Plasma dragged microparticles as a method to measure plasma flows

PHYSICS OF PLASMAS 13, 103501 共2006兲

Plasma dragged microparticles as a method to measure plasma flows Cătălin M. Ticoşa兲 and Zhehui Wang Plasma Physics Group P-24, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Gian Luca Delzanno and Giovanni Lapenta Plasma Theory Group T-15, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

共Received 18 April 2006; accepted 25 August 2006; published online 5 October 2006兲 The physics of microparticle motion in flowing plasmas is studied in detail for plasmas with electron and ion densities ne,i ⬃ 1019 m−3, electron and ion temperatures of no more than 15 eV, and plasma flows on the order of the ion thermal speed, v f ⬃ vti. The equations of motion due to Coulomb interactions and direct impact with ions and electrons, of charge variation, as well as of heat exchange with the plasma, are solved numerically for isolated particles 共or dust grains兲 of micron sizes. It is predicted that microparticles can survive in plasma long enough, and can be dragged in the direction of the local ion flow. Based on the theoretical analysis, we describe a new plasma flow measurement technique called microparticle tracer velocimetry 共mPTV兲, which tracks microparticle motion in a plasma with a high-speed camera. The mPTV can reveal the directions of the plasma flow vectors at multiple locations simultaneously and at submillimeter scales, which is hard to achieve by most other techniques. Thus, mPTV can be used to study plasma flows produced in the laboratory. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2356316兴 I. INTRODUCTION 1

Plasma flow can occur in coaxial guns, plasma processing reactors,2 tokamaks3 or Hall thrusters.4 Understanding of the plasma flow phenomena depends critically on measurements. Reliable and unambiguous measurements of plasma flows with good spatial resolution remain a challenge in most laboratory plasmas. Directional Langmuir probes 共or “Mach probes”兲,5 as one of the most frequently used local flow measurement techniques, yield the average flow speed, but quite often their results have to be supplemented by spectroscopic measurements for increased accuracy.6 A basic Mach probe incorporates two collecting tips,7 one facing upstream and the other facing downstream in the flow. More complex designs include an array of tips which provide only point measurements with a limited spatial resolution.8 It is therefore difficult to measure flow patterns created by turbulent or sheared particle transport which require an analysis of the flow at a microscopic scale over a large volume. The primary motivation of this work is to examine in detail the physics of microparticle motion within flowing plasmas under some commonly occurring laboratory conditions, i.e. comparable electron and ion densities of the order of 1019 m−3, electron and ion temperatures which do not exceed 15 eV, and plasma flow velocities on the order of the ion thermal speed v f ⬃ vti. The drag exerted on the microparticles due to momentum transfer from ions through Coulomb interactions and direct impact collisions can accelerate them in the direction of the local ion flow. The motion of a microparticle is determined by the charging and heating mechanisms. The charges collected on the particle surface affect the ion-dust Coulomb collisions, the impact collisions, as well as the electron emission from the dust surface when thermionic and secondary emission a兲

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become appreciable. Heating can potentially modify the size of the dust if the temperature increase leads to vaporization. The drag exerted on micron-size dust grains by flowing ions has been observed in recent experiments performed in thermionic filament discharges, in which a collimated low density ion beam was focused on free falling dust grains.9,10 The vertical trajectories of the grains were slightly deviated in the direction of the ion beam. At ion beam densities of about 1014 m−3 the ion drag force scaled linearly with the ion energies which ranged in value from 3 to 45 eV. A much stronger force on the dust is expected for flowing plasmas with ion densities 4 to 5 orders of magnitude higher and similar temperatures, as it is the case, for example, in plasmas produced in coaxial guns.1 A new technique called microparticle tracer velocimetry 共mPTV兲 has been proposed for plasma flow measurement.11 In mPTV, the microparticles 共or dust grains兲 of microns in size are released in the path of the flowing plasma. The dust grains are then electrically charged and dragged in the direction of the plasma flow. Simultaneous tracking of many dust grains distributed over a relatively large volume of the plasma flow can provide information about the ion flow structure at a scale as small as a few hundred microns. An optical method based on laser scattering can be used to detect the dust trajectories. A high-speed videocamera equipped with a narrow-band filter, which allows the passing of only the dust-scattered laser light, records the grain trajectories over large areas of the flow. Direct visualization of the grain trajectories can provide the direction of the local ion flow speed vectors, while a more sophisticated analysis of the grains dynamics corroborated with a suitable ion drag model is expected to give the local ion flow speed. We extend our previous work11 by studying the dust grain motion in plasma flows with further considerations of the electron and ion physics. In particular, we have used the

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© 2006 American Institute of Physics

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orbital motion limited 共OML兲 theory12 taking into account the ions drift, and the collisionless sheath-limited 共SL兲 theory13 for electrons and ions, when the size of the grains is either small 共OML applies兲 or large 共SL applies兲 compared with the Debye length. The results for the grain potential are consistent with kinetic particle-in-cell 共PIC兲 simulations14–16 in the limit where there is no relative speed between the plasma and the dust grains. The rest of the article is organized as follows: In the first part of Sec. II the models for the dust charging and heating are presented. The equations are solved for the plasma parameters mentioned above. The calculated dust floating potential is then used to obtain the plasma flow induced drag force, taking into account the change in the dust radius due to vaporization at the dust surface. The grain trajectory and speed as functions of time are then obtained in the last part of Sec. II. In Sec. III, a setup for mPTV is discussed, followed by an estimate of the signal to noise ratio of an optical system used to track the microparticles, which are illuminated with a laser beam. Conclusions are drawn in Sec. IV.

formula 共2兲 is readily done by arranging conveniently the exponential term and integrating, first, over the angular coordinate:18 2 I共f兲 i = ␲rdeni

A. Charging of dust grains

In this study, a fully ionized plasma with equal electron and ion temperatures Te = Ti ⱕ 15 eV and densities ne = ni = 1019 m−3, with a duration of a few milliseconds is considered. The flow speed v f is smaller than the ion thermal speed vti = 冑2kBTi / mi 共where mi = 1.67⫻ 10−27 kg is the ion mass兲. The Debye shielding length in the plasma is on the order −2 −2 −1/2 + ␭Di 兲 ⬇ 5 − 6 ␮m 关␭De共i兲 of a few microns: ␭s = 共␭De 2 1/2 = 共␧0kBTe共i兲 / ne共i兲e 兲 , and e = 1.6⫻ 10−19C兴. The charging of grains with radii rd Ⰶ ␭s can thus be well approximated by the orbital motion limited 共OML兲 theory.12 The ion current to the dust grain surface is deduced assuming a drifting Maxwellian particle distribution and the existence of a collection radius bc = rd冑1 − 2eVd / miv2i , where vi is the speed of ions far from the dust grain. The collecting surface ␲b2c can be defined for values of the grain potential Vd, relative to the local plasma potential, which satisfy eVd ⱕ miv2i / 2. A finite streaming speed v f on the z axis is included in the distribution function of ions: f i共vi兲 = ni

冉

mi 2 ␲ k BT i

冊 冋 3/2

exp −

册

mi共vi − zˆv f 兲2 . 2kBTi

共1兲

冏冋

I共f兲 i =e

冕

⬁

␲b2c vi f i共vi兲d3vi ,

2 I共f兲 i = ␲rdeni

where d ␪兲, vi = 兩vi兩, and zˆ · vi = vi cos ␪ is the projection of the ions velocity on the z axis. The lower limit of integration is vmin, which is either 0 for Vd ⬍ 0, as ions are accelerated towards the grain, or vmin = 冑2eVd / mi, when Vd ⬎ 0. Here vmin is the minimum speed of ions required to overcome the dust potential barrier. The integral in 3

vi = 2␲v2i dvid共−cos

1/2

1 vf

冉

⬁

dviv2i 1 −

vmin

mi共v2i + v2f − 2viv f cos ␪兲 2kBTi

2eVd miv2i

册冏

冊

cos ␪=1 cos ␪=−1

冑 冋冑 冉

. 共3兲

冊

8kBTi ␲ 2eVd 1 + 2u2 − erf共u兲 ␲mi 4u k BT i

册

1 exp共− u2兲 . 2

共4兲

In the case of a positive grain 共Vd ⱖ 0兲, the ion current is a complicated function of u, Vd and the normalized minimum speed um = vmin / vti:19 2 I共f兲 i = ␲rdeni

冑

8kBTi ␲mi

再

冑␲ 8u

冉

1 + 2u2 −

⫻关erf共u − um兲 + erf共u + um兲兴 + ⫻exp关− 共u − um兲2兴 +

冎

2eVd k BT i

冊

冉 冊

1 um 1+ 4 u

冉 冊

1 um 1− 4 u

⫻exp关− 共u + um兲2兴 .

共5兲

It can be easily verified that for vmin = 0 共i.e., um = 0兲, formula 共5兲 for the ion current is identical to formula 共4兲. For negligible plasma flow, i.e., u Ⰶ 1, the error function can be approximated as erf共u兲 / u ⯝ 2 / 冑␲, and by expanding in Taylor series the terms erf共u ± um兲 and exp关−共u ± um兲2兴, the wellknown OML formula for the ion current in the case of a stationary plasma is recovered:

共2兲

vmin

−

冊 冕

The ion current for a negative grain 共Vd ⱕ 0兲 can be expressed in terms of the normalized ion flow speed u = v f / vti, the dust potential Vd, and the error function erf共u兲 = 2 / 冑␲兰u0exp共−x2兲dx:17

Ii = ␲r2deni

The ion current can be deduced following Ref. 17,

mi 2 ␲ k BT i

⫻ exp

+ II. INTERACTION OF DUST GRAINS WITH FLOWING PLASMA

冉

Ii = ␲r2deni

冑 冉

8kBTi eVd 1− ␲mi k BT i

冑

冊

冉 冊

8kBTi eVd exp − ␲mi k BT i

for Vd ⱕ 0,

for Vd ⱖ 0.

共6兲

共7兲

Here a plasma flow speed much smaller than the electron thermal speed is considered, and therefore the electron current is the same as in the case of stationary plasma, given by the OML model:17

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Plasma dragged microparticles as a method…

Ie = − ␲r2dene

Ie = − ␲r2dene

冑

冉 冊 冑 冉 冊 8kBTe eVd exp ␲me k BT e

for Vd ⱕ 0,

共8兲

8kBTe eVd 1+ ␲me k BT e

for Vd ⱖ 0.

共9兲

When a dust grain is hit by energetic plasma electrons, secondary electrons may be emitted from its surface. The secondary emission yield is20

␦se共E兲 = 7.4␦m

冉 冑 冊

E exp − 2 Em

E , Em

共10兲

where E is the energy of the incident electrons. The parameters ␦m and Em depend on the type of material; for carbon ␦m = 1 and Em = 250 eV. The secondary electron current Ise can be deduced as:17 Ise =

8␲2r2de m2e

冕

⬁

E␦se共E兲f se共E − eVd兲dE

if Vd ⱕ 0, 共11兲

0

and Ise =

8␲2r2de m2e ⫻

冕

⬁

冉

exp −

eVd kBTse

冊冉

1+

eVd kBTse

E␦se共E兲f se共E − eVd兲dE

冊 if Vd ⱖ 0.

共12兲

f se共E − eVd兲 = 共me / 2␲kBTe兲3/2 exp关−共E − eVd兲 / kBTe兴 is the distribution function of the incident electrons. The energy of each secondary electron leaving the dust surface is taken to be Tse = 3 eV. As the dust grain temperature rises due to the interaction with plasma, the thermionic emission becomes significant and must be taken into account. At elevated dust temperatures the thermionic current can induce a reversal of the sign in the charge accumulated on the dust surface.14–16 The thermionic current Ith is given by the Richardson-Duschman formula, taking into account the effect of the charged dust surface on the emitted electrons:15 16␲2r2deme共kBTd兲2 h3

冦冉

冉

exp −

⫻

1+

W k BT d

冊

冊 冉

W + eVd eVd exp − k BT d k BT d

冊

if Vd ⱕ 0 if Vd ⱖ 0,

冣

共13兲

where Td is the dust temperature and W is the work function of the dust material. For carbon W = 4.7 eV. For large dust radii 共rd ⱖ ␭s兲 the electron and ion fluxes to the grain surface are deduced in the sheath-limited 共SL兲 theory.13 The formation of a screening sheath around the dust grain is associated with a negatively charged dust surface, relative to the surrounding plasma. In this case the ion current is13

共14兲

where cs = 冑k共Te + Ti兲 / mi is the ion sound speed at the sheath edge, and ns ⬇ 0.5ni. The formula for the electron current is similar to the one given by the OML model for negatively charged grains. The occurrence of magnetic fields 共B ⱖ 10−2 T兲 in flowing plasmas is ubiquitous. The magnetic field does not affect dust motion;21 however, the magnetic field will affect the motion of electrons and ions. We may neglect the magnetic field for ion motion because the gyroradii of ions 共⬇105 ␮m for B ⬃ 10−2 T兲 are large compared to the dust sizes considered here 共rd = 1 to 30 ␮m兲. However, the gyroradii of electrons can be comparable or even smaller than the dust grain radii. For the sake of simplicity, however, the presence of a magnetic field and thus its influence on the dust charging mechanism is neglected in our present study. Electric fields which are associated with the flowing plasma within a magnetic field are negligible and are therefore neglected. The charging equation for a spherical dust grain is Cd

eVd

Ith =

Ii = 4␲r2denscs ,

dVd = Ie + Ii + Ise + Ith , dt

共15兲

where Cd is the grain capacitance. In the OML model Ii is equal to I共f兲 i given by Eqs. 共4兲 and 共5兲, while in the SL model Ii is given by Eq. 共14兲. The capacitance of a spherical dust grain, deduced in the approximation of a radial screened potential given by the Debye-Hückel formula, is Cd = 4␲⑀rd 共1 + rd / ␭s兲.22 A crucial parameter that characterizes the plasma-dust interaction is the dust charging time. The time needed by a grain with rd = 1 ␮m to accumulate negative charges on its surface until it reaches the equilibrium in a stationary plasma is estimated to be ⬀␧0kBTe / 共e2rdnivTi兲 ⬇ 10−9 s for the plasma parameters previously mentioned, and when no electron emission from the grain surface is present.17 When thermionic and secondary emissions are taken into account, the charging time becomes much longer with values of up to 10−4 s, as it will be demonstrated by our simulations. Nevertheless, dust present in plasmas which last several milliseconds are still expected to get charged almost “instantaneously,” and remain so during the lifetime of the plasma.

B. Heating of dust grains

An important aspect relevant to dust grain survival and dust size change, which needs to be accounted for, is the heating of dust due to intense plasma particle fluxes. It is known that in the core of fusion plasmas dust grains do have a short lifetime compared with the duration of the plasmas.23 The dust destruction process can be exploited and used for example as a plasma diagnostic tool for magnetic field mapping, if dust grains are injected fast enough and in small amounts compared to the plasma particle inventory.21,24,25 The flowing plasmas of interest in this study have parameters which are rather similar to those measured within the sheath surrounding a tokamak plasma.26 However, even for these

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conditions the grains can be heated up to high temperatures close to the melting point, and therefore partial destruction of the grains may not be neglected.27,28 The change in temperature depends on the heat flux transferred to the grain surface by direct impact with the plasma particles. If a spherical grain is assumed, whose volume is heated uniformly, then the equations of heat balance and dust radius variation are

冑 2 ⌫共f兲 i = ␲rdni共kBTi兲

再冉

2eVd 5 vti u + u2 − 2 u k BT i

⫻exp共− u2兲 + 冑␲

冉

冋

eVd k BT i

⫻ 1 + 2u2 −

冊

3 eVd + 3u2 + u4 − 4 k BT i

冊册

冎

erf共u兲 ,

共20兲

while for Vd ⱖ 0 it is dTd ⌳s ˙ 兩, = ⌫e + ⌫i − ⌫se − ⌫th − ⌫rad − 兩m m dc d dt ␮ d

共16兲

˙ d兩 兩m drd =− , dt 4␲r2d␳d

冑 2 ⌫共f兲 i = ␲rdni共kBTi兲

再

⫻ A+ + A− +

共17兲

冉

⫻ 1 + 2u2 − where is the mass of the grain and ␳d is its density, cd is the specific heat of the dust material, Td is the dust temperature, and ⌫i,e,se,th are the heat fluxes of incident ions and electrons, emitted secondary electrons, and emitted thermionic electrons, respectively. In the OML model the ion heat flux ⌫i becomes ⌫共f兲 i , which includes the effects of an 4 兲 is the radiated heat ion drift speed v f . ⌫rad = 4␲r2d␴共T4d − Twall −8 2 4 flux 共␴ = 5.67⫻ 10 W / m K , and Twall = 290 K兲. The last term of Eq. 共16兲 represents the heat lost due to the sublimation of dust material, which has a sublimation heat ⌳s, and a ˙ d is the rate at which the dust mass molecular weight ␮. m sublimates, and is inferred by integrating the ClausiusClapeyron equation.29 Assuming that the dust mass is lost uniformly in the radial direction, then

md = 4␲r3d / 3␳d

˙ d兩 = 4␲r2dm0 兩m

冑

冋 冉

1 2kBTd n0T0 ⌳s 1 exp − ␲m0 Td R T0 Td

冊册

,

共18兲

where m0 is the mass of a dust molecule, n0 is the vapors density of the dust material at a specific temperature T0, and R = 8.31 J / mol K is the gas constant. For carbon m0 = 12 amu, ⌳s = 7.15⫻ 105 J / mol, ␮ = 12⫻ 10−3 kg/ mol, n0 = 1.83⫻ 1024 m−3, and T0 = 4000 K. It is assumed that the kinetic energy carried by the plasma ions and electrons hitting the dust grain is deposited as heat onto the dust surface. The secondary and thermionic electrons remove energy as they are emitted from the dust surface. Taking into account the ion drift speed v f , the heat flux collected from ions in the OML model is

冕

⬁

冉

冊

miv2i ␲b2c vi − eVd f i共vi兲d3vi , ⌫共f兲 i = 2 vmin

共19兲

where f i共vi兲 is the ion distribution function given in Eq. 共1兲. The integration is performed from vmin = 0 or vmin = 冑2eVd / mi, for a negatively or positively charged grain, respectively. The results are long expressions which can be arranged elegantly in terms of the error function and the parameters u, um, and Vd. Thus, the ion heat flux for Vd ⱕ 0 is

vti u

冑␲ 2

冋

eVd k BT i

3 eVd + 3u2 + u4 − 4 k BT i

冊册

冎

关erf共u + um兲 + erf共u − um兲兴 , 共21兲

where A± =

冋 −

3 u3 3um um u 2u m u 2 兲+ ⫿ + 共5 + 2um ⫿ 2 2 4 4 2

册

eVd 共u ⫿ um兲 exp关− 共u ± um兲2兴. k BT i

共22兲

It can be verified that for vmin = 0 共which implies um = 0兲, Eq. 共21兲 is the same with Eq. 共20兲. In the case of a stationary plasma 共u Ⰶ 1兲, the error function and the exponential terms can be expanded in Taylor series and the heat flux of ions can be put in the following simplified forms:

⌫ i = k BT i

冢

⌫i = 2kBTi

冣

eVd kBTi eVd Ii − eVd kBTi e 1− k BT i 2−

Ii e

for Vd ⱖ 0.

for Vd ⱕ 0,

共23兲

共24兲

⌫i given by Eqs. 共23兲 and 共24兲 is expressed in terms of the OML ion current in a stationary plasma, given by Eqs. 共6兲 and 共7兲. The heat fluxes of the incident or emitted electrons can also be expressed in terms of the corresponding OML currents. For Vd ⱕ 0 it is found that30,31 ⌫e = 2kBTe

兩Ie兩 , e

⌫th = 2kBTd

Ith , e

⌫se = 2kBTse

共25兲

Ise , e

while for Vd ⱖ 0,

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FIG. 1. 共a兲 Curves showing the ratio of ion currents to a dust grain in a flowing 共I共f兲 i 兲 and stationary 共Ii兲 plasma for eVd / kBTi = 0.1 and eVd / kBTi = −1. 共b兲 Curves showing the ratio of ion heat fluxes to a dust grain in a flowing 共⌫i共f兲兲 and stationary 共⌫i兲 plasma for the same dust potentials. u = v f / vti is the normalized ion drift speed.

⌫ e = k BT e

冢 冢 冢

冣 冣 冣

eVd 2+ kBTe eVd 兩Ie兩 , + eVd kBTe e 1+ k BT e

⌫th = kBTd

eVd kBTd eVd Ith + , eVd kBTd e 1+ k BT d

⌫se = kBTse

eVd 2+ kBTse eVd Ise . + kBTse e eVd 1+ kBTse

2+

共26兲

C. Results on dust grain charging and heating

In the SL model only a dust grain charged negatively is considered. In this case the heat fluxes of the incoming electrons and ions for a perfectly absorbing and collisionless sheath are given by13,26 ⌫e = 2kBTe

In Fig. 1共a兲, the ratio I共f兲 i / Ii is plotted as a function of the normalized flow speed u, for a negative grain with eVd = −kBTi, and for a slightly positive grain with eVd / kBTi = 0.1. Figure 1共b兲 shows the ratio of heat fluxes ⌫共f兲 i / ⌫i for the same dust potentials. For a negative grain, I共f兲 i is only a few percent above Ii, while ⌫共f兲 increases with 50% from ⌫i, i when u → 1. A large increase is seen in ⌫共f兲 for a positive i grain. Significant differences are found especially in the ion heat flux when u → 1.

兩Ie兩 , e 共27兲

Ii ⌫i = 共2.5kBTi − eVd兲 . e Notice that the acceleration of ions in the vicinity of the grain is taken into account as Vd ⬍ 0, and also the acceleration within the presheath, designated by the term 0.5kBTi 共Te = Ti兲, is included. The heat fluxes ⌫th, ⌫se carried by the emitted thermionic and secondary electrons, respectively, are the same as in the case of the OML model, when the dust charge is negative. The inclusion of the ion drift speed in the calculations of the ion current and heat flux for the OML model, results in changes that can be inferred by relating these quantities to their stationary values, for the plasma parameters of interest.

The equilibrium charge and temperature of a grain are deduced by solving Eqs. 共15兲–共17兲 with the following parameters: ␳d = 2230 kg/ m3 共for carbon grains兲, cd = 750 J / kg K and mi / me = 1836. The plasma flow speed v f = 30 km/ s considered here has a value close to Mach probe measurements performed in a coaxial plasma gun.32 Two sets of results are produced for plasma temperatures Te共i兲 = 10 and 15 eV, and densities ne共i兲 = 1019 m−3. The initial potential and temperature of the dust grain in the simulations are taken to be 0 V and 290 K, respectively. The results are sensitive to the dust size, plasma parameters, and the charging model used. For the two cases when rd ⱕ ␭s and rd ⱖ ␭s the electron and ion currents to the grain surface are given by the OML and SL models 共␭s = 5.2 and 6.4 ␮m for the two plasma temperatures mentioned above兲. For Te共i兲 = 10 eV, all grains stay negatively charged, as shown in Fig. 2共a兲 by curves 1–4, which correspond to dust radii rd = 1, 3, 15, and 30 ␮m, respectively. OML predicts the equilibrium dust potential Vd = −16.5 V, while SL gives Vd = −21.2 V, independent of grain radius. The grains with rd = 1 and 3 ␮m reach an equilibrium temperature Td = 2880 K, as can be seen in Fig. 3共a兲. Heating of the two larger grains is slower due to the larger grain mass. The dashed-dotted and dotted curves of Fig. 3共a兲 show the evolution in time of the temperature, for these grains. Although neither curve reaches an equilibrium value, the dotted curve

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FIG. 2. Dust potential as a function of time: 共a兲 is for Te共i兲 = 10 eV and 共b兲 for Te共i兲 = 15 eV. The plasma density is ne共i兲 = 1019 m−3. Curves 1 and 2, represented by continuous and dashed lines, are given by the OML model for a grain with rd = 1 and 3 ␮m, respectively, while curves 3 and 4, represented by dashed-dotted and dotted lines, are given by the SL model for a grain with rd = 15 and 30 ␮m, respectively. The dots represent the dust potential obtained by the PIC code.

has a less abrupt increase, indicating the need of a longer time to heat up the grain. The temperature of the 15 ␮m grain is Td = 1700 K, while the 30 ␮m grain reaches Td = 1050 K, after the time period of 5 ms considered in the simulation. These temperatures are too low to cause an appreciable sublimation of the dust material and the grain radii remain almost unchanged, as shown in Fig. 3共b兲. Sublimation of carbon takes place at temperatures of the order of or higher then 3500 K. A decrease of only ⬇3% in radius is noticed, however, for the grain with rd = 1 ␮m. This situation changes considerably when the plasma temperature is Te共i兲 = 15 eV, at the same plasma density. For the two small grains, the thermionic emission becomes dominant as they both reach the equilibrium temperature Td = 3255 K, represented by the continuous line in Fig. 3共c兲, and induces a positive charge on the grains. Their potential stabilize at Vd = 0.16 V, as indicated by curves 1 and 2 in Fig. 2共b兲. The ablation at the surface of these grains is intense. After 5 ms the radius of the 1 ␮m grain decreases to only 17% of its initial value. The grain with rd = 3 ␮m grain is less affected by the strong heating as 76% of its initial radius survives. The variation in time of the dust radii is plotted in Fig. 3共d兲. The grains with radii rd = 15 and rd = 30 ␮m accumulate on their surface more negative charges, as it is expected, resulting in higher absolute values for their potential Vd. Their potential is stable and given by Vd = −29.6 V, as presented in Fig. 2共b兲. Due to continuous heating, the temperature of the grain with rd = 15 ␮m increases up to Td = 2815 K when thermionic emission of electrons becomes noticeable and its potential start to increase. Heating of the two large grains is relatively intense, as reflected by the steeper slopes of the dashed-dotted and dotted curves shown

Phys. Plasmas 13, 103501 共2006兲

FIG. 3. Dust grain temperature and radius as functions of time. 共a兲 and 共b兲 Te共i兲 = 10 eV; 共c兲 and 共d兲 Te共i兲 = 15 eV. The plasma density is ne共i兲 = 1019 m−3. The continuous, dashed, dashed-dotted, and dotted lines correspond to the curves 1, 2, 3, and 4, respectively, of Fig. 2.

in Fig. 3共c兲, compared to those of Fig. 3共a兲. However, ablation at their surface remains low and their radii do not change after 5 ms, as shown in Fig. 3共d兲. It is important to mention here that all the OML results 共f兲 were obtained by using I共f兲 i and ⌫i in Eqs. 共15兲 and 共16兲, which take into account the ion drift speed v f = 30 km/ s. The same simulations were performed by replacing them with their stationary expressions Ii and ⌫i, and similar results were obtained. For example, the OML model without ion drift predicts for a grain with rd = 1 ␮m equilibrium potentials of −17.4 V and 0.15 V, and temperatures of 2850 K and 3250 K, at plasma temperatures of 10 eV and 15 eV, respectively. Based on this observation we may conclude that for u ⱕ 0.6, the stationary ion current and heat flux are good approximations to be used in describing the charging and heating of dust grains in the type of plasma flows considered in this paper. A more sophisticated method for calculating selfconsistently the grain charge is based on the kinetic particlein-cell 共PIC兲 simulation approach, where computational particles are used to represent a large number of physical particles.33 In the PIC simulation, the plasma particles that reach the dust grain are removed from the calculations, whereas their charge is transferred to the grain. The charges of these trapped particles accumulate on the grain surface and modify the resulting shielding grain potential. A suitable source of plasma particles must be injected into the system in order to compensate for the particles collected by the grain, and allow the system to relax to an equilibrium state. The PIC code used here is described in Refs. 14–16. In the code, a grain is allowed to emit electrons due to thermionic emission, photoemission and secondary emission. For the present study, a hydrogen plasma is considered where the electrons and ions have their real physical mass, and photoemission is neglected. All the simulations are made with an initial number of computational particles, Ne = Ni = 2 ⫻ 106, located on a uniform grid with Ng = 1000 grid points. The stability

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Plasma dragged microparticles as a method…

condition33 is well satisfied by choosing the time step of the simulation ⌬t = 3 ⫻ 10−14 s. The physical parameters of the system are the same as in our analytical model. The Maxwellian ion distribution function used in the PIC code to inject the particles neglects the drift of ions. While the inclusion of the ion drift speed v f would give a more accurate description of the physical system, the differences in the dust potential are expected to be very small for u ⯝ 0.6, as it has been demonstrated by the simulation, for the case of the OML model. The plasma system is allowed to relax self-consistently until the charge on the grain and the shielding potential around it reach a steady state. Once the equilibrium is attained, the floating potential fluctuates due to the continuous collection of plasma particles. We calculate, therefore, the mean value in time defined as the average 冑 2 over the last 80 ␻−1 pe seconds 共␻ pe = nee / ␧0me兲 of the simulation. This time period is sufficiently large compared with the grain charging time, and provides a filter for the high frequency oscillations. For the case of a 1 ␮m carbon grain, we have performed simulations with the dust temperature Td = 3255 K and W = 4.7 eV. In this case and for Te共i兲 = 15 eV, the thermionic emission becomes sufficiently important and the time averaged floating potential is Vd = −0.08 V. This value agrees reasonably well with the one obtained by the analytical model, Vd = 0.16 V. For a grain with rd = 15 ␮m, we have used Td = 2815 K. In this case, the electron emission from the grain is less important with respect to charge collection from the background plasma and the time averaged floating potential is Vd = −28.9 V. One can notice that this value is close to the predictions based on the SL model 共Vd = −29.6 V兲. For Te = 10 eV, the PIC results are also in good agreement with the OML and SL models: Vd = −17.2 V in the case of rd = 1 ␮m 共Td = 2880 K兲, while Vd = −21.7 V is obtained for rd = 15 ␮m 共Td = 1700 K兲, respectively. Based on the values for the grain potential Vd provided by the OML and SL models, the rate of momentum transfer from the plasma to the dust grain is deduced. The equation of motion for the grain is then integrated and the average “dragging” distance as well as the speed in the direction of the plasma flow are calculated. It is found that dust motion is dominated by the interaction with the ion flow, whereas electron momentum transfer can be neglected.

D. Equation of motion for dust grains

To evaluate the drag exerted on the dust grains, we consider the inelastic direct impact force Fcol and the long-range Coulomb force FCoul of singly ionized ions 共H+兲 and electrons:

Fdrag =

兺 共Fcol + FCoul.兲.

共i,e兲

共28兲

For the two plasma components the forces are explicitly34,35

FIG. 4. The drag and electric forces normalized to the dust weight vs dust radius with Vd given by OML and SL. The continuous and dashed lines correspond to Te共i兲 = 10 eV and Te共i兲 = 15 eV, respectively, and plasma density ne共i兲 = 1019 m−3. The dotted line shows the drag force after 5 ms, when the grain has lost mass due to vaporization. The electric force is calculated considering an electric field E = 500 V / m.

Fcol共i,e兲 = 2␲r2dkBTi,eni,eG0共si,e兲, 2 FCoul共i,e兲 = 2␲r2dkBTi,eni,e␾i,e ln共⌳i,e兲G2共si,e兲

with

␾i,e =

⌳i,e =

eVd , kBTi,e

冑

si,e =

␧0kBTi,e ni,e

冑

共29兲

mi,e共v f − vd兲2 , 2kBTi,e 3

冉 冊

rd e兩␾i,e兩rd 1 + ␭s

.

Here vd is the dust grain speed. The functions G0 and G2 are G0共s兲 =

冊 冑 冉 冉 冊 8s

3 ␲

G2共s兲 = s

1+

9␲ 2 s 64

3 1/2 3 ␲ +s 4

1/2

,

共30兲

−1

.

共31兲

In the case of a large dust grain 共rd  ␭s兲, surrounded by a sheath, only the direct-impact force is considered. Also it is considered that all the plasma particles entering the sheath are further collected by the dust grain. An evaluation of the drag forces gives Fdrag共e兲 / Fdrag共i兲 ⬇ 3 ⫻ 10−2, therefore the electron drag force can be neglected. The ion drag force normalized to the force of gravity for grains made of carbon is a function of the dust radius and it is plotted in Fig. 4 for two plasma temperatures. A small discontinuity at rd ⬇ ␭s is present in the curves, which is the transition from small to large grains. Large grains are dragged only by direct collisions with the ions. For small grains the Coulomb force is about the same order of magnitude as the direct impact force. The key observation is that

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Ticos et al.

the ion drag force acting on the grains is much stronger than the force of gravity. The pressure of the surrounding gas is considered small 共less than a few mTorr兲 and therefore the drag due to the neutral atoms is insignificant and disregarded in the overall balance of forces. It can be noticed in Fig. 4 that the drag on small grains 共in the OML limit兲 at Te共i兲 = 15 eV marked with a dashed line, is lower than at Te共i兲 = 10 eV, shown by the continuous line. This is in agreement with the charging state of these particles. As they lose negative charge due to electron emission, their potential decreases in absolute value and stabilizes near zero. The Coulomb drag then becomes negligible and only direct collisions with ions supply momentum. Also, as the particles lose mass due to ablation, the ratio between the drag force and their weight increases in time. This situation is represented in Fig. 4 by the dotted curve, which gives Fdrag / mdg after 5 ms of the dust-plasma interaction. For comparison, the electric force acting on charged grains, associated with the electric fields due to the plasma flow, is also shown in Fig. 4. For example, probe measurements indicate the presence of internal plasma electric fields of a few hundred volts per meter in FMP-like coaxial-gunproduced plasmas.32 The electric force as a function of the dust radius and normalized to the dust weight, Fel / 共mdg兲, is plotted only for Te共i兲 = 10 eV. The charge on the grains is calculated based on the values of the dust potential given by the OML and SL models. For Te共i兲 = 15 eV, the electric force is almost zero for small grains, given the small potential obtained by OML, while for large grains it is about the same order of magnitude as in the case of 10 eV. Figure 4 shows that the ion drag force is also orders of magnitude stronger than the electric force. The electric field is taken as E = 500 V / m, although even for higher values of up to 105 V / m the electric force will not overcome the ion drag. It is therefore expected that the trajectory of dust grains will be mostly along the direction of the plasma flow, assuming they are initially at rest. E. Results of dust grain trajectory and speed

The distance traveled by a dust grain and the instantaneous speed are deduced by solving numerically the equations of motion: dxd = vd , dt md

vd = Fcol共i,e兲共rd, vd兲 + FCoul共i,e兲共Vd,rd, vd兲, dt

共32兲

共33兲

where rd = rd共t兲, vd = vd共t兲, and Vd = Vd共t兲. In the case of large grains 共rd ⬎ ␭s兲 only the first term of Eq. 共33兲 is used in the simulation, which corresponds to drag exerted by direct collisions with ions. The distance and speed obtained for grains with rd = 1, 3, 15, and 30 ␮m are presented in Fig. 5, for plasma temperatures of 10 eV and 15 eV, and density 1019 m−3. As expected, the plasma flow has a stronger impact on the smaller grains by applying to them a larger acceleration. It can be noticed in the figure how the speed increases steadily due to the plasma acceleration. This feature can be

FIG. 5. Distance xd traveled by carbon dust grains in the direction of the plasma flow and their speed vd in 共a兲 and 共b兲 for Te共i兲 = 10 eV, and in 共c兲 and 共d兲 for Te共i兲 = 15 eV. The plasma density is ne共i兲 = 1019 m−3. The continuous, dashed, dashed-dotted, and dotted lines correspond to the curves 1, 2, 3, and 4, respectively, of Fig. 2.

favorable in a real experimental situation since as little as 3 high-speed snapshots of the dust trajectories will be sufficient to estimate the average dust acceleration. The grains are dragged in the direction of the plasma flow over a distance which depends on their size. For large grains, the dragging distance is of the order of a few millimeters, while for small grains it is about tens of centimeters. These values can be easily measured by a high-resolution CCD camera.

III. DESCRIPTION OF MICROPARTICLE TRACER VELOCIMETRY

Based on the above discussion, the mPTV technique is in general feasible when the plasma-flow induced drag force dominates over other forces acting on the dust grain, such as the force of gravity, the electrical force, and the force of friction with the neutral atoms. The drag force is proportional to the plasma density and it is a function of the relative speed between the plasma flow and the dust grain.

A. Proposed mPTV setup

An example of a setup for sampling plasma flows with dust grains is shown in Fig. 6. In this proposed scheme the plasma flow produced in a coaxial gun is ejected into a cylindrical vacuum tank provided with lateral ports on which diagnostic devices are mounted.1 Dust can be easily supplied from a reservoir at the top of the tank by shaking the reservoir with a piezotransducer controlled remotely.25 The grains are illuminated with a vertical laser sheet as they free fall into the plasma. A high speed camera mounted at a right angle with respect to the incident laser beam is used to record the light scattered by the dust grains.

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TABLE I. SNR for different single spherical dust grains, for Te共i兲 = 10 eV and ne共i兲 = 1019 m−3. The values in parentheses are given for Te共i兲 = 15 eV and the same plasma density. A reduction in size is considered at the higher plasma temperature, for the small grains with rd = 1 and 3 ␮m. Their radii are multiplied with time averaging factors of 0.6 and 0.9, respectively. The laser has a power of 0.1 W at 488 nm, and 0.5 W at 650 nm. SNR 共P0 = 0.1 W兲

SNR 共P0 = 0.5 W兲

1 3

0.7 共0.2兲 5.7 共4.7兲

4.4 共1.6兲 38.7 共31.8兲

15

134.2 共136.2兲

784.6 共791.1兲

30

487.6 共493.6兲

2243.3 共2253.1兲

Dust radius rd 共␮m兲

FIG. 6. Proposed mPTV setup for measuring plasma flows: the system includes a laser, a dust dispenser, and a high speed camera for dust detection.

B. Signal to noise ratio for dust detection

Micron-size dust grains have a relatively large lightscattering cross section and therefore a low power incident laser beam is usually sufficient to illuminate slow moving or stationary dust grains. However, fast moving grains will stay in the viewing region of a detector for only a short period of time and the intensity of the scattered light will be greatly diminished. As the signal of a detector becomes noisier with short exposures, higher incident laser intensities will be needed in order to obtain good quality images. Another factor which can affect the quality of the detection and needs to be considered is the plasma background radiation. The performance of the imaging system can be evaluated by calculating the signal to noise ratio 共SNR兲 of the charge-coupled device 共CCD兲,36 which is defined as: SNR =

I sQ Et d

, 2 2 冑共Is + Ib兲QEtd + npx共␥dn + ␥rn 兲

共34兲

where Is is the flux of dust-scattered photons, Ib is the flux of bremsstrahlung plasma photons, QE is the quantum efficiency of the detector, ␥dn is the dark current noise, ␥rn is the readout electronics noise, npx is the number of pixels, and td is the exposure time. The flux of photons scattered in a solid angle ⌬⍀ by a single dust grain, illuminated with a continuous-wave 共CW兲 laser with power P0 and beam section SL, and collected by a detector is I s = P 0␩

␴d ␭ ⌬⍀ , SL hc 4␲

共35兲

where ␩ is the transmission efficiency of the optics, ␴d is the scattering cross section of the illuminated dust grain, and ␭ is the wavelength of the laser. The bremsstrahlung photon flux

per unit frequency, collected from a plasma volume with section Sview and path length D along the observation direction is Ib ⬀ Sview2D⌬⍀n2e ␭ / 冑Tehc,6 where ne and Te are the plasma electron density and temperature, respectively. Table I gives the SNR for different dust sizes, calculated for ne = 1 ⫻ 1019 m−3 and Te = 10 eV. The results for Te = 15 eV are given in parentheses. For the later temperature value, the radii of the two small grains are multiplied with a time averaging factor, which accounts for radius reduction due to dust ablation, 0.6 for the grain with rd = 1 ␮m and 0.9 for the grain with rd = 3 ␮m. The two sets of results presented in Table I correspond to laser powers P0 = 0.1 W at a wavelength of 488 nm and P0 = 0.5 W at 650 nm, respectively. The laser fluences for these beam powers and an illumination area SL = 3 ⫻ 10−5 m2, considering a thin sheet of light with 1.5 mm thickness, are less than 2 ⫻ 104 W / m2 and therefore laser heating of particles is negligible during a period of a few milliseconds.37 The scattering cross section of the dust grains is deduced from the Mie theory:38 ␴d = 4.27 ⫻ 10−12, 3.76⫻ 10−11, 9.05⫻ 10−10, and 3.56⫻ 10−9 m2 for rd = 1, 3, 15, and 30 ␮m, respectively. Also D = 0.5 m, ⌬⍀ = 4 ⫻ 10−3 sr, ␩ = 0.64, QE = 40%, and Sview = 10−3 m2. Furthermore, to minimize the plasma radiation, a narrowband interference filter with ⌬␭ = 2 nm, centered at the wavelength of the laser, is considered. The camera integration time in the calculations is taken to be 0.5 ms. A noise level ␥dn = 1 electrons rms per pixel and ␥rn = 7 electrons rms per pixel is considered for a CCD with npx = 8 ⫻ 104 pixels. The quantitative estimations presented in Table I suggest that good imaging of fast moving dust grains is in principle possible for grains with radii larger than 3 ␮m, if the laser power used to illuminate the grains exceeds 100 mW. The laser power required by the mPTV technique is significantly higher than in typical dusty plasma experiments, where a laser beam with only a few tens of mW is sufficient to illuminate the dust. This is a direct consequence of the short exposure times and also of the more intense plasma background radiation. IV. CONCLUSIONS

The principles of a diagnostic method for determining plasma flow vectors which uses micron-size dust grains as tracers, called mPTV for microparticle tracer velocimetry, are discussed. Dust grains released in plasma with ion and

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103501-10

electron temperatures less than 15 eV and density of about 1019 m3 accumulate charges on their surface. The charge on each dust grain which is smaller or larger than the Debye screening length, is calculated using the OML and SL theories, respectively. The values agree well with the results provided by PIC simulations. The temperature of dust grains is also evaluated by solving the heat exchange equation with the plasma particles. Vaporization at the grain surface due to plasma heating is significant only for small grains 共rd ⬇ 1 to 3 ␮m兲 and Te共i兲 = 15 eV, for a plasma that lasts a few milliseconds. In this case partial destruction of the grain can occur. Ions flowing past the charged dust grains can transfer momentum to the grains via direct collisions and Coulomb interaction. The displacement and speed of dust grains in the direction of the plasma flow are calculated for the total drag force acting on the grains within the time interval of 5 ms. It is shown that dust grains with radii rd = 1 to 30 ␮m can be dragged in the direction of the ion flow over distances ranging from a few millimeters for the large grains, to tens of centimeters for the small grains. The dust grains can reach speeds between 2 and 230 m / s for plasma flows v f ⬃ 0.6vti. The signal to noise ratio of an imaging system suitable for detecting the moving dust grains is evaluated for two different incident laser powers. Micron-size particles with radii larger than a few microns can be therefore used as tracers for measuring plasma flows. 1

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