

E. Stoffels et al/Trapping and processing of dust particles in a low-pressure discharge

proceeds. In terms of the Mie theory, this implies that the particle size decreases in time. Using the numerical model the time dependent particle size as a function of time is deduced. At a given power level, the particle size variation is fairly linear with time, and the corresponding power-dependent etch rate can be determined. The densities of reactive species in the discharge increase nearly linearly with increasing power, and the same trend is expected for the etch rate. The particle shown in Figure 3 is injected into a 1.5 W oxygen discharge, and processed for about 30 minutes. At this power level only slight changes in the angle-resolved scattering intensity can be observed, which implies that the particle radius hardly changes (etch rate is 0.06 nm/s). After about 30 minutes the plasma power is increased to 5 W. At this power level etching of the particle proceeds faster (0.10 nm/s), which is reflected by time variations of the angle-resolved scattering intensity. Next, the plasma power is increased to 7 W. The plasma activity is further enhanced, resulting in an increased etch rate of 0.13 nm/s. Similarly, varying the pressure of the processing gas allows to etch the particles at a desired rate. By tuning of the plasma parameters, the etch rate can be varied from 0 up to 1 nm/s with an accuracy and reproducibility of about 10%. Similar etch rates were found for particle clouds (10). Thus, the proposed experiment of single particle processing is a simple and accurate model system for studying large scale particle processing. In fixture it will be extensively used to control particle coating in deposition plasmas.

PRODUCTION OF PARTICLES IN THE PLASMA Hybrid layers, containing a particle suspension, find many applications in the modem coating technology. Here we describe a part of the work aiming on fabrication of hybrid titanium nitride coatings with embedded molybdenum sulfide particles (11). The lubricating properties of M0S2 in combination with wear-resistant properties of a TiN film results in a unique self-lubricating hard coating. Production and codeposition of nanometer particles in CVD processing as well as in a plasma environment is a challenge, as such small grains are not commercially available and very difficult to handle ex situ. Moreover, when externally produced particles are used, there are always problems with contamination with water and oxygen. Therefore, an integrated process is required, combining particle formation, layer growth and seeding in one closed system. Particle production is generally performed in a different chemical and physical environment than the deposition process. Typically, particles are formed at high pressures, either using thermal plasmas or ovens, while for PE-CVD processes the pressure must be kept low. Thus, new recipes should be developed for particle formation, which can be incorporated in a PE-CVD environment. Particle formation at low pressures encounters many difficulties. First of all, the reaction rates are lower as a consequence of lowered densities of reagents. Especially three-body association reactions, which are essential for the formation of large molecules and their condensation into particles, are inefficient at low pressures. Moreover, in the commonly used flowing systems the residence times of the species

E. Stoffels et al /Trapping and processing of dust particles in a low-pressure discharge

183

FIGURE 4. SEM micrograph of M0S2 particles produced in the plasma at 0.5 mbar, using H2S, M0CI5, H2 and Ar. The micrograph shows particles, which are conglomerated into larger structures. The diameter of individual particles is smaller than 100 nm.

in the reactor are limited, typically under one second. This implies that many-step processes, leading to particle growth, are hindered. A possible solution to these problems is the use of a low-pressure plasma. Rapid formation of macroscopic particles in low-pressure discharges has been observed in many chemical systems (2,4). The non-equilibrium character of such plasmas and the participation of charged species in the formation processes facilitate particle production. The non-equilibrium character is reflected by a large difference between the neutral gas temperature (ambient) and electron temperature (a few eV). All chemical processes are initiated by electrons, which have sufficient energy to dissociate molecules and a high mobility to undergo frequent collisions. Other problems of particle formation at low pressures, namely the residence times which are too short to allow for particle growth, can be also overcome. In the plasma the particles acquire a negative charge at a very early stage of their growth. By means of Coulomb interactions they remain trapped in the plasma sufficiently long to reach even sub-millimeter sizes. The size can be adjusted to one's wishes by controlling the plasma duration time, plasma power and gas flows. In the considered example, M0S2 particles are produced from M0CI5 and H2S in the following reaction: M0CI5 + 2H2S + I/2H2 ^ M0S2 + 5HC1. In the neutral gas, no particles can be produced at pressures below 10 mbar. In contrast, abundant particle formation has been observed already at 0.5 mbar in the plasma containing evaporated M0CI5, H2S and excess of hydrogen. This is a typical pressure used for PE-CVD of TiN layers, which will facilitate codeposition of these species. The quality of plasma produced particles meets the requirements for the production of hybrid coatings: they have a monodysperse size distribution and the desired size of < 100 nm. During plasma operation the particle suspension levitates above the substrate, and particles can be easily collected on the surface after plasma termination. A typical SEM micrograph of plasma-produced particles is shown in Figure 4. The procedure of embedding particles in the coating will thus consist of M0S2 production and collection, and switching the chemistry for the TiN deposition.

CONCLUSIONS In this paper we discuss handling and processing of particles in the discharge, relevant for industrial particle tailoring. First, fundamental aspects, like particle trapping and etching have been treated, and an experimental model system for the

184

E. Stoffels et al /Trapping and processing of dust particles in a low-pressure discharge

Study of particle processing has been described. We have shown that a single particle can be efficiently trapped in a discharge, and its size can be accurately regulated. We discuss etching as an example of such controlled particle treatment, but the presented technique opens many other possibilities. Particles can be coated in deposition plasmas, e.g. using silane or methane. Fine tuning of the particle size, at a welldefined rate, has undeniable benefits. For example, particle modification process can be studied on a relative simple, single particle system, and extended to the situation of particle clouds. Moreover, the forces acting on a particle are size-dependent, so an accurate determination and modification of particle size can prove extremely helpful in a study of the force balance and particle motion in the plasma. Furthermore, we discuss the advantages of production of nanometer particles using a low-pressure discharge. Advanced coating technology involves co-deposition of small grains within a layer, under low-pressure conditions. An integral process of in situ particle production and seeding in a PE-CVD reactor has been proposed and efficient production of M0S2 particles has been achieved.

ACKNOWLEDGMENTS We appreciate the contribution of Drs. F. Rossi and G. Ceccone. This work is supported by the European Commission under Brite- Euram contract BRPR CT97 0438 (HALU), by the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) and by the Dutch Technology Foundation (STW). The research of Dr. W. W. Stoffels has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW).

REFERENCES 1. D.A. Mendis, C.W. Chow, P.K. Shukla, Proc. 6* Workshop The Physics of Dusty Plasmas, La JoUa, Califomia, 1995, publ. World Scientific, Singapore, 1996, ISBN 981-02-2644-6. 2. Dusty Plasmas, ed. by A. Bouchoule, to be published by Wiley&Sons, 1999. 3. H. Kersten, P. Schmetz, G.M.W. Kroesen, Surface and Coatings Technology 108-109, 507 (1998). 4. Proc. Dusty Plasmas '95 Workshop on Generation, Transport and Removal of Particles in Plasmas, Wickenburg, Arizona, 1995, publ. in J. Vac. Sci. Technol A14(2), 1996. 5. P. Roca i Cabarrocas, P. Gay, A. Hadjadj, J. Vac. Sci. Technol. A14, 655 (1996). 6. H. Suhr, R. Schmid, W. Sturmer, Plasma Chem and Plasma Proc. 12, 147 (1992). 7. E. Stoffels, W.W. Stoffels, PhD Thesis, Eindhoven University of Technology, The Netherlands (1994). 8. W.W. Stoffels, E. Stoffels, G.H.P.M. Swinkels, M. Boufhichel, G.M.W. Kroesen, Phys. Rev. E59, 2303 (1999). 9. E. Stoffels, W.W. Stoffels, D. Vender, M. Kando, G.M.W. Kroesen, F.J. De Hoog, Phys Rev. E51, 2425 (1995). 10. G.H.P.M. Swmkels, E. Stoffels, W.W. Stoffels, N. Simons, G.M.W. Kroesen, F.J. de Hoog, Pure and Applied Chemistry 10(6), 1151-1156 (\99Sy 11. E. Stoffels, W.W. Stoffels, G. Ceccone, F. Rossi, R. Hasnaoui, H. Keune, G. Wahl, accepted for publ. in J. Appl. Phys. (1999).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukia (eds.) © 2000 Elsevier Science B.V All rights reserved

135

Effects of Gravity, Gas and Plasma on Arc-Production of FuUerenes Tetsu Mieno Department of Physics, Shizuoka University, Shizuoka-shi 422-8529, Japan Abstract. In orda^ to improve production efficiencies of many kinds of ftillerene families, production characteristics of fullerenes in an arc discharge are investigated. Effects of gravity, impurity and magnetic field in the arc production are investigated and, key conditions and more efficient methods to produce fullerenes in the arc discharge are found.

INTRODUCTION Mtoy kinds of fullerenes and nanotubes are produced in large quantity by an arc discharge in helium gas atmosphere. While, their molecular process of self-organization in arc region is not made clear because many compUcated reactions are mixed and many parameters influence on the reaction. Usually, the reaction process is decided by arc condition and control of reaction parameters such as reaction time, reaction volume, reaction temperature is not carried out well. Here, microscopic and macroscopic properties of fuUerene synthesis are investigated and important basic parameters are pointed out. !» ^ In order to improve tiie synthesis, effects of gravity, gas pressure and impurities, and magnetic field are investigated and effective results to improve fuUerene production have been obtained. Efficient production of these new carbon materials is demanded for various kinds of applications.

BASIC CONDITION OF FULLERENE SYNTHESIS Basic configuration of an arc reactor to produce fullerenes is shown in Fig. 1. 3 Empirically production rates of fullerenes strongly depend on many parameters of arc discharge. For example, production characteristics change with discharge current, discharge voltage, gap distance, diameters of electrodes, gas pressure, gas species, impurity density in the gas, etc. 1* ^ By arc heating, sublimated carbon atoms from an anode react in hot gas atmosphere and they flow upward by heat convection and deposit on a upper wall of the chamber. Elucidation of basic growing process of fullerenes in the arc discharge is hard because their reaction occurs in high pressure and high temperature condition, and synthesized molecules are in high density. Instead of the arc discharge, molecular process in laser-ablated carbon gas, "^ which is expected to have similar growing process, is precisely investigated by use of the mass-spectrometric technique and the ion drift methods and important result is obtained by Bower's

186

T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes

VACUUM CHAMBER (STAINLESS STEEL)

FIGURE 1. Example of an arc reactor to produce fuUerenes.

group. 5, 6 j h e evaporated carbon ions are mass separated by a sector type magnet and selected mass species is injected into a drift cell, in which molecules of different shapes are separated like a chromatography method and difference of transit time signal is recorded. From this result and theory of the molecular stabiUty, shapes of synthesized ion species for each mass are determined. As a result, it is found that carbon molecules grow via chain, ring and multiple ring structures into fuUerene type cage structure. For cluster size of more than 50, more than 90 % of produced clusters have fuUerene structure. This growing process is simulated by the molecular dynamics method by Maruyama' s group and similar result is obtained theoretically. ^ These clusters are partly stable and alive in the air condition, because by the pentagon rule of molecular stability, ^60' ^70 and C2n (n> 35) are only stable in the air ^ and collected as fuUerene samples. In order to clarify the region where the growing reaction takes place in the arc discharge, a simple experiment is carried out A large carbon block is located on an arc flame and its distance from the arc center ZCB is changed. ^ Cgo content in produced soot versus Z^g is measured and shown in Fig. 2, where/?= 300 Torr and 7^= 60 A. At ZcB< 2.5 cm and temperature of higher than 900 ''C, Cgo content seriously decreases, which means that fuUerenes are effectively synthesized only in the arc region at higher than 900 ^C. From the investigation of the molecular process and result of macroscopic experiment, important conditions of high-efficiency production of fuUerenes in the arc discharge are obtained as hollows, 1) High sublimation rate of carbon atoms from anode material by controlling a discharge current and anode size. 2) Long reaction time of carbon molecules in high temperature He gas, which is

T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes

15

187

1500

' I • ' • • I ' ' ' ' I

^X(C6o) CARBON BLOCK

5

1000

10

o

t

o

(+)

(-)

^v-=r

4^500

>


0

10 P(H2) (Torr)

FIGURE 9. Production rate of C^Q versus H2 partial pressure. Total pressure ;?= 300 Torr, /j= 70A, and4od= 6.5 mm.

T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes

191

/(j= 70 A, and anode diameter (irod= 6.5 mm. When more than 5 % of H2 gas is included, fuUerene production is strongly obstructed. Impurities of N2 and Si also reduce the production rate, while impurities of O2, Al, Fe, W, Au do not reduce it. Near the normal carbon stars, hydrogen content in their atmosphere is large, which would obstruct fuUerene synthesis in space, though much abundance of Cgg"*" ions are reported to be observed in space. 13

EFFECT OF MAGNETIC FIELD In order to increase the production efficiency of fuUerenes and reduce the deposition of carbon material on a catiiode, a steady magnetic field {^ 30 Gauss) is apphed perpendicular to the arc current By the JxB force, arc plasma and carbon gas are jetted out to the JxB direction. 14, 15 By this method, production rate of C50 considerably increases and deposition rate to the cathode decreases. By using the JxB arc method, chip-carbon material injection type fuUerene automatic producer has been produced as shown in Fig. 10.1^ An anode is made of a carbon crucible, in which carbon chip or carbon grain materials are dropped in and they are submitted by the/xfi arc discharge. By applying the magnetic field, the production rate is about 6 times increased. Using this machine, fuUerene production from plant materials and used carbon-based materials are examined, and from charcoal, activated carbon, used synthetic rubber, and used toner, CQQ, C70 and higher fullerenes are successfully produced.

VACUUM CHAMBER

CARBON ROD

FIGURE 10. Schematic of a carbon-chip-injection type /xfi arc reactor.

192

T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes

SUMMARY 1) Fullerenes are synthesized from carbon atoms via chain and ring structures in arc region. Reaction temperature is between 1,200 - 5,000 K. 2) High subhmation rate of carbon atoms from anode, long reaction time of carbon molecules in high temperature gas, high collision frequency with He to be annealed, low impurity density in space, are important for efficient production. 3) By means of a 12 m VST, 13 min of the integrated gravity-free discharge time is obtained. Collection rate of La@Cg2 at the gravity-free discharge is about 14 times higher than that of the normal discharge. 4) Hydrogen impurity strongly obstructs fuUerene synthesis. 5) Magnetic field improves production efficiency of fullerenes.

ACKNOWLEDGMENTS I would like to thank Professor Eiji Oosawa of Toyohashi University of Science and Technology for useful comments.

REFERENCES 1. T. Mieno and D. Yamane, J. Plasma Fusion Res. 74, 1444 (1998). 2. T. Mieno, H. Takatsuka, E. Kumekawa A. Sakurai and T. Asano, J. Plasma Fusion Res. 69, 793 (1993) [in Japanese]. 3. T. Mieno, J. Phys. Soc. Jpn. 62, 4146 (1993). 4. H. W. Kroto, J. R. Heath, S. C. O'Brein, R. F. Curl and R. E. Smalley, Nature 318, 162(1985). 5. G. vonHelden, M. Hsu, N. Gotts andM. T. Bowers, J. Phys. Chem. 9 7 8 , 8182(1993). 6. N. G. Gotts, G. von Helden and M. T. Bowers, Int. J. Mass Spectrum. Ion Proc. 149/150,217(1995). 7. Y. Yamaguchi and S. Maruyama, Chem. Phys. Lett. 286, 336 (1998). 8. R. E. Smalley, Ace. Chem. Res., 25, 98 (1992). 9. S. Aoyama and T. Mieno, Jpn. J. Appl. Phys. 38, L267 (1999). 10. S. Usuba etal, Proc. 14th FuUerene Sympo., Okazaki, 1998, p.24 (in Japanese). 11. T. Mieno, Jpn. J. Appl. Phys. 37, L761 (1998). 12. T. Mieno, Jpn. J. Appl. Phys. 35, L591 (1996). 13. B. H. Foing and P. Ehrenfreund, Nature 369, 296 (1994). 14. T. Mieno, FuUerene Sci. Technol. 3, 429 (1996). 15. T. Mieno, A. Sakurai and H. Inoue, FuUerene Sci. Technol. 4, 913 (1996). 16. T. Mieno, T. Asano and A. Sakurai, Advanced Materials '93,1/B, 1201 (1994).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

193

Formation of Dust and its Role in Fusion Devices J. Winter Institut fur Experimentalphysik II, Ruhr-Universitdt, D 44780 Bochum, Germany

Dust particles are produced in fusion devices by various plasma-surface-interaction mechanisms. This paper discusses properties of dust collected from fusion devices, some dust generation mechanism and the interaction of dust with the cold edge of a fusion plasma. The potential problems for operation and performance associated with the dust-plasma-interaction are outlined. First attempts aiming at the in situ observation of dust in fusion plasmas are presented.

1. Introduction Although it is known since long that dust particles are formed by plasma-surfaceinteractions in fusion devices (1), the problems associated with their presence have not been addressed in detail until recently (2-5). In the framework of the ITER project (Intemational Thermonuclear Experimental Reactor) it became evident that for conditions of large plasma fluences (steady state operation) the fomiation of dust raises serious safety problems (6,7). ITER, as most of the present large devices, will have carbon based wall components (graphite, Carbon Fiber Composites). The incorporation of tritium in carbon dust can be as high as 2 T atoms/C atom (6) giving rise to T inventories of up to several kg. Reactions of the highly activated radiation damaged dust with steam from a leaking cooling line may liberate H in large amounts creating an explosion hazard. In addition, the existence of dust may also influence the plasma operation and performance of the device (4). It is not only the dust particles formed during the specific discharge which may cause problems (1,4) but also the repeated interaction with dust particles accumulating on the bottom of the device. Experiments (8) and theoretical considerations (9) on the problem of dust shedding and levitation make it very likely that some of the existing particles may be levitated and sucked into a discharge. Depending on the confinement in the plasma edge, dust may even accumulate in certain areas of the cold edge plasma during the plasma discharge (2,4).

2. Characterization of dust and of its sources The particles collected from existing fusion devices with carbon wall components TFTR, D-III-D, JET, TEXTOR (10, 11,2) span a wide size range from several 10 nm up to the mm range showing essentially a log-normal distribution (10). Dust in the context of this paper is defined as loose material with particle diameters up to about 0.1mm. In the case of the D-III D tokamak, the mass concentration was found to be in

194

J. Winter / Formation of dust and its role infusion devices

the range from 0 . 1 - 1 ^lg cm"^ on vertical surfaces, 10-100 |xg on the floor and lower horizontal surfaces. The total amount is estimated to be between 30 - 120 grams (10) for an integrated plasma exposure time of less than a few hours. The dust composition is dominated by carbon but may also include all other materials used inside the vessel or those for wall conditioning (B, Si (12)). Scanning Electron Microscopy (SEM) on the dust from TEXTOR reveals that the majority of particles are flakes of failed films produced by redeposition. The average H (D) flux at the leading edge of the graphite limiter of TEXTOR is 10^^ m"^ s\ The averaged erosion yield (including both sputtering and chemical erosion) is 2x10'^ resulting in a primary loss rate of 2x10^^ m"^s C atoms. This corresponds to a gross erosion of Im per year plasma exposure at the limiter tip. Almost all of this material is redeposited. Since the location of redeposition is not identical to that of erosion redeposited layers will accumulate. C incorporates hydrogen upon redeposition. The material is brittle with a layered or columnar structure and many stress induced cracks. Finally flakes fall off the surfaces. Figure 1 shows a SEM picture of a large flake from the deposition dominated area of the limiter. The majority of flakes found on the bottom of TEXTOR is similar to that in figure 1. This source of dust increases proportional to the plasma fluence. In the case of TEXTOR about 15% of the dust was

0.1 mm FIGURE 1. Large flake of redeposited carbonaceous materialfroma TEXTOR limiter

found to be ferromagnetic. Metal atoms are enriched in the flakes to preferential reerosion of the pure carbonaceous matrix (4). Flakes of typically a few 100 nm thickness may be liberated from spalling of thin films deposited for the purpose of wall conditioning (12). The films usually have a very good adhesion but may flake when the vessel is exposed to moist air during an opening. Following plasma operation may then liberate flakes by plasma-surfaceinteractions. In TEXTOR a significant number of almost perfect metallic spheres with diameters from 0.01 to 1 mm were identified. The metal spheres are formed most probably by coagulation of metal atoms on hot graphite surfaces (13). They are first evaporated

J. Winter / Formation of dust and its role infusion devices

195

from metal wall elements by unipolar arcing, transported via the plasma to the limiter surface and, as their surface mobility increases when the limiter gets hot from to the thermal load of the plasma, they coagulate. They are then released by plasma-surfaceinteraction from the limiter. Arcing occurs mostly during the start-up phase of the plasma or at disruptions (uncontrolled rapid loss of the plasma). Electric fields are strongest during these phases leading to an easy local breakdown of the plasma sheath. Coagulation of particles from the vapor phase during arcing may be another possible mechanism for formation of the metal spheres (see below). Thermal fatiguing and thermal overloading of wall components is another source of dust. There exist power transients in fiision devices. Edge LocaUzed Modes (ELM's), plasma pressure gradient driven instabilities occurring at the plasma edge of divertor tokamaks can periodically deposit 2-5 % of the total stored plasma energy [14]. In the case of JET this was found to be up to a few GWm"'^ during 100 |LIS occurring at a rate of up to 100 Hz (15). Another transient is disruptions during which the total stored plasma energy is deposited on part of the wall. The deposited power in ITER is estimated to be lOOMJm''^ during 1 ms., i.e. power densities of 100 GWm"'^ on the divertor plates are expected. This leads to local evaporation of large quantities (several kg) of material. Even in present large devices the material loss during disruptions is significant. The failure mode for repeated exposure to low powers is a loosening of the material structure by propagating cracks as consequence of the large compressive stress. For graphite finally the ejection of grains may occur. In the TEXTOR dust graphite grains were easily distinguished from flakes by their facetted appearance. They have an average diameter of typically 5-20 |im. For higher power fluxes the evaporation or sublimation of material is dominant. The vapor has a high density close to the surface and the particles have consequently a short mean free path. Coagulation processes from supersaturated vapor leads to the formation of small particles (16,17) with average diameters of several 10 to 100 nm. This mechanism was verified in a laboratory experiment in which different carbon materials were repeatedly exposed to power loads of 0.5 - 1.5 GW m"^ using a high power electron gun (18). Transmission electron microscopy of the released material shows small globular clusters, see figure 2 left side, and evidence for the formation of fiiUerene like materials. Quantitative image analysis shows that most of the particles have sizes below 500 nm. Interestingly, a similar type of particles is identified in the TEXTOR dust (figure 2 right side) consisting of agglomerates of individual single globular particles of about 100 nm diameter. They too may have been formed by coagulation from C vapor during transients. Another possible mechanism is the growth of dust in the scrape off layer (2,5) involving negative hydrocarbon ions and multiple ion-neutral reactions. Hydrocarbons are released from the wall as a consequence of chemical erosion and their concentration may be as high as 10 % of the plasma density. The electron temperatures can be below 5 eV and electron densities about 10^^ m'^ close to the wall. Under these conditions the formation of negative ions by dissociative electron attachment is likely and conditions are close to that of a methane process plasma in which the formation of nanoparticle precursors was studied recently (19). Negatively

196

J. Winter / Formation of dust and its role infusion devices

30 nm

30 nm

200 nm

FIGURE 2 . Left side: Transmission Electron Micrograph of a globular carbon particle likely to be formed by coagulation from dense C vapor (from ref. 18). Right side: Coagulated dust particle collected from TEXTOR (4).

charged ions and agglomerates are confined in the plasma edge: the sheath potential repels them from the surface, confinement in radial direction is provided by the magnetic field. A fiiction force from the background plasma is acting on these particles, driving them away from the stagnation point. Thus probable locations of dust particles are close to limiter-like protrusions of the wall, where an effective trap from the superposition of these forces exists (4). The high flux of UV photons in the fiision plasma tends to photo-detach the excess electrons, however. The balance of attachment and detachment rates will critically depend on geometrical factors and is difficult to assess a priori. No experimental proof of this mechanism exists to date.

3. Possible impact of dust on the performance of fusion plasmas Large particles falling into a fiision plasma can induce a disruption. Narihara et al.(3) used the Thomson scattering set up in JIIPT-2U to study the influence of small carbon particles with diameters < 2|Lim on the performance of tokamak discharges. Dust was dropped from the top of the machine and the authors concluded that an amount of about 10 particles of 2 ^im do not affect a fiiUy developed discharges, but that such particles existing in the main volume before start up of the plasma lead to increased initial impurity concentrations. Most particles fall to the bottom of the device at the end of a fiision plasma discharge. After some period a significant reservoir may have accumulated which increases with plasma fluence. Light particles may be re-injected into thefiisionplasma by magnetic and also by electric forces when dust flakes are charged by plasma contact. They may then be levitated close to the wall (8,9,4). Magnetic particles experience a VB-force and may be sucked into the main vessel volume upon rising the toroidal magnetic field. Plasma

J. Winter / Formation of dust and its role infusion devices

197

breakdown and bumthrough may be scuuu&iy impeded under these conditions. The intense impurity radiation often observed during the start-up phase of tokamak plasmas may be due to levitated dust. Erosion rates are often determined involving measurements of the line radiation of neutral atoms or ions. In building up clusters by agglomeration, neutral impurity atoms or molecules will be consumed before they can radiate, thus escaping emission spectroscopy. Divertor armor madefi-omberyllium was deliberately overheated in JET and Be atoms evaporated at a large rate. The data show still unexplained large differences between gross erosion and spectroscopically measured fluxes (20). This discrepancy may very well be due to agglomeration and Be dust formation. Fine particles can be transported far away from their point of origin and are subject e.g. to thermal forces, unlike e.g. massive splashes of molten metal, which are deposited close by. Dust particles are a sink for electrons and, for large concentrations, will change the balance between electrons and protons in the edge plasmas. This will result in a different sheath potential and heat transmission factor and in a different dynamics of the edge plasma.

4. Laser light scattering from particles in TEXTOR The experimental set up for the light scattering experiment on TEXTOR is described in detail in ref (5). The observation concentrated on the plasma edge region. Not knowing the location, the spatial distribution and density of the particles, a volume as large as possible was investigated with adequate spatial resolution compromising however the laser power density. The beam of an Ar-ion laser (70 mW cw) was guided through a fiber to a tilting mirror on top of TEXTOR sweeping it across a poloidal cross section (diameter 1.04 cm) with a width of about 2 cm at the bottom of the vessel. The observation of the scattered light fi-om almost the fiiU poloidal cross section was made through a tangential port, toroidally about 1.5 m away by a CCD camera with Hght amplification and a narrow band interference filter for the laser wavelength. The geometry was dictated by the available existing ports. When the vessel was vented to nitrogen moving particles levitated from the bottom by gas convection, were observed in the lower 1/3 of the vessel when the pressure reached values > 400 mbar. Light scattering by particles was also observed in the initial pump down phase from atmospheric pressure. Attempts to measure scattering signals during the presence of tokamak discharges failed. The background intensity of plasma light is orders of magnitude larger than the intensity from the previously identified scattering events. However few scattering signals from particles in the lower part of the vessel could be identified uniquely before discharge initiation and also right after the discharge. Though the data are scarce, the results strongly suggest that existing dust is levitated and may interact with the fiision plasma.

198

J. Winter / Formation of dust and its role infusion devices

5. Conclusions Dust particles found in existing tokamaks can be correlated with various production mechanisms involving plasma-surface-interactions and gas phase agglomeration. Plasma induced growth is a potential mechanism, too. Failed redeposited layers are a main source of dust. The dust quantity increases with plasma fluence. In addition to safety hazards, dust is likely to impact operation and performance of fusion devices. Dust is thus an important issue for future long pulse devices.

Acknowledgement The author greatfully acknowledges the help of G. Gebauer in performing the laser light scattering experiment.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Ohkawa, T., Kako Yugo Kenkyo 37, 117(1977) Winter, J., Proc. 24* EPS Conf. Controlled Fusion and Plasma Physics, EPS Conf. Abstr. 21 A, 4,1777(1997) Narihara, K., Toi, K., Yamada, Y., et al., Nucl. Fusion 37, 1177 (1997) Winter, J., Plasma Phys. Contr. Fusion 40, 1201, (1998) Winter, J., Gebauer, G., Proc. 13* Int. Conf. Plasma Surface Interactions in Controlled Fusion Devices, May 18-22, 1998, San Diego, USA, J. Nucl. Materials, in press Federici, G., et al., 13* Int. Conf. Plasma Surface Interactions in Controlled Fusion Devices, May 18-22, 1998, San Diego, USA, J. Nucl. Materials, in press Piet, S.J., Costley, A., Federici, G., et al., Proc. 17* lEEE/INPSS Symp. Fusion Engineering, Oct. 6-10, 1997, San Diego, USA, IEEE 97HC35131,Voll, 167 (1998) Sheridan, I.E., Goree, J., Chiu, Y.T., Rairden, R.L., Kiessling, J.A., J. Geophys.Res. 97, A3, 2935 (1992) Nitter, T., Aslaksen, T.K., Melandso, F., Havnes, O., IEEE Trans. Plasma Science 22, 159 (1994) McCarthy, K.A., et al., Fusion Technology 34, 728 (1998) Peacock, A.T., et al., 13* Int. Conf. Plasma Surface Interactions in Controlled Fusion Devices, May 18-22, 1998, San Diego, USA, J. Nucl. Materials, in press Winter, J., Plasma Phys. Contr. Fusion 38, 1503 (1996) Behrisch, R., Borgesen, P., Ehrenberg, J., et al, J. Nucl. Materials 128+129, 470 (1984) Leonhard, A.W., Suttrop, W., Osborne, T.H., et al., J. Nucl. Materials 241-243, 628 (1997) Lingertat, J., Tabasso, A., Ali-Arshad, S., et al., J. Nucl. Materials 241-243, 402 (1997) Schweigert,V.A., Alexandrov, A.L., Morokov, Y.N., Bedanov, V.N., Chem.Phys.Lett. 235, 221 (1995) Schweigert,V.A., Alexandrov, A.L., Morokov, Y.N., Bedanov, V.N., Chem.Phys.Lett. 238, 110 (1995) Bolt, H., Linke, J., Penkalla, H.J., Tarret, E., Proc. 8* Int. Workshop Carbon Materials, Sept. 34, 1998, Jiilich, FRG, Physica Scripta, in press, and private communication Winter, J., Leukens, A., Proc 14* Int. Conf. Plasma Chemistry, August 2-6, 1999, Praha, Czech Republic Campbell D.J. and the JET Team, J.Nucl. Materials 241 -243, 379 (1997)

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V. All rights reserved

199

The Formation and Behavior of Particles in Silane Discharges Alan Gallagher JILAy University of Colorado and National Institute for Standards and Technology Boulder, CO 80309-0440 ABSTRACT: Particle growth in silane RF discharges and the incorporation of particles into hydrogenated-amorphous-silicon (a-Si:H) devices is described. Measurements of particle density and growth in a silane RF plasma, for particle diameters of 8-50 nm, are described. A decrease in particle density during the growth indicates a major flux of these size particles to the substrate. Particle densities are a very strong function of pressure, film growth rate and electrode gap, increasing orders of magnitude for small increases in each parameter. A full plasma-chemistry model for particle growth from SiHm radicals and ions has been developed, and is outlined. It yields particle densities and growth rates, as a function of plasma parameters, which are in qualitative agreement with the data. It also indicates that, in addition to the diameter >2 nm particles that have been observed in films, a very large flux of SixH^ molecular radicals with x >lalso incorporate into the film. It appears that these large radicals yield more than 1% of thefilmfor typical device-deposition conditions, so this may have a serious effect on device properties.

INTRODUCTION In silane (SiH4) and SiH4-H2 discharges, a-Si:H film deposition is initiated by electron collisions that dissociate SiH4. This yields primarily neutral-radical fragments, but is accompanied by a small fraction of positive ions (cations) and a very small fraction of negative ions (anions). The neutral radicals Si, SiH, SiH2, and H rapidly react with SiH4 to form stable molecules or SiH3, while SiH3 reacts only with the a-Si:H film. Thus, film growth is primarily due to the reactions of SiHs radicals with the surface. Cations induce only a few percent of the growth, but as they typically strike the surface with energies in the 1-30 eV range this can be important. The anions can not reach the growing film due to sheath fields, so for many years these were neglected as a contributor to film electronic properties. However, it has now become clear that anions can profoundly influence the film, for they cause Si particles to grow in the plasma. Since anions (SixHn~) are trapped in the plasma, there is plenty of time for SiH3anion reactions to induce their growth, and this can lead to enormous values of x, the number of Si atoms. Once x exceeds perhaps 30, these anions are beUeved to be relatively spherical with a structure similar to that of the a-Si:H film: clustered Si at near-crystalline density with occasional H inside and an H terminated surface. Thus, these larger clusters are generally described as "particles." From standard "particle-in-plasma" theories, particles with x>200 (radius Rp>l nm) are expected to be negatively charged and consequently trapped in the plasma by the sheath fields.^ (The number of negative charges increase Unearly with Rp.) With gas flow, these charged particles are dragged away to the pump once they grow to 0.1-1 |im size (x=10^-10^^) and the film does not suffer. However, we discovered several years ago that particles of 2-15

200

A. Gallagher/The formation and behavior ofparticles in silane discharges

nm diameter escape the plasma and incorporate into the films.^ These particles constituted 10"'-10' of the film volume, and their surface bonds to the surrounding film exceed the electronic defect density of a-Si:H by many orders of magnitude. Here I report that when smaller particles not visible in that experiment are included, particles probably constitute 110% of the film volume and their bonds to the surrounding film are a significant fraction of all Si-Si bonds within the film. These particles are a major concern, as it appears likely that voids and strained bonds occur at particle-film interfaces.

MEASUREMENTS OF PARTICLES IN SILANE PLASMAS Since the original detection of particles in silane plasmas by Roth et al.,^ the most common method of measuring particle suspended in plasmas is still Hght scattering. However, important insights have also been gained from measurements of negative ions escaping the plasma afterglow,"^ and from scanning tunneling microscope^ and transmission electron microscope (TEM) measurements of particles collected on the electrodes.'^ Unfortunately, most of these measurements were not carried out in pure SiH4 or SiH4-H2, or for conditions that yield device-quality a-Si:H films. Thus, to understand particle growth under device production conditions, I have utilized primarily the insights, not the particle data, from these experiments. In my laboratory we have concentrated on the conditions and gases used to produce devices. Initially, we studied the accumulation of larger particles (Rp >50 nm) at the downstream end of the RF discharge and their escape to the pumps.^ These particles can influence the discharge operating conditions, but they do not incorporate into devices. Thus, in the last few years we have studied^ the very small particles that form directly below the substrate and are indicative of those that incorporate into films. It is difficult to study the very small particles (Rp l negative charges become stable, these additional charge states are included. Iteratively calculating only x-x-i-1 steps is an approximation, as some collisions add more than one Si atom. However, it greatly simplifies the calculation and is reasonable because particle growth is dominated by collisions with SiH4 for x100. An example result is shown in Fig. 2 for discharge conditions that are typical of those in Ref. 7. In Fig. 2, first consider the neutral particles that are labeled n(z=0) and flux(O). These neutral particles are molecular radicals, with x=l representing SiH3. The x=2 radicals result from collisions between a pair of SiH3, and the x=3 primarily result from collisions between x=l and x=2 radicals. As x increases from 1 to 2, n(x,z=0) and F(x,0) drop a factor of -2.5 and -3.5 respectively. As x further increases, a smaller decrease per x-x+1 step occurs, due to an increase in the rates of growth/diffusive loss. Even before considering the larger particles, the heavy-radical incorporation indicated in Fig. 3 could have significant effects on film properties. (This has been pointed out and modeled previously.^) The rate of F(x,0) falloff versus x is very sensitive to plasma parameters; it slows if G, L or P is increased and visa versa. For x>20, n(x,0) begins to follow the negative ion density, n(x,-l), as these populations become strongly coupled by electron and cation collisions. Turning now to the anions, labeled n(-l) in Fig. 2, their density is almost independent of X for x » - ^ 109 h : - - - > ^ K a - \ - 1 ••^?^=??=0r*^ 108

E o CO

c Q o

10^ r" 10^

0.1 0.01

••^

r

I

1 10

'*••.. Total/nm

^

"1

***'*••-

—

''*""*"-J ~H

i^V"'^'^v*^^?>^J°^^' ^^^ ^-- ' ^ s > ^ 1.^ ^^^^^S^::>^ ^^ ^ O v T ^ ^ ^ ^

1 1 —

' '^

^*l

-2j

r

J

V. • • , ^Vk. — ^" •' • ^ v

^

^'"""^Nniiii^

1 L—

10^ 100 10 1

1

P V ^^ ~ —"->.••.> ^ h ^

CO

^—

H

X-'^-^-'^^^ ' ^^^

L_J 100

X (Dimensionless)

1

"^

^^ -4

—^-

j **^^'""' 1

1 ^ ^ ^ 1 1*10^

ino""

Figure 3. Calculated particle densities, n(x,2), as a function of x, the number of Si atoms, and 2, the charge. Also shown is density per nm of pardcle radius and pardcle flux to surfaces.

204

A. Gallagher / The formation and behavior ufparticles in silane discharges

Considering next the cations, labeled n(+l) and flux(+l) in Fig. 2, the density of x=l-3 is severely lowered by very rapid growth reactions with SiH4. However, these reactions slow at x=3 and essentially terminate by x=6, and only slower reactions with radicals cause further cation growth. The result is a peak in n(x,l) at x=5. For x>30, the cations also become closely connected to the z=-l and 0 particles by charge-changing collisions, and all three densities fall together. The cation-silane reaction rates for x10'^ is a very sensitive function of P,G, and L, as is shown in Fig. 4, where n4 is the total particle density per nm of radius at x=10'^. Since the total particle negative charge cannot exceed the cation density, most particles become neutral once the particle density exceeds n+. Particles then agglomerate and escape the plasma with a high probabiUty, so this sets a Umit on total particle density. The data is plotted versus P«L^^ because this reduces results at different P and L values to a relatively common curve. The extreme sensitivity to the plasma parameters is consistent with the results of the experiment described above, as is the range of P,L and G that yield the measured particle densities. However, the exact values of P,L and G for which the model yields these densities is not meaningful, as many collisional rates must be assumed.

A. Gallagher/The formation and behavior of particles in silane discharges

205

CO

E o

"^ c o

0.2

0.4

0.6

0.8

P L^-^ (Torrcm^-^)

Figure 4 LogioCnVcm''^) as a function of RF discharge parameters. G=film growth rate in A/s. L=electrode gap in cm, P=silane pressure in Torr at 300 K, and n4 is the density of 3.6 nm radius (x=10'^) particles per nm of radius.

CONCLUSIONS In order to mitigate particle incorporation into a-Si:H devices, we first set out to understand why this incorporation occurs. Our model has shown that, in addition to the 2-15 nm radius particles that can incorporate at lO'"^ to 10"^ of the film density, large SixHm complexes with x>10 incorporate into films as more than 1% of the volume. It seems likely that some of the interfaces between these particles or large complexes and the remaining film are not well structured or passivated. Particle growth in the plasma is an inevitable consequence of the strong electron-attachment energies of Si-containing molecules and particles. SiHs" production and growth into SiiooHm" (Rp=0.8 nm) occurs within 1 ms of discharge initiation, and cannot realistically be avoided. Particle growth from x=100 to 10"^ (Rp = 3.6 nm) or larger can be mitigated by using on-off modulation or low P,L and G, but this is achieved by allowing the growing particles to escape to the electrodes before they reach "large" size. As there is not much one can do to prevent the growth of particles, it is important to ask why they incorporate into the film, and if this might be avoided. Alternatively, if one wishes to incorporate crystalline siHcon particles into the film, can we control this? Only neutral or positively charged particles diffuse to the electrodes, so it is important to understand why a large fraction of the particles are neutral or positive. The reason is that n+ and n. are much larger than Ue; this occurs because n+ and n+ build up to a value where cationanion collisions balance the rate of producing anions. There doesn't seem to be much one can do about this in SiH4 or SiH4-H2 gases; a large fraction of particles will be neutral and will escape to the surrounding surfaces before being dragged to the downstream end of the reactor.

206

A. Gallagher/The formation and behavior ofparticles in silane discharges

There may be something one can do about where the particles escape to, which need not be equally to all surfaces or regions of the electrodes. One possibility is to drive particles to the RF electrode with the thermophoretic force, which causes all size particles to drift toward colder regions with a velocity proportional to the temperature gradient. (The very small ones partially back diffuse against this drift.) Since the substrate is normally heated to --2400, this temperature gradient would naturally occur if the RF electrode were cooled. A second method of controlling where particles escape is to design advantageous variations in plasma potential. If the discharge runs harder in some regions between the electrodes, the plasma potential is larger in these regions, yielding a potential well for negatively charged particles in the plane between the electrodes. It may be possible to guide particles out to the plasma edges with certain electrode structures.

ACKNOWLEDGEMENTS I wish to thank M. A. Childs and A. V. Phelps for many valuable ideas and suggestions. This work has been supported in part by the National Renewable Energy Laboratory under contract DAD-8-18653-0L

REFERENCES 1. Daugherty, J. E., Perteous, R. K., Kilgore, M. D. and Graves, D. B., J. AppL Phys. 11, 3934 (1992). 2. Tanenbaum, D. M., Laracuente, A. L., and Gallagher, A., Appl Phys. Lett. 68, 1705 (1996). 3. Roth, R. M., Spears, K. G., Stein, G. D., and Wong, G., Appl. Phys. Lett. 46, 253 (1985). 4. Howling, A., Sansonnens, A. L., Dorier, J.-L., and Hollenstein, Ch., J, Appl. Phys. 75, 1340(1994). 5. Boufendi, L. Plain, A., Blondeau, J. Ph., Bouchoule, A., Laure, C., and Toogood, T., Appl. Phys. Lett. 60, 169 (1992). 6. Jelenkovic, B. M., and Gallagher, A., J. Appl. Phys. 82, 1546 (1997). 7. Gallagher, A., Barzen, S., Childs, M., and Laraquente, A., "Atomic Scale Characterization of Hydrogenated Amorphous SiUcon Films and Devices" NREL/SR-520-24760, Golden, Colorado (June 1998); Childs, M. A., and Gallagher, A., in preparation. 8. Gallagher, A., J. AppL Phys. 63, 2406 (1988). 9. Perrin, J., Leroy, O., and Bordage, M. C , Plasma Phys. 36, 1 (1996). 10. Haller, I., J. Vac. ScL Technol. Al, 1376 (1983). 11. Gallagher, A., and Scott, J., "Diagnostics of a Glow Discharge Used To Produce Hydrogenated Amorphous SiUcon Films," Annual Report to Solar Energy Research Institute (now NREL) for Contract XJ-0-9053-1, Golden, Colorado (April 1981). * Member, Quantum Physics Division, NIST. E-mail address: [email protected]

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

207

Dust particles influence on a sheath in a thermoionic discharge M. Mikikian, C. Amas, K. Quotb* and F. Doveil Equipe Turbulence Plasma - Lab. de Physique des Interactions loniques et Moleculaires, UMR 6633 - CNRS / Univ. de Provence, Centre de Saint Jerome, case 321, Avenue Escadrille Normandie Niemen, 13397 Marseille cedex 20, France *Laboratoire d'Astronomic Spatiale, Traverse du Siphon -12eme Arr./BP 8, F14476 Marseille cedex 12, France

Abstract. The basic characteristics of dust particles trapped in the sheath of a plate embedded in a plasma produced by emissive hot filaments, are reported. For different plate biases, taking into account the measured plate sheath potential profiles, the high negative dust charge is determined. This charge value is compared to the one predicted by the Orbital Motion Limited model when the contribution of the primary electrons emitted by the filaments is considered. When several dust particles levitate, a sheath potential steepening is observed. The analysis of a collision between two dust particles yields the shielding length at the levitation height.

INTRODUCTION We report experiments on dust particle levitation in the sheath of a conducting horizontal disc plate embedded in a continuous discharge plasma. The ionization sources are hot filaments emitting energetic primary electrons. In these conditions, using the Orbital Motion Limited (OML) model (1), we show that the main contribution of the high negative dust charge is due to the primary electrons. Using a differential emissive probe diagnostic, we measure the sheath-presheath profile of the disc plate, for different negative plate biases. Taking into account the obtained potential distributions, the charge of an isolated dust particle is established by measuring the levitation height h. When the negative plate voltage is increased, h increases too. For any plate bias, the equilibrium height corresponds to the same sheath potential (same electric field) where the balance between the electric and gravitation forces is achieved. The charge found in this way is compared to the value predicted by the OML model. Using the non-intrusive Laser Induced Fluorescence (LIF) diagnostic, we observe that: i) the dust particles levitate below the sheath edge, in a layer where the ions are supersonic and ii) their presence produces a steepening of the sheath potential profile. Using the classical Coulombian scattering relations for a collision analysis between two dust particles and taking into account the measured parameters like scattering angles.

208

M. Mikikian et al /Dust particles influence on a sheath in a thermoionic discharge

impact parameter, minimum approach radius, we can find the shielding length at the levitation height.

EXPERIMENTAL ARRANGEMENT The experiments are performed in a multipolar device in series with a continuous discharge circuit, operating at low argon pressure: Par =10" mbar (coUisionless plasma). The primary electrons emitted by hot tungsten filaments are accelerated toward the grounded wall by a negative voltage VD = - 4 0 V (the plasma appears at VDO =-35 V). In the plasma center, we have set a conducting disc plate which can be left to the negative floating potential or can be more negatively biased by an extemal power supply. The dust particles are hollow glass micro-spheres of radius: rd = (32 ± 2) |im, with a mass density: pa ~ 110 kg/m^. In our standard conditions, the dust particles levitate in an horizontal plane (2), parallel to the plate (liquid phase). Their trajectories are rectilinear, with sudden direction changes in the horizontal plane due to their collisions. In order to establish the potential profiles perpendicularly to the plate, we use two diagnostics: i) the differential emissive probe (DEP) system (3), displaced perpendicularly to the plate, for five different plate biases and ii) for a given plate bias, the non intrusive LIF diagnostic (4), in presence of dust particles. In this case, the laser beam (3 mm diameter) propagates perpendicularly to the plate and the collection optics system is displaced vertically. A CCD camera is used to measure the levitation heights.

DUST PARTICLE CHARGE A dust particle acquires a negative charge when the balance of the ion and electron currents on its surface is reached (5,6). These currents are given in a reasonable approximation by the OML model where it is assumed that the dust grain behaves like a spherical probe in a coUisionless plasma and verifies: r^ « X^D, A.D being the Debye length. So, the dust charge must fulfill the following electrostatic equilibrium equation: I i ( V ) + I e ( V ) + Ipe(V) = 0

(1)

where Ii is the ion current (monoenergetic ions flowing from the plasma to the plate), Ig is the current of the background plasma electrons (Maxwellian population) and Ipe is the current of the primary electrons well represented in our experimental conditions, by an isotropic drifting MaxweUian population (2). V is the difference between the dust surface potential Vd and the local potential. The charge of the grain simply is: (Qd)oML= CVd where C is the capacitance of a sphere with radius r^j. The standard conditions of our experiments give: V(i= -17 V. This high negative value is mainly due to the contribution of the primary electrons and for r^= (32 ± 2) [im, the resulting negative charge is: (Qd)oML= (3.7 ±0.3) 10'e-

(2)

M Mikikian et al/Dust particles influence on a sheath in a thermoionic discharge

209

EXPERIMENTAL RESULTS Experimental dust particle charge Figure 1 gives different potential profiles in function of the distance to the plate in mm, obtained with the DEP diagnostic, for five plate biases: the floating potential -26.5 V and for-32,-38,-44 and-50 V.

presheath

fitting functions

• Vbias = -50 V o -44 V ° -32 V V -26.5 V -60

^ -38 V

H — I — I — \ — I — I — \ — \ — I — I — I — I — I — I — \ — I — I — I — I — h

0

1

2

3

4

5

6

7

8

Distance from the plate (mm)

9

10

FIGURE 1. Plate sheath-presheath vertical profiles, for five plate biases. The symbols are the measurements and the curves, the best fits. The plate biases are: floating potential -26.5 V and applied voltages- -32 -38 -44 and-50 V. & . , ,

The effect of increasing the plate voltage is to shift the potential distribution toward the plasma reached, at about 20 mm. Each set of measurements is well fitted by a function: V(z) = -a.exp(-b.z) + c

(3)

where z is the vertical direction. The parameters a and b depend on the plate voltage and c has a common value equal to the plasma potential: Vp = -1.5 V.

FIGURE 2. Superimposition of five CCD camera frames showing the levitation height increase of the same dust particle when the plate voltage is increased from -26.5 V to -32, -38, -44 and -50 V.

210

M Mikikian et al /Dust particles influence on a sheath in a thermoionic discharge

TABLE 1. For each plate bias, dust height, sheath potential and electricfieldand dust charge values Vbias (V)

-26.5

-32

-38

-44

-50

hexp (mm)

1.90

2.33

2.62

2.90

3.34

Veq (V)

-3.88

-3.60

-3.89

-4.25

-4.40

Eeq (kV/m)

2.11

1.62

2.11

2.17

2.40

Qd (10' e-)

4.3

5.6

4.3

4.2

3.8

The charge can now be estimated by measuring the levitation height. Using the empirical function (3), the distance from the plate gives directly the sheath potential where the electric force Fg due to the negative plate balances the gravitation force Fg. Because we can inject a unique dust particle in the plasma, we can study its position change over the plate. In Figure 2, five CCD camerafi-amesare superimposed, each one showing the position of the same levitating isolated dust grain for a given plate voltage. We observe that when the plate bias is increased, the levitation height rises. For each Vbias? we have reported in Table 1 the measured levitation height hexp. Each one corresponds to almost the same equilibrium sheath potential Vgq and local electric field Egq. For each case, the dust charge found by writing Fg = Fg is calculated and provides approximately the same value. So, we conclude that an isolated dust particle of radius 32 |im is in equilibrium in the plate sheath where the mean local potential is Veq = (-4.01 ± 0.12) V. Its mean charge is Qd = (4.4 ± 1.2) 10^ e". This value is slightly higher than the one predicted by the OML model given by solution (2). We assume that this discrepancy can originatefiromtwo facts: i) the use of a DEP diagnostic could induce an artificial flattening of potential profiles, producing a decrease in the local electric field and (or) ii) a local sheath potential steepening in presence of dust particles could appear. So, with our measurements, in order to balance the gravitation effect, the dust particle charge must be higher. We have checked the assumption ii) through measurements with the LIF diagnostic when several dust particles levitate.

Sheath potential steepening in presence of dust particles When the optics collection system of the LIF diagnostic is displaced in the sheath plate, perpendicularly to the plate, biased at Vbias = -38 V, the ion velocity distribution function f(vi) is shifted toward higher ion velocities. We measure the ion drift velocity v^ax corresponding to the maximum of f(vi), versus the distance h to the plate (Figure 3-a), firom 2.3 mm to 3.2 mm, the sheath layer where dust particles are trapped. The full circles are the data with dust particles and the open triangles without dust particles. For a given h, the drift velocity is higher when dust grains levitate (ion acceleration). The error bar, due to the estimated error of v^ax position is the same for all the data and it is given at h = 2.6 mm. The dust particles behavior in this experiment was the following: i) at each time, they were six to eight dust grains crossing the laser beam and ii) they were oscillating in a vertical plane (vertical oscillation observed for high primary electron emission).The mean levitation height was: ho=(2.7±0.05) mm and the oscillation amplitude: Ah = 0.5 mm.

M Mikikian et al /Dust particles influence on a sheath in a thermoionic discharge

I - ^ With dust particles 3.0E+03

A NO dust particle

J

211

f-^ With dust particles A NO dust particle

Vbias = -38V T

^ ^A

1 I 2.5E+03

•

A

T

A " " «

j 2.0E+03 -1 2.2

A - ^ .

a) 1 - H — —t 2.4

h-

2.6

1

1

1

2.8

h (mm)

1

11

3.2

2.4

2.6

2.8

h (mm)

FIGURE 3 a). Ion drift velocity vmax, versus the distance to the plate (full circles: with dust particles, open triangles: without dust particle), b) corresponding sheath potential profiles (full circles: with dust particles, open triangles: without dust particle).

This oscillation amplitude corresponds approximately to the width of the sheath layer (oscillation influence) where the curves presented in Figure 3, are separated. Moreover, for Te = 1.8 eV, the ion sound speed is Cg ~ 2.1 10^ m/s. The ions reach this velocity at a plate distance of the order of 3.1 mm. So, the dust particles levitate in a sheath layer where the ions are supersonic. The corresponding potential profiles are found writing that the energy of an ion of velocity Vmax? in the potential V is: E = (m^ v^^x) / 2 + V . The ion energy conservation yields: E = eVp. So, V can be deduced and it is given in Figure 3-b) versus h. For a given h, the potential is higher in absolute value with dust particles than without. In other words, the slope of the curve with dust particles is higher than the slope of the one without dust particles (except at each end where both curves begin to superpose). This result is the signature of an electric field increase in this region.

Local shielding length determined through collisions The analysis of a collision (7) between two dust particles can provide the local shielding length. In the presented example, before the collision, one of the dust grain is moving (dust 1) while the second one is motionless (dust 2). Making the following assumptions: a) the collision is elastic (no charge lost) and b) the interaction force between both particles is a central force, we have checked the momentum and energy conservation during the collision. In the laboratory frame, the classical scattering of a charged particle on another, provides the equation: tg(Xo) =

sin(2yo) mi Ivci2 -COS(2V|/Q)

(4)

linking the projectile deviation angle Xo and the target trajectory angle VJ/Q, after the collision, mi,2 are respectively, the mass of the dust grain 1 and 2. Measuring (Xo ? H^o)' we

212

M Mikikian et al /Dust particles influence on a sheath in a thermoionic discharge

have checked that (4) is fulfilled. According to assumptions a) and b) given previously, the following equation also must be fulfilled:

1-^-4 1

2

r^

=0

(5)

where p is the impact parameter of the dust grain 1, |i is the reduced mass of the dust particle system, r^ is the minimum approach radius, vi the dust grainl velocity before the collision and (|)(r) the interaction potential between both particles. Equation (5) must be fulfilled whatever the variation law of ^(r). Taking a screened Coulombian potential interaction, ^(r) is given by: (^(r) = ^ ^ e x p ( - - L )

(6)

where Zi^2^ are respectively the charge of dust grain 1 and 2 andX^g is the local shielding length. Measuring p, r^, vi and calculating the dust charges with the OML model, the only unknown parameter in equation (6) is X^.ln this experiment, where the plate bias is: Vbias ^ -50 V and the levitation height: h ~ 3.3 mm (no vertical oscillations), we find: X^ = (0.46 ±0.12) mm. The measured plate sheath potential profile allows us to estabUsh the electron and ion densities at any position from the plate and then the corresponding electron and ion Debye lengths. At h - 3.3 mm, X^ is close to the electron Debye length.

ACKNOWLEDGEMENTS The authors would like to thank V. N. Tsytovich and S. V. Vladimirov for helpful discussions during their visits in Equipe Turbulence Plasma. This work was supported by a grant from the Direction des Recherches et Etudes Techniques.

REFERENCES 1.Bernstein, I. B . , and Rabinowitz, I. N., Phys. Fluids 2, 112-121 (1959). 2. Amas, C , Mikikian, M., Quotb, K., and Doveil, F., "Dustparticle levitation experiment in a hot cathode discharge at low argon pressure", presented at the ICPP&25^ EPS Conf on Contr. Fusion and Plasma Physics, Praha, 29 june-3 July, ECA 22C, 2493-2496 (1998) 3. Yao, W. E., Intrator, T., and Hershkowitz, N., Rev. Sci. Instrum. 56, 519-524 (1985) 4. Goeckner, J., Goree, J., and Scheridan, T. E., Phys. Fluids B 4, 1663- 1670 (1992) 5. Whipple, E. C , Rep. Prog Phys. 44, 1197-1250 (1981) 6. Havnes, O., Aanesen, T. K., and Melands0, J. Geophys Res. 95, 6581-6585 (1990) 7. Konopka, U., and Ratke, L., and Thomas, H. M., Phys. Rev. Lett. 79, 1269-1272 (1997)

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) 2000 Elsevier Science B.V

213

PLASMA DEPOSITION OF SILICON CLUSTERS: A WAY TO PRODUCE SILICON THIN FILMS WITH MEDIUM-RANGE ORDER ? P. ROC A i CABARROCAS, Laboratoire de Physique des Interfaces et des Couches Minces, (UMR 7647 CNRS), Ecole Polytechnique, 91128 Palaiseau, Cedex, France ABSTRACT The growth of hydrogenated amorphous silicon films is often explained by the arrival of SiHx radicals on the substrate and the subsequent cross-linking reactions leading to an homogeneous material which can be described by a continuous random network. Here we summarize our recent work on a new class of silicon thin films produced under plasma conditions where silicon clusters and radicals contribute to the deposition. The main aspects are: i) silicon clusters with sizes of the order of 1-5 nm are easily formed in silane plasmas; ii) these silicon clusters can contribute to the deposition and lead to the formation of films with mediumrange order ("polymorphous silicon"); iii) despite their heterogeneity, the films have improved transport properties and stability with respect to a-Si:H. The excellent transport properties are confirmed by the achievement of single junction p-i-n solar cells with efficiencies close to 10 %. INTRODUCTION Among thin film semiconductors, hydrogenated amorphous silicon (a-Si:H) has experienced a rapid development in the last three decades. Many of today's ubiquitous consumer products (e.g. solar cells, flat panel displays, and photodetectors) were made possible by the development of this material. Even though much progress has been achieved in the understanding of film growth, two fundamental questions still remain open: 1. What is the structure of a-Si:H ? Two main descriptions have been proposed: on the one hand, a-Si:H is considered an homogeneous material which can be described by a continuous random network (CRN) [^]; on the other hand, a-Si:H is often viewed as an heterogeneous material in which silicon clusters are embedded in an hydrogenated amorphous matrix [\\'*]. 2. What are the growth mechanisms of a-Si:H films ? They have been largely discussed on the basis of the physics and chemistry of silane plasmas and the current models consider SiHs to be the main film precursor [W even though there is some controversy on this subject [\ In fact, the two questions are related. The understanding of the growth mechanisms, sketched in Figure 1, can indeed throw some light on the film structure. Focusing on the gas phase reactions, one usually considers the primary processes, i.e. the dissociation of the silane molecules after inelastic collisions with electrons [\ Because of the complex plasma chemistry most of the models used to describe the a-Si:H deposition are limited to these primary processes. However, even in this simple case the increase of atomic hydrogen flux towards the substrate achieved through: i) a high dissociation of silane [\ ii) a high dilution of silane in hydrogen ["^], iii) the use of a layer-by-layer technique [ \ results in the growth of an heterogeneous material with long-range order: microcrystalline silicon (|ic-Si:H). Many models have been proposed to explain the growth of this latter material, but most of them fail to explain the formation of crystallites and the long term evolution of the film properties [^% Now, given these two materials (a-Si:H and |ic-Si:H) which can be deposited in the same reactor, one can ask whether there is a sharp transition between the amorphous and the microcrystalline silicon materials. In other words, is it possible to grow silicon-thin-film materials with different degrees of medium-range order ? One might think that just by choosing plasma conditions at the border between a-Si:H and |ic-Si deposition, silicon films with medium-range order should be obtained. However, thermodynamic considerations suggest a

214

P. Roca i Cabarrocas /Plasma deposition of silicon clusters

discontinuous disorder/order transition, taking place for crystallite sizes of about 3 nm \^\ Nevertheless, some experimental observations support the existence of ordered domains with sizes below 3 nm {'\'W In order to progress in the understanding of the deposition and structure of silicon thin films, we like to stress the importance of secondary reactions. Indeed, the SiHs hypothesis which may be reasonable under "soft" plasma conditions, becomes less plausible under conditions of high silane dissociation (needed to achieve high deposition rates) where secondary reactions and powder formation take place ]^\ This phenomenon, which is a real drawback in the microelectronics industry because it can lead to a marked decrease in production yield, has attracted much attention in the last five years [^\ Detailed studies of powder formation in silane plasmas have allowed to separate the two main stages of the process, namely the initial formation of a high density of nanosized silicon particles followed by their coalescence to produce powder into five steps \^\ Moreover, studies based on TEM analysis of the particles produced in silane-argon discharges at room temperature before the coalescence stage have shown that the 2-nm silicon particles are crystalline [^"]. Therefore, the incorporation of these crystallites in an a-Si:H matrix should allow to synthesise a new class of silicon thin films with a degree of order intermediate between that of a-Si:H (short-range order) and crystalline silicon (long-range order). Here we give evidence of the contribution of clusters formed in the plasma to the deposition of silicon films consisting of an amorphous matrix in which these nanoparticles are embedded. Such films will be referred to in the following as polymorphous silicon (pm-Si) [^'].

Electrical Power Electron density Energy distribution

SiH^

[-•(

Primary reactions

e-+SiH4

)^[Pumps

Secondary reactions

Increasing RF power. Pressure, Geometry, Flow

.•

SiHx+SiH.

SinHm

Clusters, Crystallites polymers. Powder

|a-Si:H ^ic-Si pm-Si

Surface mobility Chemical equilibrium

Substrate Temperature Figure 1. Schematic representation of the plasma processes (gas phase) and solid phase reactions involved in the growth of silicon thin films.

215

p. Roca i Cabarrocas /Plasma deposition of silicon clusters EXPERIMENT

The films were produced by the decomposition of silane in a multiplasma-monochamber reactor [^^]. Different process parameters (substrate temperature, RF power, total pressure, and dilution of silane in H2, Ar, or He) have been studied in order to achieve plasma conditions where silicon clusters but not powder are formed. The plasma conditions necessary to be at the onset of the formation of powder were determined from Mie scattering measurements. The optical properties of the films were characterized by optical transmission, spectroscopic ellipsometry in the UV-visible range, and photothermal deflection spectroscopy (PDS) measurements. The hydrogen content and hydrogen bonding were deduced from infrared transmission measurements [^^]. High Resolution Transmission Electron Microscopy (HRTEM) measurements were performed in selected samples to confirm the presence of ordered domains, which was also deduced from the kinetics of crystallization of the films. The defect density of the films was studied by PDS measurements and their transport properties by dark and photoconductivity measurements, from which the r^t product was derived. Further information concerning the transport properties and stability of the polymorphous silicon films are given in references ^^ and ^^ respectively. In order to confirm the device quality of the polymorphous silicon films, single junction p-i-n solar cells have been produced and their stability has been studied. The accelerated light-soaking tests on films and solar cells were performed at 80 °C under a 350 mW/cm^ light from a 1-kW Xe arc lamp filtered by- a 150-nm a-Si:H film to ensure uniform creation of defects through the film thickness. RESULTS Low temperature growth: From powders to nanoparticles Figure 2 shows a schematic diagram of the genesis of powder formation. The evolution of the size of the particles, related to the time elapsed from the beginning of the discharge, is very fast. Detailed studies have shown that the time scale is in the range of seconds to minutes and that the increase in size is accompanied by a drastic decrease of the particle density, from lO^^ cm-^ at the beginning down to 10^ cm"^ after agglomeration [^^ ^\ A general review on particle formation, from monomers to macroscopic particles, has been given by J. Perrin and Ch. Hollenstein in the chapter II of the book « dusty plasmas » (Wiley & Sons, 1999). Radicals Molecules

Macromolecule

X^

Clusters, Crystallites

Agglomeration

A

^Sf 1 nm

Hr

M\y—I

10 nm

V ^

Powder

0.1 ^m

h*

1 |iim

Figure 2. Schematic representation of the genesis of the formation of powder. Because of its simple implementation, we have used Mie scattering to determine the plasma conditions (pressure, power, temperature...) which are close to the limit of powder formation

P. Roca i Cabarrocas /Plasma deposition of silicon clusters

216

) ^ \ Moreover, because powder formation is favoured at low temperatures, we have performed studies at 30 °C in silane-argon mixtures, a largely investigated system for which the kinetics are known in great detail [20]. Figure 3 gives an example of the evolution of the scattered light intensity as a function of the plasma duration in a 7 % silane-in-argon discharge at room temperature with an RF power of 20 W. One can see that the increase of pressure results in a sharp transition between a pristine and a powdery plasma. Moreover, the formation of powder is an extremely fast process, as indicated by the strong increase of the signal after a few seconds at 67 Pa. Now, the fact that a scattered light is not detected does not necessarily imply the absence of particles in the plasma. However if there are any, they are present in a very low density or, rather, they are too small to produce a detectable signal. To prove the presence of particles at 20 Pa, we used modulated discharges with a microscope grid located on the substrate holder. As shown on figure 4, nanometer-size crystalline particles are evidenced on HRTEM micrographs in these conditions (silane-argon discharge at room temperature). Similar nanometric particles have been evidenced by other groups using different plasma conditions ) ^ \ ]^\ Although individual silicon particles with sizes in the nm range are not easily detected, this is precisely the range of sizes we are interested in, because their incorporation into the growing film leads to polymorphous films (with a medium-range order), intermediate between a-Si:H and |jc-Si.

0

10

20

30

40

50

60

Time (s) Figure 3, Mie scattering measurements performed in a silane-argon discharge at 30 °C with an RF power of 20 W.

Figure 4. HRTEM of a microscope grid

An important step towards the deposition of polymorphous silicon was the demonstration that films with similar optical properties can be deposited under continuous and modulated plasma conditions [28]. As a matter of fact, this result is implicit in the data shown in Figure 3. Because the nanoparticles are formed before the transition to macroscopic powder and do not grow to form powder, they are not trapped in the discharge and can therefore contribute to the deposition. For this to be possible, one has to admit that the silicon clusters are not always negatively charged, or else agglomeration could not happen. The contribution of nanoparticles to the deposition of a-Si:H has also been observed by scanning tunnelling microscopy ^ \

217

p. Roca i Cabarwcas /Plasma deposition of silicon clusters

At this stage we have shown that the deposition of siUcon thin films can result from the contribution of radicals and positive ions (left side of Fig. 1) as well as from that of silicon clusters and crystallites (the products of secondary reactions in Fig. 1). Therefore, under the latter conditions the growth of heterogeneous materials can take place. Now, what about the optoelectronic properties of such films ? The optimization of a-Si:H deposition conditions has been often driven by the desire to obtain dense and homogeneous films where all kinds of microstructure are avoided ^\ Moreover, there is a general consensus about deposition at room temperature leading to poor quality a-Si:H, even though low-defect density a-Si:H films have been obtained at 50 °C ['']. Figure 5 shows the infrared transmission spectra of films deposited at 30 °C by the dissociation under an RF power of 20 W of a 7 % silane-in-argon mixture to which an increasing amount of hydrogen was added. The film deposited without addition of hydrogen shows two strong absorption bands at 840-890 cm~^ a signature of the presence of (SiH2)n groups, while the Si-H stretching band is shifted towards 2100 cm"^ The addition of hydrogen to the silane-argon mixture produces a dramatic decrease of the (SiH2)n groups as can be seen in the spectra of the films deposited with the addition of 50 - 100 seem of hydrogen. Their stretching bands are shifted to 2010 cm*^ while the bands at 840 - 890 cm"^ are absent. As a matter of fact, these spectra are comparable to those of polymorphous silicon films deposited at 250 °C [^]. On the contrary the spectra of the films produced from argon or with only 10 seem of hydrogen added to the argon-silane mixture correspond to those of a-Si:H films which are generally considered as porous [^^]. However, very similar spectra have been reported for microcrystalline silicon films produced by reactive sputtering at 100 K ^% Despite the large changes in the hydrogen bonding configurations, the hydrogen content of the films, deduced from the integrated absorption of the wagging band at 630 cm-^ remains in the range of 21 % to 24%. Ar + 100 seem H.

c^ '-^w 2 1 % /

A

Ar + 50 seem H AT+10 seem H

B

'^24

%//

e - \l\ y i c

^^^^Ar

y^

o

S h

2

H

•

:3

\22%/

500 600 700

800 900 1000 1100 1200

1900

2000

2100

2200

Wavenumber (cmO Wavenumber (cm"^) Figure 5. IR transmission spectra of a-Si:H films deposited at 30 ^Cfrom a7 % silane-inargon gas mixture to which an increasing flow ofH2 was added. The above results show that even at 50 °C it is possible to produce dense a-Si:H films. Note that no oxygen band (• 11(X) cm"^) related to the ambient contamination [35] is detected in these IR spectra.

P. Roca i Cabarrocas /Plasma deposition of silicon clusters

218

The changes in the hydrogen bonding are also reflected in the absorption of the films deduced from PDS measurements (Fig. 6). One can see that the addition of hydrogen produces a shift of the high energy part of the spectra towards lower energy. 100 seem H, annealed 100 seem H^ as-deposited Argon

S

0.8

1.2

1.4 1.6 1.8 Energy (eV)

2.2

Figure 6. Absorption coefficient measured by PDS in samples deposited at 30 ^C by the decomposition ofa7% silane-in-argon gas mixture.

The absorption coefficient of the film deposited without hydrogen addition has a larger optical gap (Eo4 = 2.15 eV) than the film produced with the addition of 100 seem of hydrogen. As for the high absorption coefficient values in the 1.3 - 1.7 eV range, they could be attributed to a high defect density. However, they can also be due to the presence of crystallites in the film. The fast decay for photon energies below 1.2 eV, along with the fact that annealing the film for 1 hour at 200 °C did not change the absorption coefficient (not plotted for clarity), are in favor of that hypothesis. A quite different behaviour is observed in the film produced under hydrogen dilution. This film has a smaller optical gap (E04 = 2 eV) and we observe a strong effect of annealing during 1 hour at 200 °C. Indeed, the absorption coefficient values strongly decrease over almost the whole spectral range. While the decrease in absorption in the 0.8 - 1.6 eV range can be related to the annealing of deposition-induced metastable defects [17], the decrease in the exponential region indicates that large structural rearrangements have occurred. The standard analysis [37] of the absorption spectra in the annealed state gives a defect density of 3.10^^ cm'^ and a disorder parameter of 60 meV. The different shapes of the absorption edge as well as the different evolution of both types of films under annealing could be explained by the different size of the ordered domains. In the argon case stable silicon crystallites are formed in the plasma (Fig. 4), while smaller (unstable) paracrystallites may be formed with the addition of hydrogen, known to reduce powder formation. Therefore, and not surprisingly, the addition of hydrogen results in denser a-Si:H films with lower defect density. The question is whether nanoparticles are still present in the film, or whether the addition of hydrogen has completely suppressed their formation. Figure 7 shows the HRTEM micrograph of a film deposited at 50 °C from a hydrogen-silane mixture in its asdeposited state and after annealing in the microscope at 425 °C. One can see that nanometersize ordered domains with circular and fringe-like features are indeed present in the film. These nanoparticles can act as seeds for the crystallization of the film at a temperature as low as 425 °C. Interestingly enough, HRTEM studies performed on silicon powders produced from pure

p. Roca i Cabarrocas /Plasma deposition of silicon clusters

219

silane and annealed between 300 °C and 600 °C for 1 hour [15] have shown the formation of ring-like and fringe-like contrast features similar to those reported in Figure 7. Moreover, we have reported similar results for films deposited from silane-hydrogen mixtures at 100 °C [38, 39].

Figure 7. HRTEM of a silicon film produced by the decomposition of a 2 % silane4nhydrogen mixture at 50 °C in its as-deposited state (left) and after annealing in the microscope chamber at 425 ^Cfor one hour. The above results show that the formation of nanometer-size ordered particles in silane plasmas is a quite general feature. As a matter of fact, care must be taken to choose plasma conditions where no secondary reactions take place, though this tends to decrease the deposition rate. Therefore, we tend to think that many studies dealing with a-Si:H films deposited at thigh rates are actually related to nanostructured or polymorphous films as, in fact, claimed by some authors [2-4]. Towards high temperature growth of polvmorphous silicon films Even though low-defect density films can be deposited at 50 °C, their high hydrogen content makes them unstable and fragile. So far we have focused on the formation of polymorphous films at low temperatures. According to Figure 1, the change from a pristine plasma to a plasma containing silicon clusters or powder can be obtained in different ways. As a matter of fact, lowdefect density a-Si:H films have been produced at 250 °C under conditions of powder formation [17]. By adjusting the plasma parameters just below the onset of the formation of powder, it is possible to produce pm-Si films in a wide range of deposition conditions. Figure 8 shows the effect of pressure on the imaginary part of the pseudo-dielectric function (epsilon 2) of silicon films deposited at 250 °C from the decomposition of a 2 % silane-inhydrogen gas mixture under an RF power of 20 W. The increase of the pressure from 66 Pa up to 160 Pa results in strong changes of epsilon 2 spectrum. In particular, the shoulder at 4.2 eV, characteristic of pc-Si films, disappears for the films deposited at high pressures. The analysis of the data by within the framework of the Bruggeman effective medium theory gives a crystalline

P. Roca i Cabarrocas /Plasma deposition of silicon clusters

220

volume fraction of 67 % for the film deposited at 66 Pa. For the films deposited at high pressures, even though their spectra are similar to those of dense a-Si:H films, we cannot model the whole spectrum by a mixture of amorphous silicon and voids, in particular the low energy part (interferences). We have recently shown [34] that nanometer-size ordered domains are also present in these films, with features similar to those presented in figure 7. Moreover, as in the case of the deposition at 50 °C, the existence of ordered domains is revealed by the crystallization of these films at lower temperatures, or equivalently, by their faster kinetics of crystallization. In Figure 9 we compare the evolution of the conductivity of silicon thin films produced at 250 °C (under the same conditions as for the films in figure 8) to that of a standard a-Si:H film; the conductivity was measured at 560 °C in order to determine the kinetics of crystallization. We can see that in the case of standard a-Si:H films there is a long incubation time (to) during which the conductivity does not change. On the contrary the conductivity of the |ic-Si film produced at 120 Pa starts to increase from the beginning of the experiment, i.e. there is no incubation phase. As for the polymorphous films, they display an intermediate behaviour which appears as a signature of the presence of domains with medium-range order. The solid-phase crystallization kinetics shown in Figure 9 can be simulated by two characteristic times: the incubation time to and the crystallization time tc which represents the time above which 63 % of the material has been crystallized [^^]. The inset in Figure 9 shows that the incubation time is reduced in the pm-Si films, but that there are large fluctuations in the crystallization times. While in the case of the film deposited at 133 Pa both to and tc are shorter than for standard a-Si:H, tc is higher in the pm-Si films deposited at 160 Pa and 213 Pa. This suggests that, as the pressure increases, there are changes in either the density or the nature of the domains (ring-like versus fringe-like?). Then, even though the presence of domains with medium-range order (clusters or paracrystallites) reduces the incubation time, their interaction with the growing crystallites in the solid phase crystallization process may result in a slowingdown of the kinetics. 10

—I—I—I—I—I—I—I—I—I—r-—t—I—\—I—I—I—I

1—

P(Pa) 5 120 133 160 213 t^^(s) 9485 — 2103 3891 2130 t"(s) 6852 2926 4952 1060 9919 a-Si:H (5 Pa)

2.5

3

3.5

4

4.5

Photon energy (eV)

Figure 8. Effect of pressure on the of films produced from the dissociation of 2% films silane-in-hydrogen at 250 °C and 20 W,

10000 15000 Time (s)

20000

Figure 9. Evolution of the conductivity during the crystallization of silicon thin at 560 °C under vacuum.

Because most of the results presented above were obtained with silane-hydrogen discharges, one could suspect hydrogen of being responsible for the formation of domains with medium-

p. Roca i Cabarrocas /Plasma deposition of silicon clusters

221

range order; i.e. consider that the ordered domains have nothing to do with gas phase reactions. Indeed, |ic-Si films are formed through hydrogen-mediated solid-phase reactions leading to the complete rearrangement of the a-Si:H matrix. As a matter of fact, we have shown that there are two routes for the production of nanostructured silicon films ^^\. i) The use of a high hydrogen flux with respect to the flux of SiHx radicals (solid phase reactions in Fig. 1). ii) The direct deposition of SiHx radicals and silicon crystallites resulting respectively from the dissociation of silane and from the reactions of the radicals in the gas phase (gas phase reactions in Figure 1). These processes are schematically described in Figure 10. Hydrogen dilution has been reported as a way to improve the quality (decrease the disorder) in a-Si:H films ^\ Moreover, the use of high dilution of silane in hydrogen allows to produce |ic-Si:H films. The control of the flux of atomic hydrogen with respect to that of silicon radicals is easily achieved by the layer-by-layer technique [11]. As shown in Figure 10, the increase of the hydrogen exposure time (dilution) results in the formation of a highly porous and disordered film, before the transition to |Lic-Si growth occurs; as a matter of fact, the nucleation of crystallites takes place in the highly disordered porous phase ^% Moreover, a further increase of the hydrogen exposure time (dilution) results in no film deposition on a glass substrate [11], but in the formation of a more disordered material (a film with medium-range order) when the process is performed on an a-Si:H substrate. This is explained by the long-range effects of hydrogen which induces the rearrangement of the whole a-Si:H substrate ]^]. PLASMA Gas phase reactions (^•TT Crystallites Clusters .

I2 o

— Hz -^^Ar -o-He

-

W) 1 o

C/3

|iC-Si

pm-Si

Flux of atomic Hydrogen Solid state reactions (Layer-by-layer growth) Fig. 10. Schematic diagram of the disorder in silicon thin films as a function of the flux of polyatomic hydrogen impinging on the growing film.

1 . . .

0.8

1 . . .

1 . . .

1 . . .

1 . . .

1.2 1.4 1.6 Energy (eV)

1 . .

.

1.8

Fig. 11. Absorption coefficient measured by photothermal deflection spectroscopy on morphous films produced by the dissociation of silane diluted in either H2, Ar ,or He.

Now, to further support the assumption of the contribution of clusters formed in the plasma to the deposition, as opposed to solid phase reactions, we substituted hydrogen by either argon or helium because in these cases the smaller concentration of hydrogen makes solid state reactions driven by hydrogen improbable. Then we adjusted the plasma conditions in order to be close to the onset of powder formation. It is striking to see that the three films deposited at 250

P. Roca i Cabarrocas /Plasma deposition of silicon clusters

222

°C from different gas mixtures have almost the same Urbach energy (52 - 55 meV), subgap absorption (defect density • LIO^^ cm-^), as well as transport properties and stability (see Fig. 12). We suggest that since in the case of Ar or He dilution hydrogen can hardly be accounted for in solid phase reactions, the growth of films with similar properties is indeed related to the gas phase reactions, i.e. the contribution of clusters produced in the gas phase to deposition. However, further studies are still necessary to analyze in detail the structure of the 1-2 nm clusters (paracrystallites) as a function of the gas used. Stability and Devices Further support for the existence of medium-range order in polymorphous silicon films is given by the study of their metastability [25]. Figure 12 shows the evolution of the r||ax product as a function of the light-soaking time for different silicon thin films. The initial r||iT value of the pm-Si films lies between that of standard a-Si:H and that of |ic-Si films with crystalline fractions above 60 % [25]; i.e. irrespective of the gas added to silane (H2, Ar or He), the pm-Si films have higher r||XT values than standard a-Si:H films. If we now consider the evolution with time of the r||iT product, we see that the |Lic-Si film with a crystalline fraction of 96 % is stable, while the standard a-Si:H film experiences a decrease of the r[\xx product, following a stretchedexponential law. The kinetics of the pm-Si films are quite different from those of a-Si:H. We observe a fast decrease of the r||iT product in the initial stages of light-soaking. However, despite this strong light-induced decrease of the T^T product, by about a factor of 100, the steady-state y\\ix values of the polymorphous films produced with a He or a H2 dilution remain higher than the annealed-state value of a-Si:H. Another interesting aspect is the evolution of the r||iT product in the microcrystalline film with a crystalline fraction of 30 %. Its x\\ix product shows no degradation during the first 100 s of light-soaking, then decreases very fast and stabilizes at a value close to that of the pm-Si film produced with hydrogen dilution. In our opinion this is a further indication of the presence of medium range-order in the pm-Si films. 12

10^ic-Si (Fc = 96 %) -oo - - - O- - -O - - O- - O- - -O , - 0- - O- - - • - pm-Si (Ar)

10-i-40

pm-Si (0.45 nm)

10

- RQ^ and nd{r) ^ 0, QEr = Ar for r < RQ, In other words, the plasma-dust cloud is not a diffuse object with smooth density profile, but it is a compact formation with a sharp boundary. To determine the radius of the dust cloud we consider the force balance inside it. Assuming that Ro ^ ^D^ we implement Eq. (26) and the plasma neutraUty condition Yla ^cx'^a — Q'^d = 0 to rewrite Eq. (23) as 1 d ^dndjr) r^dr dr

^

ndjr) b^ '

^

'

where

!>' = ^

(28)

fixes the spaticil scale of the cloud. According to the estimations written above, b^ « A^/a. Eq. (27) is easily solved resulting in , ,

N

Mr) = 4^.^3

sm(r/b)

V^

„

- < ^-

,^^,

(29)

Since the dust density is nonnegative, the cloud radius should be RQ = irb. The range of vedidity of the expansion in powers of the dust density may now be written as N 100 amu) because the magnetic force is insufficient to cause deflection. At very low mass (< 2 amu) the ions are deflected sufficiently to prevent detection. At intermediate masses, the collection efficiency exceeds 100% because ions are deflected into the collector that would have fallen outside the collector in the absence of the Lorentz force. The mass distribution of aerosols would be found by flying several such detectors with differing mass thresholds. The flow of air around the rocket carries with it low-mass aerosols but heavier aerosols are not entrained in the flow and may strike the surface. This effect has been studied with a gas kinetic computer code [19] but not yet for the conditions of our test flight. Preliminary work indicates that the temperature behind the shock is suffcient to cause evaporation of ice clusters below about 10 nm in radius. The collected current as a function of altitude is plotted for our test flight in Figure 3a and b. A narrow layer appears in the data at an altitude of 86.5 km, approximately the location of the mesopause. From the rocket velocity we flnd that

278

S. Robertson et al. /A rocket-borne detector for charged atmospheric aerosols

100

10

10000

1000

proton masses FIGURE 2. Collection efficiency as a function of mass for the conditions of the test ffight. Efficiency is defined as the number of particles collected divided by the number in the volume directly upstream. Positively charged particles are deflected toward the collecting surface thus the efficiency can exceed 100%.

90 85 4 km 86.5 4

km 80 4 75 4 I ' 70 -0.4 -0.2

' ' I

0

0.2

0.4

detector signal (nA)

0.6

0

0.1

0.2

0.3

0.4

0.5

detector signal (nA)

FIGURE 3. Current collected on descent as a function of altitude. The plot at right shows the detailed structure of the layer. Data points are spaced 14 m apart.

S. Robertson et al. /A rocket-borne detector for charged atmospheric aerosols

279

the charge density is 2,150 cm~^ if we assume that the flow field about the rocket does not alter the aerosol density and that the collection eflSciency is 100%. The duration of the signal, 0.23 seconds, indicates a layer thickness of 495 m. Formation of ice clusters is extremely unlikely at the latitude of New Mexico thus this layer may either be ions or meteoric dust. The standard mechanisms for ion production in the ionosphere have scale heights of many tens of kilometers, thus if the detected layer is ions some mechanism must be invoked for vertical convergence.

FUTURE WORK Work on this topic is continuing in several directions. Computations are being made of the flow fleld around the rocket for the conditions of our test flight. The goals are to find the effect of shock heating on the evaporation rate of ice clusters, and the effect of both flow and evaporation upon the detection efficiency. In the laboratory, we have constructed a facility for creating beams of ice clusters for testing and calibration of rocket instruments and for fundamental studies of clusters such as the determination of rates of electron attachment and photoionization. This facility consists of a vacuum chamber having at one end a supersonic nozzle that creates a spray of argon and water vapor that forms ice clusters during expansion into vacuum. The velocity of the clusters (~550 m/s) is comparable to the velocity of sounding rockets. A magnetic mass spectrograph has been constructed for determining the distribution in masses of the clusters. Initial results indicate that the creation of clusters with masses of four to eight water molecules.

ACKNOWLEDGMENTS The authors acknowledge support from the National Aeronautics and Space Administration and the Department of Energy. We thank Don Hassler, Southwest Research Institute, and Tom Woods, Laboratory for Atmospheric and Space Physics, for the test flight.

280

S. Robertson et ah /A rocket-borne detector for charged atmospheric aerosols

REFERENCES 1. Thomas, G. E., Rev. Geophys. 29, 553, (1991). 2. Hoppe, U.-R, T. A. Blix, E. V. Thrane, F.-J. Lubken, J. Y. N. Cho and W. E. Swartz, Adv. Space Res. 14, 139 (1994). 3. Cho, J.Y.N, and J. Rottger, J. Geophys. Res. 102, 2001 (1997). 4. Ulwick, J. C., K. D. Baker, M. C. Kelley, B. B. Balsley and W. L. Ecklund, J. Geophys. Res. 93, 6989 (1988). 5. Witt, G., Space Res. 9, 157 (1969). 6. Thomas, G. E. and C. R McKay, Planet. Space Sci. 33, 1209 (1985). 7. Thomas, G. E., J. Atm. Terr. Phys. 58, 1629 (1996). 8. Thomas, G. E., Adv. Space Research 18, 149 (1996) 9. Goldberg, R. A., E. Kopp and G. Witt, Adv. Space Res. 14, 113 (1994). 10. Liibken, F.-J., K.-H. Pricke and M. Langer, J. Geophys. Res. 101, 9489 (1996) 11. Bjorn, L. G., E. Kopp, U. Herrmann, P. Eberhardt, P. H. G. Dickinson, D. J. Mackinnon, F. Arnold, G. Witt, A. Lundin and D. B. Jenkins, J. Geophys. Res. 90, 7985 (1985). 12. Zadorozhny, A. M. A. A. Vostrikov, G. Witt, 0 . A. Bragin, D. Yu. Dubov, V. G. Kazakov, V. N. Kikhtenko and A. A. Trutin, Geophys. Res. Lett. 24, 841 (1997). 13. WachU, U., J. Stegman, G. Witt, J.Y.N. Cho, C. A. Miller, M. C. Kelley and W. E. Swartz, Geophys. Res. Lett. 20, 2845 (1993). 14. Havnes, O., J., Tr0im T. Blix, W. Mortensen, L. L Naeshim, E. Thrane and T. T0nneson, J. Geophys. Res. 101, 10839 (1996). 15. Walch, B., M. Horanyi and S. Robertson, Phys. Rev. Lett. 75, 838 (1995). 16. Robertson, S., Phys. Plasmas 2, 2200 (1995). 17. Horanyi, M., J. Gumbel, G. Witt and S. Robertson, "Simulation of rocket-borne particle measurements in the mesosphere," to appear in Geophysical Research Letters, 1999. 18. Gibbons, D. J., in Handbook of Vacuum Physics, Vol. 2., A. H. Beck, editor (Pergamon Press, Oxford, 1966), p. 301. 19. Feuerbacher, B. and B. Fitton, J. Appl. Phys. 43, 1563 (1972).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

281

Paleo-heliosphere: Effects of the Interstellar Dusty Wind Based on a Laboratory Simulation Shigeyuki Minami* and Shigeo Miono ^ *Dept. Electrical Engineering, Osaka City University, ^ Dept. Physics, Osaka City University Sumiyoshi Osaka 558-8585 JAPAN Abstract. The heliosphere is produced as an interaction between the super sonic (solar wind) plasma flow and interstellar supersonic magnetized plasma flow (local interstellar medium:LISM) in relative to the solar system. Our solar system is also moving with respect to the galactic spiral structure. The density changes in time having the period of 0.6 billion years by a differential rotation. The highest 7

-3..

neutral density of the LISM is the order of 10 cm Thus it is possible to see the temporal change of LISM. A few weight percent of dust component exists with the neutral. In this paper, the effects of such high density neutral/dust in the LISM are considered to form the heliosphere based on a laboratory

simulation

considering

the

ancient

electromagnetic

environment,

called

the

paleo-heliosphere. The neutrals of the simulated LISM can be controlled by changing the applied voltage of plasma gun. The laboratory interaction between the spherically expanding solarwind plasma flow and another uniform supersonic and super/sub Alfvenic plasma flow have qualitatively revealed the important role of the LISM magnetic field and the neutral component of the LISM on the structure of the heliosphere. The experimental result of the partially ionized LISM plasma flow suggested a lack of the pressure balance at the nose of the heliosphere and the diffusion of the magnetic field. The structure of the heliopause might be changed due to the long term variation of rich neutral and dusts of LISM because the solar system is moving through the various conditions of our galaxy

INTRODUCTION The solar wind spreads radially outwards at supersonic velocities throughout the solar system. The wind collides with the ambient interstellar medium (ISM) through which the solar system is moving with respect the galactic frame. The region is called the heliosphere. The outer boundary called the heliopause is still a postulated separatorix. The heliocentric distance is predicted to be 100 to 500 AU.

282 S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation

The local ISM parameters are also not yet known directly, especially the magnetic field intensity is uncertain to be 0.01 nT to 1 nT (1). The density of neutral and plasma are known to be 0.1 and 0.01 [cm-3] respectively. One of the most important thing about the ISM is its variability in space. The recent millimeter radio waves from the interstellar CO gas has revealed a clear evidence of irregularity forming spiral arms as shown in Figure 1(1).

"P ••••to

FIGURE 1 ••«•

»?••«

The distribution of the CO cloud in

the Galactic plane (after Dame et al.)

It is important to examine the role of neutral gas to the structure of the heliopause. It is known that our solar system is rotating as a period of 0.2 billion years, while the gas cloud on the solar system diameter takes 0.3 billion years for one rotation. So the solar system encounter the same structure of the cloud having the period of 0.6 billion years. The enhanced neutral density is regarded to be about 1 million times higher than the present local ISM. Although the electromagnetic environment of the heliosphere can not be recorded historically on the earth, it could be estimated, because the interstellar parameters especially the neutral density is recorded in space. So we named the structure as the paleo-hehosphere.

LABORATORY EXPERIMENTS Our experiment is performed using a plasma emitter powered by an intense capacitance bank power supply, which can produce luminous supersonic Ba plasma flow spherically expanding from a certain point; this flow simulates the solar wind plasma flow. The density at 5 cm from the center of the emitter inside of the terminal shock is about lO^'* cm^ which is measured without the LISM. The temperature is about 2 eV, and the obtained nose distance is about 10 cm. The velocity of the simulated solar wind (50 km/s) is measured by the time evolution of the spherically expanding plasma images taken by a time-resolved camera. The LISM (argon plasma; supersonic, but sub- or super-Alfv^enic; density of 10^^ cm'^ temperature of 2

S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation 283

BxCOIL

BxCOIL B2COIL

FUGURE 2 Experimental et up. The LISM flow comes from the left. LISM magnetic field is controlled by the two pairs of coils.

SUN PLASMA L ^

eV) is produced by a plasma LISM gun. The LISM magnetic field FLOW of a maximum 450 G can be applied in any direction to the BzCOIL ^ LISM plasma flow, permitting the experimenter to change the AlfVen Mach number. The experimental design for similar laboratory simulations of terrestrial shock formations has already been described by Minami (2). The experimental set up is illustrated in Figure 2. The time evolution of the interaction is recorded by a time-resolved camera. The variable magnetic field of 0 - 450 G can be applied to magnetize and to change the Alfvenic Mach number, MA, of the LISM. The LISM plasma flow can be magnetized because the magnetic field is applied before the particle ionization at the plasma gun. The LISM plasma flow is regarded to be a frozen-in condition. According to the measurements, the LISM can flow toward the simulated sun even when MA < 1 according to measurements. GUN

EXPERIMENT 1 (With fully ionized LISM flow) Our experiment is performed using a plasma emitter powered by an intense capacitance bank power supply, which can produce luminous supersonic Ba plasma flow spherically expanding (FIGURE 3) from a certain point; this flow simulates the solar wind flow. FIGURE 5 shows the structure of the simulated heliopause by illumination in images taken by a 1 u s time resolved camera at r = 50 /x s; (a) with the magnetic field parallel to the LISM flow and MA =0.3, (b) With the LISM magnetic field tilted (about 30 degrees) toward the LISM plasma flow and MA = 0.1, and (c) without the magnetic field, then MA = 10. The fast-mode Mach number, MA = vo / V" ( (VA^VS^) , is used for case (c) because this Mach nimiber controls the shock structures, where the quantities vo, VA, VS are the solar wind, Alfv^en, and the sonic velocities, respectively. The LISM comes from the left. The emitted plasma that simulates the solar wind interacts with the magnetized LISM plasma flow (3).

284 S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation

The illumination created by the Ba plasma line simulating the solar wind plasma so that the boundary of the outflowing solar wind plasma, the heliopause, can be seen optically. The heliopause structure obtained by this illumination shows a clear axiasymmetric structure to the LISM flow when MA < 1 and the LISM magnetic field is tilted. The LISM plasma density is almost constant for the different LISM magnetic fields.

FIGURE 3 Simulated solar wind plasma gun

FIGURE 4 photographs Expanding solar wind for 5 jU s, 15/is, 2 5 / i s after the ignition. FIGURES (a),(b),(c) Simulated heliosphere without neutral LISM.

EXPERIMENT 2 (With partially ionized LISM flow) Figure 4 shows the effects of the neutral component of the LISM showing a collapsing heliosphere. A neutral gas plume of several cm"^ at 1 bar is injected into the plasma gun. When the applied gun voltage is low, the simulated LISM flow is

S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation 285

partially ionized. The experiment is performed by applying insufficient voltage to the plasma gun. The result shows significant difference to the result of fully ionized LISM flow. At the beginning of the ionization of the neutral LISM flow, the nose of the contact surface, heliopause, is extended due to the ionization of the neutral flow by the solar wind. The initial ionization along the LISM magnetic field at the nose can be explained by the intrusion of the LISM magnetic field into the heliosphere.

FIGURE

5

Laboratory

heliosphere

structure for different ionization rate in %. LISM flow comes from the left.

0.5 H

10%

50%

100% lonizatjoi^ Rate

DISCUSSION The preliminary laboratory interaction between the spherically expanding solar wind plasma flow and another uniform supersonic and super/sub AlfVenic plasma

286 S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation

flow have qualitatively revealed the important role of the LISM magnetic field and the neutral component of the LISM on the structure of the heliosphere. It is also known that the magnetospheric protector, called the Solar magnetopause is also collapsed when the wind —> neutral is rich as shown in FIGURE 6. The center is the the simulated earth. The soalr wind comes from the left, (a) without (a) Without fast neutral. neutral, (b) with netural.

Solar wind

FIGURE 6. The simulated magnetopshere (a) without neutral flow (b) with neutral flow allowing the intrusion (b) With fast neutral.

of the solar wind partilces into the magnetopause.

FIGURE 7

Illustrated paleo-magnetosphere when the rich interstellar neutral/dust are

encoutered.

The rich neutral brings dusts together (a few perent mass weigh ratio), so the rich

S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation 287

dust particles might precipitate to the solar system and also to the earth. At that time, every day meteor shower could be seen in the sky. FIGURE 7 shows a possible electromagnetic environment of the ancient earth showing a meteor shower. The study of the paleo-heliosphere and the paleo-magnetosphere has just started. The research was supported by the Grant-in-Aid for Scientific Research (03238214 and 04222215) of the Ministry of Education and Culture in Japan.

REFERENCES 1. Dame et al., Astrophys. J., 305, 892 (1986). 2. Minami, S., Geophys. Res. Let, 21, 81-84, (1994). 3. Minami S., et al., Geophys. Res. Lett., 13, 884 (1986).

This Page Intentionally Left Blank

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

289

Current Loop Coalescence in Dusty Plasmas J. I. Sakai and N. F. Cramer* Laboratory for Plasma Astrophysics, Faculty of Engineering Toyama University, 3190, Gofuku, Toyama 930-8555, Japan * Theoretical Physics Department and Research Centre for Theoretical Astrophysics, School of Physics, The University of Sydney, N.S.W. 2006, Australia

Abstract. The weakly ionized plasmas that occur in protostellar disks and in the cores of molecular clouds generally have a dust component. We consider the effects of the presence of charged dust on magnetic reconnection process. We investigate the dynamics of two current loops coalescence, for both cases; partial and complete magnetic reconnection. It is shown that the field-free gas produced during the two loops coalescence with complete magnetic reconnection would be able to initiate star formation.

I

INTRODUCTION

The w^eakly ionized plasmas that occur in protostellar disks and in the cores of molecular clouds generally have a dust component. The ionization fraction of molecular clouds is typically only ^ 10"'', and the dust may contribute ^ 1% of the mass of the cloud. It has been shown that the ion-neutral drift leading to ambipolar diffusion can lead to the steepening of the magnetic field profile and to the formation of singularities in the current density of hydromagnetic fluctuations [1,2] and Alfven waves [3,4]. Recently Bulanov and Sakai (1998) [5] investigated in datails magnetic reconnection process in weakly ionized plasmas. In the present paper we consider the effects of the presence of charged dust on magnetic reconnection process. We investigate the dynamics of two current loops coalescence, for both cases; partial and complete magnetic reconnection. It is shown that the field-free gas produced during the two loops coalescence with complete magnetic reconnection would be able to initiate star formation.

290

J.I. SakaU N.E Cramer/Current loop coalescence in dusty plasmas

II

BASIC EQUATIONS

We consider molecular clouds consisting of neutral atomic and molecular species, the ionized atomic and molecular species, the electrons, and negatively charged dust grains. We start a 4-fluid model of the plasma, which employs the fluid momentum equations for plasma ions (singly charged), neutral molecules, charged dust grains and electrons: A f - ^ + Vz • V v J = -Vpi + UiC (E + Vi X B) -Pi^ini^i - Vn) - pmdi^i - Vd),

(1)

P„(^+v„.Vv„)=-Vp„-p,.„.(v„-v.) -Pn^nd(Vn-Vrf), 'dVd

Pdi-gf+^d'

(2)

V v J = ZdUde (E + Vd X B) -Pd^dni^d - Vn) - Pd^di{^d - VO,

0 = - n e e ( E + Ve X B ) - pe^eni^e

" V ^ ) " pe^ed{^e

" ^d) " Pe^ei{^e

(3) " V^),

(4)

where E is the wave electric field, rris is the species mass, Ps is the species mass density and V5 is the species velocity, pi and Pn are the ion thermal and neutral thermal pressures, and u^t is the coUision frequency of a particle of species s with the particles of species t. We have neglected electron inertia in (4), momentum transfer to ions from electrons in (1) and to dust grains from electrons in (3), and the dust thermal pressure gradient in (3). The parameter S = Ue/rii < 1 measures the charge imbalance of the electrons and ions in the plasma, with the remainder of the negative charge residing on the dust particles, so that the total system is charge neutral. -eue + erii - ZdCUd = 0.

(5)

A typical value of 5 for molecular clouds is (^ = 1 — 10~^. The charge on each dust grain is assumed constant, and for simplicity we also assume that 5 is constant, even though Tin and thus n^ are variable. The neutral mass density obeys the continuity equation

^

+ V • (p„v„) = 0.

(6)

To complete the system of equations. Maxwell's equations ignoring the displacement current are used, with the conduction current density given by j = e{niVi - UeVe - ridZdVd).

(7)

291

11. Sakai, N.E Cramer/Current loop coalescence in dusty plasmas where equilibrium charge neutrahty is expressed by (5). Equations (4) and (7) lead to the following generahzed Ohm's law: X. J X B X B = ^^ e

e

e

We now use the strong coupling approximation (Suzuki and Sakai 1996) [3], whereby the ion inertia term (the left hand side) and the ion thermal pressure term are neglected in (1), leaving a balance between the remaining terms. At this point it is useful to normalize the magnetic field by a reference field 5o, and define the Alfven speed based on the field BQ and the ion density: VA = Bo/{iioPi)' Eq. (1) may then be written, using (8) and Faraday's law neglecting the displacement current, - ^ ( V X B ) X B = Qrni^i

- Vrf) X B + Uin{Vi - V^) + Uid{Vi - V ^ ) ,

(9)

where fi^ = Cli{l — S)/6 and Qi is the ion cyclotron frequency, Qi = Boe/rrii. The presence of dust introduces the first and third terms on the rhs of (9). For strongly coupled dust grains and neutral gas, such that v^ = v^, the ion equation of motion gives the following expression for the ion-neutral relative drift velocity: VD - V, - V, = F^

[(V X B) X B - (i?/p)((V x B) x B) x B],

(10)

where Ui = Uin + Uid, R = ^mpn/^ip^. B^ = B^ + B^ + B^^ and F = l/{l + R^B^/p'^), For a small dust number density, Uid « i^m, so we approximate ui = Vi^. Summing the momentum conservation equations (1,2,3) of the ions, neutrals and dust, the equation of motion for the neutral velocity is obtained. We normalize the density by PQ, defining p = {pn + Pd)/po, and the pressure by po- The velocity is normalized by the Alfven velocity based on po, VA — Bo/(/ioPo)^^^7 and space and time are normalized by LQ and TA = LQ/VA- The result is p [ ^ + v - V v j =-p\/p+{V

xB) x B .

(11)

The magnetic induction equation gives, using (8) with the coUisional electron momentum transfer terms neglected, and neglecting the Hall term, dB ^ , — = V X (v X B) + ADS/

X - (((V X B) X B) X B + {R/p)B\V

where J5 = |B| and Ajj = Pn/j^i^APi-

x B) x B )

(12)

292

J.I. Sakai, N.F. Cramer/Current loop coalescence in dusty plasmas

(a)

(b)

0 -1^^^^^^P^^^H*^®*''

50 H

50 -

>^ 100 H

00 -

150 H

50 -

^-^^^H ' ^'^"^^H

-

\

^

X

50

100 Y

-0.5

0.0 Bz

150

^—

2.5 5.0 7.5 10.0 12.5 D e n s i t y and V x - V y v e c t o r p l o t s

-1.5

-1.0

0.5

1.0

1.5

(d)

(C) 0 -

50 -

!>\/ V y y ^ ^ ^ ^ ^ '^ ^ ^ ^ \l \f S W N / y w l ^ ]l\/^ i^ 1^

50

N

100 H

X ^ >j \ \/ \f sL J/\L

NZ- i^

^>i'ii^i^>\f^i4'

V \l^ ]/' ^ ^

-»^ -?

-:»

50 -A

14 ^ ^ ^ ^

X

«

y ;^ f ^ ^ '^k ^

^

^ -^^^^^^ / 7\ f^ M\f

150 H

/ /

100 H

7? ; f ;?1 ^

15 0 H

/ . >l / l ' ^ ^ f f. f yjK } ^

/v- V ^ 1^ 1^ A f f^ f f^ f n / /\ /\ /\ f ^ ^ ^ 1^ \ ^ Kj^ ^ ^ y ^ ^ f f f f f^ -t—I f ^ ^ |A- t-f^

50

100 Y

150

-t-

2.5 5.0 7.5 10.0 12.5 •1.0 Density and Vx-Vy vector plots

T

-0.5 0.0 0.5 Bz and Density Contour

50 -\

X

X

100

150 H

100 H

150

0.01

0.02

0.03

0

T

5 10 Jz Temperature F I G U R E 2. (a) Density distribution overlapped with velocity field {v^ — Vy)^ (b) magnetic filed Bz with density contour plots, (c) current J^, and (d) temperature distribution at t — 8.1r^ for complete reconnection, with self-gravity eflPect. 0.00

1.0

294

J.I. Sakai, N.E Cramer/Current loop coalescence in dusty plasmas

III

SIMULATION RESULTS

We use a 3-D simulation code of the above equations in which the numerical scheme is the modified 2-step Lax-Wendroff method. The system sizes are 0 < X = y < GirL, and 0 < 2: < O.birL in the x, y and z directions, respectively. The mesh points are A^^; = 200, Ny = 200 and A^^ == 10 in the x, y and z directions, respectively. We used free boundary conditions (first derivatives of all physical quantities are continuous) for the x and y directions, while periodic boundary condition is used in the z direction. We take an initial each current loop, which is placed along the z direction to satisfy a force-free condition. Other parameters used here are P = 0.01 and AD = 0.5. Fig. 1 shows four snapshots ait = S.ITA- (a) density distribution overlapped with velocity field {v^ — Vy), (b) magnetic filed Bz, (c) current J^, and (d) temperature distribution. As seen in Figs. 1(a) and (b), during two loops coalescence the density increases about 13 times of the initial state. This is contrast to the case of the partial reconnection, where only poloidal magnetic field produced by two loop currents can dissipate. As seen in Fig. 1(a), the density accumulation appears near the region where the magnetic reconnetion occurs and the magnetic field is almost free. Next we investigate the effect of self-gravity. Fig. 2 shows the simulation results for complete magnetic reconnection case with the self-gravity at t = 8.1r^. As seen in Fig.2 (a) there occurs strong density accumulation with about 13 times larger than the initial value near the region where the magnetic fields are almost free, due to complete magnetic reconnection. The field-free gas produced during the two loops coalescence with complete magnetic reconnection could be able to initiate star formation.

IV

CONCLUSION

We investigated the effects of the presence of charged dust on magnetic reconnection process for both complete and partial magnetic reconnection during two current loops coalescence. We showed that the field-free gas produced during the two loops coalescence with complete magnetic reconnection could be able to initiate star formation. We thank the Cosel and Densoku company for the support.

REFERENCES 1. Brandenburg, A. and Zweibel, E., Ap. J. 427, L91 (1994). Brandenburg, A. and Zweibel, E., Ap. J. 448, 734 (1995). 2. Mac Low, M. M. et al., Ap. J. 442, 726 (1995). 3. Suzuki, M. and Sakai, J.I., Ap. J. 465, 393 (1996), 4. Suzuki, M. and Sakai, J.I., Ap. J. 487, 921 (1997). 5. Bulanov, S. V. and Sakai, J.I., Ap. J. Suppl. 117, 599 (1998).

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V. All rights reserved

295

Jeans-Buneman instability in non-ideal dusty plasmas S.R. Pillay\ R. Bharuthraml'^ F. Verheest^ N.N. Rao^ and M.A. Hellberg^ ^Department of Physics, University of Durban-Westville, S.A., "^M.L. Sultan Technikon, Durban, S.A., ^Sterrenkundig Observatorium, Universiteit Gent, Belgium, "^Physical Research Laboratory, Ahmedabad, India, ^Faculty of Science, University of Natal, Durban, S.A.

Abstract. The behaviour of Jeans-Buneman instabilities in a non-ideal dusty plasma is examined, by deriving the dispersion law and solving it numerically for a range of realistic parameters. The thresholds and growth rates for the Jeans and Buneman instabilities are discussed in terms of the relative drift between ions and charged dust, of the dust-to-ion temperature ratio and of the deviations from the ideal gas law for the dust. We find that for weak dust self-gravitation non-ideal effects have a significant influence on the growth rate of the pure Jeans instability, whereas for stronger dust gravitation the departure from the ideal gas behaviour is less prominent. On the other hand, non-ideal effects have little or no influence on the Buneman instability.

INTRODUCTION Dusty plasmas are encountered in a wide range of environments, from astrophysical situations to industrial and laboratory devices. Their increased observation has resulted in a growing interest in the behaviour and properties of dusty plasmas. Various reviews of the field have covered dusty plasmas form the solar system [4] to waves and instabilities in dusty plasmas [7,13]. On the other hand, the Jeans instability (JI) of a self-gravitating system has been well known for a long time [6]. In a dusty plasma, on the other hand, the JI has come into focus more recently [1,2,9,14], and it has also been found that in the presence of particle streaming, both the JI and the usual Buneman instability (BI) can overlap [8,10]. Further investigations have examined the influence of particle size distributions on the Jeans-Buneman instability, concluding that the growth rate was enhanced both for discrete particle size distributions [8] as well as for continuous size distributions [11]. One of the outstanding features of dusty plasmas is that the dust is of finite size, making these plasmas non-ideal [3,12]. The commonly used ideal gas law is primarily valid for particle sizes in the submicrometre range and for dilute dusty plasmas. However, this approximation breaks down for grain sizes in the supermicrometre

296

S.R. Pillay et al /Jeans-Buneman instability in non-ideal dusty plasmas

range, as the mean particle separation is smaller and interactions between neighbouring particles are enhanced [12]. In this paper we examine the Jeans-Buneman instability in a non-ideal dusty plasma, by incorporating the Van der Waals equation of state for the dust [12], whilst retaining the ideal gas law for the ions.

MODEL EQUATIONS AND LINEAR DISPERSION LAW Our dusty plasma consists of electrons, ions and charged dust. The electrons are light enough so that inertial effects can be neglected and their density is Boltzmann distributed. Full ion dynamics is retained, on the other hand, with a possible equilibrium streaming compared to the background dust species. Finally, for the dust both non-ideal and self-gravitational effects are incorporated. Ion self-gravitational effects are omitted, as these have been shown to be negligible [11,14]. The description of parallel electrostatic modes starts from the continuity equations for the ion and dust species, ^

+ ^ M i ) = 0

{j = hd),

whereas the electrons are assumed to be Boltzmann, i.e., rig = In addition, we need the ion and dust momentum equations, dvj dt dVd dt

dvi^ _ _ jiksTj drij _ j e^ ^dcj) ^ ' dx miTii dx rrii -^ ^-^^ dx' d^p Qd dcj)_ ^^_r_ ^ _ _ 1 9Pd _ ^iu_^^ rrid dx ndTUd dx rriddxdx dxdx

(1) Neexip(e(l)/kBTe). .^. ,^x

Here cj) and T/? are the electrostatic and gravitational potentials respectively, the labels j = e^i^d refer to electrons, ions and charged dust grains, and the other notations are standard. Finally, the description is closed by the electrostatic and gravitational Poisson's equations, ^ 0 ^ + H ^3^3 = 0.

-Q^ = AnGndTrid,

(4)

3

together with the Van der Waals equation of state for the dust {pd + Anl){l

- Brid) = Udi^BTd^

(5)

The constants A and B in (5) are calculated in terms of the critical parameters [5], by requiring that dpd/drid — 0 and d'^pd/d'^Ud — 0, yielding A = 9kdTc/Snc and B = l/3nc, where the subscript c denotes the respective values at the critical point. Using uppercase letters for the equilibrium values, charge neutrality in equilibrium requires that A^^- = Ne + ZdNd^ assuming that the dust is negatively charged and carries Zd electron charges.

S.R. Pillay et al. /Jeans-Buneman instability in non-ideal dusty plasmas

297

We linearize and Fourier transform the relevant equations, invoke 'Jeans swindle' in ignoring the zeroth order of the gravitational potential in looking at local perturbations, and obtain the general dispersion law as 1 1 + —1 ^ k^\l^

=

o;^uj'^. H! . 2i (a; - kVoY - k'Cf ^ a;2 + u;j, - k^Cj - k^Cl,'

(6) ^^

We have introduced the different plasma frequencies through LJ^J = NjOj/sorrij^ the dust Jeans frequency through CJJ^ = AnGNdrnd^ the thermal velocities through Cj = ^yjKBTj/mj and VQ is the ion drift speed. In addition, there is a non-ideal part to the dust thermal effects, given by Cj^^ = V — Q^ with V = BNdkdTd{2 — BNd)/md{l - BNdf and Q = 2ANd/md. We briefly discuss some limiting forms of this dispersion law. In the limit when (a; - kVoY ^ k'^Cf, the ions also are Boltzmann-like, i.e. ujli/{to - kViY " ^^^i — — 1/k'^Xjy-. With the help of the ion Debye length X^i we introduce a global Debye length XD through l/Xj^ = 1/A^g -|- l/Xj^-^ so that (6) then reduces to

P

l + k^Xl

+ CI + CI,,

(7)

Without dust gravitational effects this is precisely the dispersion law given by Rao [12]. On the other hand, for an ideal gas A = 5 = 0, and we have a generalization of the results of Avinash and Shukla [1] to warm dust. Expanding (6) we have the following normalised form of the dispersion relation, 0 = u^ai -f u^{-2KVoai)

+ ^'^[K^a^a^ - (1 + a2) + ax{l3'^a2 - K'^a^)]

+ u{2KVo[a2 - ai{^^a2 - K^a^)]} + K^a^[ai{f3^a2 - K^a^) - a^] -{f3^a2-^K'a3),

(8)

where ai = I + Ne/K'^Ni, a2 = ZdNdrrii/Nimd, a^ = miTd{l + e^r + ^cf)/mdTe, a4 •= V^ — Ti/Te^ and e^,^ = 7y(6 — ry)/(3 — r/)^, Q / = —9ar]/A^ represent respectively the contributions to the volume reduction coefficient and the molecular cohesive forces. Also we have that rj = Nd/ric^ a = Tc/Tdj K = kX^ and uj is the normalised frequency. The following normalisations have been introduced: distance by A^ = (soTe/iVie^)^/^, time by uj~^ and speed by c^ = {Te/miY^'^. The factor /? is the ratio of the dust Jeans frequency to the dust plasma frequency, i.e. /3 = ^jdl^pd-

RESULTS AND DISCUSSION The dispersion relation (8) is solved numerically for the following fixed parameters: milrrie = 1836, Zd = 700, Ne/Ni = 0.5 and Td/Te = 0.1. To aid our understanding of the low frequency nature of the fiuctuations, the frequencies are plotted in terms of the dust plasma frequency ujpd- Note that in all figures the continuous line represents the ideal case. Figure 1 examines the growth rate as

298

S.R. Pillay et al /Jeans-Buneman instability in non-ideal dusty plasmas

a function of the non-ideality parameter rj at kXi) = 0.2, Ti/Te = 0.1, /3 = 0.1 and a = 0.1, 1.0 and 2.0, respectively, for stationary ions (i.e. K = 0). Clearly this mode is a pure JI. We note that at a = 0.1, the growth rate decreases with increasing rj. For both a = 1.0 and 2.0 there is an initial increase in the growth rate with r/, before decreasing below that observed for an ideal gas. The growth rate remains greater than that of an ideal gas over a larger range of rj for higher a. This can be interpreted as follows. With Td/Te fixed, increasing a corresponds to an increase in the critical temperature, Tc. Hence one notices that for a greater critical temperature, the combined non-ideal effects of volume reduction and cohesive forces initially enhance the growth rate of the pure JI, before exerting a strong damping influence. Furthermore, for large a, there is a large range of 77-values over which the factor ^ = (1 -h e^^r + ^c/) < 1- On inspecting figure 1 we notice that the enhancement in the growth rate in the non-ideal case corresponds to the state when (^ < 1, i.e. the cohesive forces dominate over the volume reduction effect. Figure 2 repeats the investigation illustrated in figure 1, but now at /3 = 0.5, corresponding to stronger dust gravitation. The behaviour of the curves for increasing a is very much the same as for /? = 0.1. However, the enhancement in growth rate due to non-ideal effects is significantly lower here than in figure 1. Hence it appears that in the presence of stronger dust gravitation the growth rate of the pure JI is less sensitive to the non-ideal nature of the plasma. We also note from figures 1 and 2 that the corresponding drop in the growth rate at high r/ is less significant. Figure 3 now examines the growth rate as a function of rj for an ion drift speed, Vo = 0.5. At this drift speed both the JI and the BI occur [11]. Here kXD = 0.2, /3 = 0.1 and T^/Tg = 0.1. The observed behaviour in the growth rate for increasing a is similar to that observed in figure 1. Comparing figures 1 and 3 one notes that the departure from ideal behaviour is more prominent at K = 0, i.e. in the absence of the BI. It has been shown that the transition from the JI to the BI occurs when K > Vti [11], which in normalized form implies Vo > {Ti/TeY^'^. In figure 3 we have Vo = 0.5 > \ / 0 T . Hence we conclude that the instability seen in figure 3 is the BI, whilst that seen in figure 1 (for Vo = 0) is the pure JI. As it has been shown that the BI is a much stronger instability than the JI [11], the growth rate in figure 3 (for the BI) is much larger than that in figure 1 (for the JI). Figure 4 examines the growth rate as a function of the ion drift speed K at /? = 0.1, Ti/Te = 0.1, kXD = 0.2, T] = 2.0 and a = 0.1, 1.0 and 2.0, respectively. Only marginal differences are observed in the growth rates at the respective a values. Figure 5 examines the behaviour of the growth rate under the same conditions, but over a range of low ion drift speeds (i.e. corresponding to the state when the JI is still present). One now notes a more significant dependence of the growth rate on the non-ideal parameters. Thus, when the BI dominates, as is the case at high ion drift speeds, the influence of the non-ideal parameters is negligible. Figure 6 examines the growth rate as a function of kXD for 1^ = 0, /? = 0.1, Ti/Te = 0.1, 7/ = 2.0 and a = 0, .1, 1.0 and 2.0, respectively. Clearly, this is once again the pure JI. Here, the departure from ideal gas behaviour is more prominent for low a, whereas at high a one approaches the ideal gas limit. There is a maximum

S.R. Pillay et al. /Jeans-Buneman instability in non-ideal dusty plasmas 0.080

299

0.222

i

0.221

0.075

1.0

1.5

2.0

2.5

Figure 1 :Normalized growth rate y/cop4 vs. r|

Figure 2:Normalized growth rate y/©^ vs. r\

forVo/Cs=0.1andp=0.1.

for V/C==0 and (3=0.5. O S

^

0.130 0.129 0.128 0.127 0.126 0.125 Figure 3 :Normalized growth rate y/© . vs. r|

Figure 4:Normalized growth rate y/o)^ vs. V^ /C^

forV /C =0.5and|3=0.1.

for |3=0.1aiidr|=2.0.

O S

>^

0.10

0.085

i

0.075

0.10

Figure 5:Normalized growth rate y/Wp^ vs VyC^. a values are indicated on the curves.

0.05

Figure 6:Normalized growth rate y/cOp^ vs. k?i^. a values are indicated on the curves.

300

S.R. Pillay et al /Jeans-Buneman instability in non-ideal dusty plasmas

kXn value, beyond which the JI does not exist due to gravitational collapse. This limit decreases only marginally with decreasing a.

CONCLUSION We have investigated the influence of non-ideal effects on the Jeans-Buneman instability. We note that for weak self-gravitation of the dust, non-ideal effects have a significant influence on the growth rate of the pure JI. There is a marked damping of the growth rate of the JI at high rj and low a {= Tc/Td). For stronger dust gravitation, the departure from the ideal gas behaviour is less prominent. With the onset of the BI, the departure from ideal gas behaviour becomes negligible. Hence non-ideal effects in the dust component have little or no influence on the BI. As in the ideal gas case, we have a critical Jeans length for the JI, which decreases marginally for low a.

Acknowledgements This work was supported by the Flemish Government (Department of Science and Technology) and the (South African) Foundation for Research Development in the framework of the Flemish-South African Bilateral Scientific and Technological Cooperation on the Physics of Waves in Dusty, Solar and Space Plasmas.

REFERENCES 1. Avinash, K. and Shukla, P.K., Phys. Lett A 189, 470-472 (1994). 2. Bliokh, P.V. and Yaroshenko, V.V., Sov. Astron. 29, 330-336 (1985). 3. Fortov, V.E. and lakubov, I.T., Physics of Nonideal Plasmas^ New York : Hemisphere, 1990. 4. Goertz, C.K., Rev. Geophys. 27, 271-292 (1989). 5. Joos, G., Theoretical Physics^ New York: Dover, 1986, p. 497. 6. Kobb, E.W. and Turner M.S., The Early Universe^ Reading: Addison-Wesley, Reading, 1990, p. 342. 7. Mendis, D.A. and Rosenberg, M., Annu. Rev. Astron. Astrophys. 32, 419-463 (1994). 8. Meuris, P., Verheest, F. and Lakhina, G.S., Planet. Space Sci. 45, 449-454 (1997). 9. Pandey, B.P., Avinash, K. and Dwivedi, C.B., Phys. Rev. E 4 9 , 5599-5606 (1994). 10. Pandey, B.P. and Lakhina, G.S., Pramana 50, 191-204 (1998). 11. Pillay, S.R., Bharuthram, R. and Verheest, F., Proc. 1998 Int. Conf. Plasma Physics^ Geneva: EPS, 1998, 22C, pp. 2497-2500. 12. Rao, N.N., J. Plasma Phys. 59, 561-574 (1998). 13. Verheest, F., Space Sci. Rev. 77, 267-302 (1996). 14. Verheest, F., Shukla, P.K., Rao, N.N. and Meuris, P., J. Plasma Phys. 58, 163-170 (1997).

Part V. Basic Experiments

This Page Intentionally Left Blank

FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved

303

Waves and Instabilities in Dusty Plasmas N. D'Angelo Department of Physics and Astronomy The University of Iowa, Iowa City, Iowa 52242

Abstract. The first section of the paper presents a summary of the work performed by our group at the University of Iowa on waves and instabilities in dusty plasmas, emphasizing the very close connection between wave phenomena in these plasmas and those in plasmas with large percentages of negative ions. The second section examines the effects of negatively charged dust grains on an ionization instability in plasmas of low degree of ionization, in which electrons are present which have an energy slightly above the ionization energy of the neutral gas.

I. This section of the paper presents a summary of the experimental and theoretical work on v^aves and instabilities in dusty plasmas, performed by our group at the University of Iowa during the last decade. Our work has generally proceeded along two parallel lines, with the same wave modes and/or instabilities being studied both in dusty plasmas and in plasmas with large percentages of negative ions. Since in typical laboratory dusty plasmas the dust grains are negatively charged, the close connection between wave phenomena in these plasmas and those in negative ion plasmas should not come as a surprise. Dusty plasmas, on the other hand, in addition to much smaller charge-to-mass ratios have certain properties which are not shared by negative ion plasmas. In the first place, in a dusty plasma many charge-to-mass ratios for the dust grains occur at the same time, unless one is dealing with the so-called "monodisperse" grains. Secondly, in the presence of a wave the dust grain charge may be variable, as pointed out, e.g., by (1). And finally, in a steady state plasma, the ions lost to the grains must be replaced by ions produced by ionization of a neutral gas. This introduces the wave-damping mechanism referred to as "creation" damping (2). Table 1 lists papers by our group, divided according to the particular wave mode investigated, for both negative ion plasmas and dusty plasmas. If we confine our attention to "low-frequency" electrostatic modes in magnetized dusty plasmas, we find (3) that four such modes exist, two being of an ion-acoustic

N. D'Angela/Waves and instabilities in dusty plasmas

304

TABLE 1. Negative ion plasma Dusty plasma Wave mode/ (refs.) (refs.) Instability 4,11 2, 9,10 Ion-acoustic (fast) 4 10, 12, 13 Ion-acoustic (slow) 10,18 15, 17 EIC (fast) 19 15, 17 EIC (slow) 23,24 22 PVSI 26 Rayleigh-Taylor 25 28 29 PRI 31 Shocks

type and the other two of an electrostatic ion cyclotron (EIC) type. The first two modes correspond to the so-called "fast" and "slow" modes in a negative ion plasma analyzed by D'Angelo et al (4) and experimentally investigated by Wong et al (5), Sato et al (6), and Nakamura et al (7). The "fast" mode in a dusty plasma is simply an ion-acoustic mode modified by the presence of a nearly stationary dust. It is also called a DIA mode (dust ion-acoustic mode). The presence of negatively charged dust increases the phase velocity of the wave, thereby decreasing the wave coUisionless (Landau) damping (see, e.g., Rosenberg (8). Properties of grid-launched DIA waves are illustrated in Figure l(a,b) which show (a) the wave (normalized) phase velocity and (b) the ratio of spatial damping rate to wave number, Ki/Kr^ as functions of eZ or (eZ^), where e = rid/n^ is the ratio between dust density and positive ion density, while Z = Q/e (or Zd) is the ratio between a dust grain charge and the electron charge (9,10). The variation of the phase velocity and of the damping rate with 6, for the case of grid-launched waves in a negative ion plasma is shown in Figure 2(a,b) (11). 1.5

>^ H U

O J W

>

UJ C/)

< X cu

1.4

\J

T -

"•'

1 DIA

• 1

'

• /

• /

^ -

/

/•

-

/

1

* /

/ • /'

1.2

/ :

^ v^ •

y%

•'-''

— .

I

1 \

'

1

/

'

V'

I

-]

^ X ^

1

(a)

„^

1.04 r ^ » " ' — -

i

;

/ /

l.i

'

I

0.5

^

.

•

•

1

ZZr

FIGURE 1. Grid-launched dust ion-acoustic waves (a) phase velocity vs. eZd and (b) spatial damping rate/wavenumber, Ki/Kr, vs. eZd (ref. 10).

N. D'Angelo/Waves and instabilities in dusty plasmas

F

5|

»

1

•

t

•

1

*

— KINETIC THEORY - - FLUID THEORY

>

1

•

]

. ^ \^)

(

• 20 kHz A 50 kHz • 100 kHz

3H

0.10.

Xx:

100 kHz

(b)

0.08.

3

\\

ii ^V

0.061

N^

•* — 0.04-'

2

i

3

£1 ex.

^ •

•

1—

0.0

• — ' »


10^). On the other hand, for much smaller m_/m^^ negative ions have a destabilizing effect on the PVSI. For example, for fixed values of the plasma

A^. D'Angelo/Waves and instabilities in dusty plasmas

—r

T

•-T

T

1

"T "

307

r

r

•i-

•/ •/ • / •/ •/ ••7 /

K*

•

» .—

0.1

1

1

1

1

•

1

•

-Ji^-""'^'''^

•

1

•

1

1

r

••

-

3

•

2 h\

xK*

• 1

0

0.2

1

1

0.4

1

1

0.6

1

1

0.8

0.011

1

1.0

FIGURE 5. The EIC wave in negative ion plasmas. The wave frequency vs. e, for both the fast and slow mode (ref. 17).

L

0.2

_J

0.4

I

I

0.6

I

L-

O.B

1.0

FIGURE 6. The critical drift velocities for ^^^ ^'^^ ^^^ ^^^ EIC modes in negative ion P^^"^^ ^s- ^ ^'^^- l^)-

density gradient transverse to the magnetic field, of the parallel and perpendicular wavenumber and of the velocity shear parameter S = {l/ujc+)dv+o^/dx (where Uc+ is the angular ion gyrofrequency and dv+ojdx is the variation of the ion flow velocity, V+Q^, parallel to the magnetic field B = Bz in the transverse, x, direction) the instability growth rate was shown to increase with increasing e. This and other theoretical predictions were verified in a Q machine experiment with K+ positive ions and SFg ions, by An et al. (24), using essentially the same experimental setup employed many years before by D'Angelo and von Goeler (21) (see Fig. 8). An analogous experiment has been planned for a Q machine dusty plasma. The Rayleigh-Taylor instability may occur when a magnetic field acts as a light fluid supporting a heavy fluid (the plasma). If either negative ions or very massive negatively charged dust grains are added to the plasma, the range of unstable wavelengths is reduced, that addition thus having a stabiUzing effect (25,26). The potential relaxation instabihty (PRI) is excited when the cold endplate of a single-ended Q machine is biased positively (27). For suflaciently large positive biases, coherent oscillations are excited at a frequency on the order of a few kilohertz. Time-resolved measurements of the plasma potential oscillations show that they are associated with a moving double layer that appears when the endplate is biased sufficiently positive. When negative ions (SFg) are added to the plasma in sufficient concentration, the PRI is quenched, but a current-driven ion-acoustic (lA) instability is observed. As the negative ion concentration is increased, the frequency of the lA waves increases and the critical electron drift needed for their

308

N. D'Angela/Waves and instabilities in dusty plasmas 3.0

EDIC Vjl. T3