Sheath characteristics in the radioSheath characteristics in the radio--frequency discharge inferred from frequency discharge inferred from

microparticlemicroparticle levitationlevitation

T. Antonova, V.V. Yaroshenko, T. Antonova, V.V. Yaroshenko, H.M. Thomas and G.E. MorfillH.M. Thomas and G.E. Morfill

MaxMax--PlanckPlanck--Institute for Extraterrestrial Physics, Institute for Extraterrestrial Physics, GarchingGarching, Germany, Germany

AbstractAbstractThe diagnostics of plasma sheath is of importance for fundamThe diagnostics of plasma sheath is of importance for fundamental ental

studies of plasma in laboratory as well as in industrial applicastudies of plasma in laboratory as well as in industrial applications. The tions. The

screening length, the electric field and particle charge as wellscreening length, the electric field and particle charge as well as the as the

vertical gradients of the field and the dust charge, are determivertical gradients of the field and the dust charge, are determined ned

experimentally in a sheath of the experimentally in a sheath of the rfrf discharge for several different gas discharge for several different gas

pressures. The plasma characteristics are derived directly from pressures. The plasma characteristics are derived directly from

measurementsmeasurements of levitation height of the particles ofof levitation height of the particles of different sizes and different sizes and

its variations due toits variations due to appearance of new dust layers and related an appearance of new dust layers and related an

additional electric field.additional electric field.

Since natural dusts are widely presented in universe (interplaneSince natural dusts are widely presented in universe (interplanetary and tary and

interstellar clouds, comet tails planetary ring etc.) our work cinterstellar clouds, comet tails planetary ring etc.) our work could be of ould be of

relevance for astrophysical research such as nearrelevance for astrophysical research such as near--surface dust clouds on surface dust clouds on

Mars, Moon etc.Mars, Moon etc.

Experimental setExperimental set--upupExperimental measurements are

carried out in the cylindrical chamber with two parallel plate electrodes of 3.8 cm diameter, distance between electrodes is 3 cm.

Upper electrode is radio-frequency driven with RF voltage of 300V pp. The lower electrode is kept floating.

Measurements are made at 30, 55 and 100 Pa Argon pressure.

Melamine formaldehyde particles with different radii (1.6 mm, 2.2 mm and 2.4 mm) are used in the experiments. They are injected in the discharge through a fine mesh of the dispenser.

The particles in the chamber are illuminated by the laser beam and their positions are recorded by a video camera at frame rate of 17 frame/second.

1) Levitation of dust 1) Levitation of dust monolayersmonolayers

The levitation height is mainly The levitation height is mainly influenced by gravitational and local influenced by gravitational and local electrostatic force electrostatic force

0 0 0 0 1 1 1 1( ) ( ), ( ) ( ),m g q z E z m g q z E z= =

Levitation of dust Levitation of dust particles of aparticles of a11=1.6=1.6mmm m (left) and a(left) and a00=2.4=2.4mmm m (right) above lower (right) above lower electrode at 100 Pa. electrode at 100 Pa. The picture size is The picture size is 10x9.8 mm10x9.8 mm2 2

subscript 0 subscript 0 refers to the dust particle sizerefers to the dust particle size aa00=2.4=2.4mmm and m and subscript 1 subscript 1 –– totoaa11=1.6=1.6mmm m

zz00zz11

1) Theoretical model1) Theoretical modelLevitation of dust Levitation of dust monolayersmonolayers: vertical gradients of electric field : vertical gradients of electric field and dust chargeand dust charge

0 0( ) (1 ( ))E z E z zγ+ −

0 0( ) (1 ( ))z z zϕ ϕ β= + −( )q z aϕ=

Since electric field nearly linear increases within the sheath, Since electric field nearly linear increases within the sheath, we present electric field as follows we present electric field as follows

Dust charge is expressed through the surface Dust charge is expressed through the surface potential which is also approximated by linear functionpotential which is also approximated by linear function

γ and and ββ are the are the vertical gradients of vertical gradients of electric field and dust electric field and dust charge, respectively charge, respectively

Measuring two levitation heights, zMeasuring two levitation heights, z00 and zand z11, and writing , and writing levitation conditions for both particle sizes gives levitation conditions for both particle sizes gives following relation between vertical gradients following relation between vertical gradients γ and and ββ

21

0 1 20 1 0

1 1 az zz z a

β γ βγ⎛ ⎞

+ + − = − −⎜ ⎟− ⎝ ⎠

Comparing the levitation height zComparing the levitation height z00 with zwith z2 2 forfor dust particles of adust particles of a22=2.2=2.2mmm we obtain m we obtain relation similar to (1) and solving them simultaneously we couldrelation similar to (1) and solving them simultaneously we could estimate vertical estimate vertical gradients of electric field and dust charge within a plasma sheagradients of electric field and dust charge within a plasma sheath at our experimental th at our experimental conditions for different pressuresconditions for different pressures

(1)(1)

30 Pa 30 Pa γγ==--3.53.5, ββ==--3.43.455 Pa 55 Pa γγ==--55, ββ==--4.74.7100 Pa 100 Pa γγ==--88, ββ==--7.87.8

The assumption about the constant dust The assumption about the constant dust chargecharge ((ββ =0) =0) within a sheath is within a sheath is questionablequestionable

2) Theoretical model2) Theoretical model2 layer system: separation of one dust layer in two layers2 layer system: separation of one dust layer in two layers

Injecting more Injecting more microparticlesmicroparticles (of (of the same kind, athe same kind, a00=2.4=2.4mmm ) in the m ) in the discharge leads to the formation discharge leads to the formation of two or even more separate of two or even more separate layers, in which the particles are layers, in which the particles are structured one over anotherstructured one over another

The existence of the shifts The existence of the shifts ddzz--11,, ddzz+1+1 means that each layer produces anmeans that each layer produces anadditional electric field due to the spatially localized highly additional electric field due to the spatially localized highly charged charged microparticlesmicroparticles(in other words, each dust layer possesses its own (in other words, each dust layer possesses its own microsheathmicrosheath as it has been as it has been assumed in assumed in E. Thomas, Phys. Plasmas 8, 329 (2001)E. Thomas, Phys. Plasmas 8, 329 (2001)) )

l

2) Theoretical model2) Theoretical model2 layer system: 2 layer system: DebyeDebye length gradient within a plasma sheath length gradient within a plasma sheath We model the additional electric field We model the additional electric field presenting the particle layer as a charged presenting the particle layer as a charged plane with an effective surface charge plane with an effective surface charge density density σσ==q/q/∆∆22, where , where ∆∆ denotes the denotes the average average interparticleinterparticle distance in the distance in the horizontal direction.horizontal direction.

The electric field produced by the The electric field produced by the jj--layer layer at some distance at some distance lljj can be approximated can be approximated byby

,j 2

4E 4 exp expj j j

jD D

l q lσ

ππσ

λ λ⎛ ⎞ ⎛ ⎞

= − = −⎜ ⎟ ⎜ ⎟∆⎝ ⎠ ⎝ ⎠

, 1 0 1(1 )D D zλ λ εδ± ±= +

The additional electric field will introduce the force from one The additional electric field will introduce the force from one particle layer to another particle layer to another neighboring one, neighboring one, FFσσ,j ,j . Writing down the levitation balance equation for dust layers . Writing down the levitation balance equation for dust layers we assume the changing of we assume the changing of DebyeDebye length at different particle positions, length at different particle positions,

1 12

0

1 ( / )

D

z zl

ε δ δλ

+ −+= −

This gives the screening length gradient This gives the screening length gradient εε=(d=(dllDD/dz)//dz)/llD0D0 within a sheathwithin a sheath

3) Theoretical model3) Theoretical model3 layer system: influence of heavier particles on light ones 3 layer system: influence of heavier particles on light ones

To proceed we injected a To proceed we injected a small number of the light, small number of the light, 1.6 1.6 mmmm--sized, particles to sized, particles to form a monolayer in the form a monolayer in the presence of the two layers presence of the two layers of the "reference", 2.4 of the "reference", 2.4 mmm, m, grains. grains. The light particles are now The light particles are now trapped at the positions trapped at the positions shifted upwards on a value shifted upwards on a value ±z ±z with respect to their with respect to their initial height initial height zz11characteristic for the single characteristic for the single layer configurationlayer configuration

Dust particles above the lower electrode , P=55PaDust particles above the lower electrode , P=55Pa

3) Theoretical model3) Theoretical model3 layer system: 3 layer system: DebyeDebye length length

Using the fact that light particles do not change the positions Using the fact that light particles do not change the positions of heavier ones but it is of heavier ones but it is working with a big efficiency other way around we write the equaworking with a big efficiency other way around we write the equation of levitation tion of levitation balance for 3 layer system and after some algebra obtain the equbalance for 3 layer system and after some algebra obtain the equation for screening ation for screening length length llD0D0

1 0 1

1 1 0

1 (1 ( )) exp(1 ( )) exp

(1 )D

D D

lz l zz z z l l

z z

βδ β δλγ δ

δ βδ λ λ− −

⎛ ⎞⎛ ⎞− + − − −⎜ ⎟⎜ ⎟

⎛ ⎞+ − ⎝ ⎠⎝ ⎠= − − +⎜ ⎟+ ⎝ ⎠With a local screening length With a local screening length

0 11 10 2

1

1 DD D

lz zz l

λδ δλ λδ+ −

−

⎛ ⎞+= −⎜ ⎟

⎝ ⎠

The estimation of the screening length in a plasma sheath has alThe estimation of the screening length in a plasma sheath has always been a ways been a challenge. The standard approach implies that in the bulk plasmachallenge. The standard approach implies that in the bulk plasma the screening is the screening is mainly due to the thermal mainly due to the thermal ions,i.eions,i.e. lD lDi. In the sheath the shielding becomes more In the sheath the shielding becomes more electronelectron--like, although the value of like, although the value of llDD could be also affected by noncould be also affected by non--thermal ions (thermal ions (A. A. PielPiel and A. and A. MelzerMelzer, Plasma Phys. Control. Fusion 44 (2002), Plasma Phys. Control. Fusion 44 (2002)))

3) Theoretical model3) Theoretical modelDebyeDebye lengthlength in the sheathin the sheath

Solving last equations the screening Solving last equations the screening length is estimated for different gas length is estimated for different gas pressurespressures

30 Pa 30 Pa llDiDi =7.5=7.5··1010--3 3 cm, cm, llDeDe =8=8··1010--22 cmcm55 Pa 55 Pa llDiDi =4.9=4.9··1010--22 cm, cm, llDe De =6=6··1010--22 cmcm100 Pa 100 Pa llDiDi =3.6=3.6··1010--22 cm, cm, llDeDe =4=4··1010--22 cmcm

The ion and electron screening lengths The ion and electron screening lengths defined in bulk plasma for our defined in bulk plasma for our experimental conditionsexperimental conditions

30 Pa 30 Pa llD0 D0 =6=6··1010--2 2 cmcm55 Pa 55 Pa llD0D0 =4.6=4.6··1010--22 cmcm100 Pa 100 Pa llD0D0 =3.3=3.3··1010--22 cmcm

Note that the values obtained from experimental data are much laNote that the values obtained from experimental data are much larger than the ion rger than the ion screening length defined in the bulk plasma, but on the other siscreening length defined in the bulk plasma, but on the other side, they are de, they are approximately 25approximately 25--20 % smaller than the electron 20 % smaller than the electron DebyeDebye length for the bulk plasma at length for the bulk plasma at the given pressures. Thus, our estimates are consistent with thethe given pressures. Thus, our estimates are consistent with the current physicalcurrent physicalrepresentations of the sheathrepresentations of the sheath

1 12

0

1 ( / )

D

z zl

ε δ δλ

+ −+= −Solving equation for the screening lengtSolving equation for the screening length gradient h gradient εε=(d=(dllDD/dz)//dz)/llD0D0

we found that its value is small we found that its value is small εε ≈≈--0.7 0.7 at low gas pressure of at low gas pressure of p p = 30 Pa; but = 30 Pa; but εε increases with gas pressure to increases with gas pressure to εε ≈≈--2.3 at 2.3 at p p = 55 Pa; and even up to = 55 Pa; and even up to εε ≈≈--4.3 4.3 at at p p = 100 Pa.= 100 Pa.

4) Theoretical model4) Theoretical modelDust charge in the sheathDust charge in the sheathThe force balance which is valid for the twoThe force balance which is valid for the two--layer system and together with the layer system and together with the expression for additional electric field produced by adjacent paexpression for additional electric field produced by adjacent particle layerrticle layer

,j 2

4E expj j

D

q lσ

πλ

⎛ ⎞= −⎜ ⎟∆ ⎝ ⎠

gives us the particle charge gives us the particle charge qq0 0 corresponding to the particlescorresponding to the particles with with radius a=2.4radius a=2.4mmmm 1/ 2

20 1

00

1 1

exp

4 (1 ( ))D

lm g zq

z z

β γ δλ

π β δ δ

+

+ −

⎛ ⎞⎛ ⎞+ ∆⎜ ⎟⎜ ⎟

⎝ ⎠⎜ ⎟⎜ ⎟+ +⎜ ⎟⎜ ⎟⎝ ⎠

With the calculated quantities With the calculated quantities llD0D0 , , γγ and and ββ above equation above equation gives the charge gives the charge number number ZZdd = = qq00/e/e, which is, which isat 30 Pa at 30 Pa ZZdd=2=2·10·1033

at 55 Pa at 55 Pa ZZdd=1.4·10=1.4·1033

at 100 Pa at 100 Pa ZZdd=1.2·10=1.2·1033

Comparison with weakly Comparison with weakly collisionalcollisional theory theory ((S.A. Khrapak, S.V. Ratynskaia, A.V. S.A. Khrapak, S.V. Ratynskaia, A.V. ZobninZobnin, A.D. , A.D. UsachevUsachev, V.V. Yaroshenko, , V.V. Yaroshenko, M.H. Thoma M.H. Thoma et.alet.al, Phys. Rev. E 72, , Phys. Rev. E 72, 016406 (2005)016406 (2005)) using corresponding ) using corresponding parameters in the plasma bulk gives good parameters in the plasma bulk gives good agreement at 30 Pa (agreement at 30 Pa (ZZdd=2.5·10=2.5·1033), but ), but disagrees at higher pressures (i.e. for disagrees at higher pressures (i.e. for 100 Pa theory gives 100 Pa theory gives ZZdd=1.9·10=1.9·1033). ).

5) Theoretical model5) Theoretical modelSheath electric fieldSheath electric fieldThe sheath electric field required to support The sheath electric field required to support dust particles with a=2.4dust particles with a=2.4mmmm radius against radius against gravity is found as gravity is found as EE00 = = mm00g/qg/q00..Its local value is estimated to be Its local value is estimated to be at 30 Pa at 30 Pa EE00 =26 =26 V/cmV/cmat 55 Pa at 55 Pa EE00 =38 =38 V/cmV/cmat 100 Pa at 100 Pa EE00 =45 =45 V/cmV/cm

1.0 1.5 2.0 2.50

10

20

30

40

50

60

elec

tric

field

, v/c

mDistance from lower electrode, mm

30pa 55pa 100pa

The electric field profiles The electric field profiles E E ((zz))≈≈EE00(1+(1+γγ(z(z--zz00)))) at at different gas pressuresdifferent gas pressures

The basic sheath parameters derived from the The basic sheath parameters derived from the dust particles positionsdust particles positions

20 30 40 50 60 70 80 90 100 110

1.01.52.02.53.03.54.04.55.05.56.06.57.0

E(x

101 V

/cm

), Z d(x

103 e)

, λD(

x10-2

cm)

Pressure, Pa

Debye length Charge Electric field

Summary:Summary:The micrometerThe micrometer--sized particles externally injected into plasma have been used sized particles externally injected into plasma have been used as a as a

diagnostic tool for a sheath regiondiagnostic tool for a sheath region

The screening length, the electric field and the particle chargeThe screening length, the electric field and the particle charge as well as the vertical as well as the vertical gradients of these quantities are determined experimentally withgradients of these quantities are determined experimentally within a sheath of the in a sheath of the RF discharge for several different gas pressuresRF discharge for several different gas pressures

The sheath structure reveals vertical The sheath structure reveals vertical inhomogeneityinhomogeneity with significant local gradients of with significant local gradients of the main plasma parameters (electric field, the main plasma parameters (electric field, DebyeDebye length) and dust chargelength) and dust charge

The screening length in the vertical direction within the sheathThe screening length in the vertical direction within the sheath is found to be more is found to be more electronelectron--like than ionlike than ion--like and the value of the difference (like and the value of the difference (llDeDe--llDD) decreases with ) decreases with increase of the gas pressureincrease of the gas pressure

The decrease of the particle charge in the The decrease of the particle charge in the rfrf sheath with respect to the theoretical sheath with respect to the theoretical estimates might be attributed to the peculiarities of the sheathestimates might be attributed to the peculiarities of the sheath region as well as to region as well as to collisionalcollisional regime of the experimentsregime of the experiments

Since our method depends only on one experimentally determined qSince our method depends only on one experimentally determined quantity, the uantity, the levitation height (which could be measured with good spatial reslevitation height (which could be measured with good spatial resolution), this olution), this minimizes the errors in the estimates due to measurement uncertaminimizes the errors in the estimates due to measurement uncertainties.inties.