A. Samarian, O. Vaulina, W. Tsang, B. JamesSchool of Physics, University of Sydney, NSW 2006, Australia

2

Introduction

Various self-excited motions are considered in dusty plasma with Various self-excited motions are considered in dusty plasma with spatial charge gradientspatial charge gradient

Two basic types of instabilities in these systems were studied Two basic types of instabilities in these systems were studied numerically and analytically. The basic attention is given to the cases numerically and analytically. The basic attention is given to the cases of vortex motions of dust particlesof vortex motions of dust particles

Conditions suitable for forming of considered instabilities in Conditions suitable for forming of considered instabilities in discharge dusty plasmas are discussed. It was shown that discharge dusty plasmas are discussed. It was shown that dust charge dust charge gradient is an effective mechanism to excite the dust motion, which gradient is an effective mechanism to excite the dust motion, which allows explanationallows explanation of considerable range of phenomena observed in of considerable range of phenomena observed in the inhomogeneous laboratory dusty plasmathe inhomogeneous laboratory dusty plasma

The results of experimental observations of the horizontal and vertical The results of experimental observations of the horizontal and vertical vortices in the planar capacitive RF discharge are presented vortices in the planar capacitive RF discharge are presented

3

Dispersion relations for non-conservative systems

Analysis of the roots Analysis of the roots (k),(k), of equation of equation L(L(,k)=0,k)=0 allows the existent allows the existent region of nontrivial and unstable solutions of the wave equations to be region of nontrivial and unstable solutions of the wave equations to be determineddeterminedMathematical models developed for study of oscillations in non-Mathematical models developed for study of oscillations in non-equilibrium non-linear systems are based on analysis of differential equilibrium non-linear systems are based on analysis of differential wave equationswave equationsIn these models, there are two basic types of instabilities: In these models, there are two basic types of instabilities:

Dissipative instability for systems, where dissipation is present (case 1); Dissipative instability for systems, where dissipation is present (case 1); Dispersion instability, when the dissipation is negligibly small (case 2)Dispersion instability, when the dissipation is negligibly small (case 2)

We consider a dispersion relation We consider a dispersion relation L(L(,k)=0,k)=0 for small perturbations of for small perturbations of a stable system a stable system GG by a harmonic wave with amplitude by a harmonic wave with amplitude bb::Dispersion relation Dispersion relation L(L(,k)=0,k)=0 is the linear analogy of differential wave is the linear analogy of differential wave equation of motion. It determines the functional dependency of equation of motion. It determines the functional dependency of oscillation frequency oscillation frequency on wave vector on wave vector k:k:

= = bbexp{ikx-iexp{ikx-it}t}

4

Dispersion relations for non-conservative systems

The differential wave equations can be written in functional form as The differential wave equations can be written in functional form as GG(ik;i(ik;i;;))bb and and L(L(,k),k)det(G)=det(G)=00 will show whether the model under will show whether the model under consideration contains any decay termsconsideration contains any decay termsWhen attenuation is present (When attenuation is present (case 1case 1), ), L(L(,k),k) will be complex both for will be complex both for stable stable ((I<0)<0) and for unstable and for unstable ((II >0)>0) states of system. The roots will states of system. The roots will also be complex (i.e. also be complex (i.e. ==RR+i+iII). And hence:). And hence:

==bbexp{ikx-iexp{ikx-iRRt}exp{t}exp{IIt}t}

For For II>0, the solution will increase in time and will be unstable. The >0, the solution will increase in time and will be unstable. The point where point where II changes sign is the point of bifurcation in the system changes sign is the point of bifurcation in the system

For For case 2case 2, the dispersion relation is a real function. But roots can be a , the dispersion relation is a real function. But roots can be a complex conjugate pair: complex conjugate pair: ==RR i iII. Hence:. Hence:

==bbexp{ikx-iexp{ikx-iRRt}exp{t}exp{IIt}t}

and the solution will increase exponentially for any and the solution will increase exponentially for any II00

For the stable solutions For the stable solutions II=0=0, harmonic perturbation will propagate , harmonic perturbation will propagate dispersively instead of attenuating as in a dissipative systemdispersively instead of attenuating as in a dissipative system

5

Lets consider the motion of Np particles with charge Z=Z(r,y)=Zoo+Z(r,y), in an electric field , where r=(x2+z2)1/2 is the horizontal coordinates in a cylindrically symmetric system. Taking the pair interaction force Fint, the gravitational force mpg, and the Brownian forces Fbr into account, we get:

where l is the interparticle distance, mp is the particle mass and fr is the friction frequency Now is the interparticle potential with screening length D, and e is the electron charge. Also is the total external forceSo total external force and interparticle interaction are dependent on the particle’s coordinate. When the curl of these forces 0, the system can do positive work to compensate the dissipative losses of energy. It means that infinitesimal perturbations due to thermal or other fluctuations in the system can grow

Now is the interparticle potential with screening length D, and e is the electron charge. Also is the total external forceSo total external force and interparticle interaction are dependent on the particle’s coordinate. When the curl of these forces 0, the system can do positive work to compensate the dissipative losses of energy. It means that infinitesimal perturbations due to thermal or other fluctuations in the system can grow

D

l

l

yreZD exp

,

),()(},)({ yreZrEjgmyreZyEiF pext

So total external force and interparticle interaction are dependent on the particle’s coordinate. When the curl of these forces 0, the system can do positive work to compensate the dissipative losses of energy. It means that infinitesimal perturbations due to thermal or other fluctuations in the system can grow

r

yeZrFD

,)(int

Dispersion relations for non-conservative systems

)()(),( rEjyEiyrE

extj

brk

frp

jk

jk

lllk

p FFdt

ldm

ll

lllF

dt

ldm

jk

)(int2

2

6

Equation of motion

Assume particle charge Assume particle charge ZZoo = = ZZoo oo + + ZZ((rroo,y,yoo) is in stable state at an extreme point in the dust ) is in stable state at an extreme point in the dust cloud in the position cloud in the position ((rroo,y,yoo) ) relative to its center.relative to its center.

• Denote 1Denote 1stst derivatives of parameters at the point derivatives of parameters at the point ((rroo,y,yoo)) as as

rr==dEdEee(r)/dr, (r)/dr, yy=-=-dEdEee(y)/dy(y)/dy rr==Z(rZ(r,,y)/y)/r, r, yy==Z(rZ(r,,y)/y)/y y

rr==EEiirr(r,y)/(r,y)/r, r, yy==EEii

yy(r,y)/(r,y)/yy andand oo==EEiirr(r,y)/(r,y)/yy EEii

yy(r,y)/(r,y)/rr

• Then the linearized system of equations for the particle deviations can be presented in the Then the linearized system of equations for the particle deviations can be presented in the form:form:

dd22r/dtr/dt22==--frfrdr/dt+аdr/dt+а1111rr++аа1212yy

dd22y/dty/dt22 ==--frfrdy/dt+аdy/dt+а2222yy++аа2211rr

where where аа1111== -eZ -eZoo{{rr--rr}}/m/mp p , а, а

1212== eZeZoooo/m/mpp ,,

аа2211==[[eZeZoooo + m + m

ppgg/Z/Zoo]]/m/mp,p, аа2222= = [[-eZ-eZoo{{yy--yy}}+ m+ m

ppggyy/Z/Zoo]]/m/mpp

• For the case of stationary stable state of the dust particle ( For the case of stationary stable state of the dust particle ( rroo==rr((tt ); ); yyoo=y(t=y(t )); ; EEee(r(roo))==

EEiirr (r(r

o o ,, y yoo)); ; EEee(y(y

oo)) E Eiiyy(r(r

oo,,yyoo)=)= m mppg/eZg/eZ

oo) in a position above center of the dust cloud ) in a position above center of the dust cloud ((rroo,,++yyoo) ) or under it or under it ((rroo,,--yyoo))

• We can obtain a “dispersion relation” We can obtain a “dispersion relation” L(L())det(G)=det(G)=0 0 from the response of system to a small from the response of system to a small perturbation perturbation ==bbexpexp{{-i-itt}},, which arises in the direction which arises in the direction rr or or yy::

44+(а+(а1111+а+а

2222--frfr22))22++(а(а

1111аа2222-а-а1122аа2121)+i)+ifrfr{{2222+а+а

1111+а+а2222}}=0=0

• It shows that the small perturbations in system will grow in two cases: It shows that the small perturbations in system will grow in two cases: Type 1Type 1 When a restoring force is absentWhen a restoring force is absent

Type 2Type 2 Near some characteristic resonant frequency Near some characteristic resonant frequency cc of the system of the system

7

Equation of motion

An occurrence of Type 1 dissipative instability is determined by the condition:

(а11а22-а12а21)0The equality of the above equation determines a neutral curve of the dissipative instability (R=0, I=0).

Taking coefficients aij into account, and assuming that ZoZoo>>Z(r,y), we can obtain:

eZo{( -)(y-y)-o 2} <or g/Zo

An occurrence of Type 2 dispersive instability is determined by the condition :

c2[4а12а21+(а11 - а22)

2]/4fr2

Thus dispersion spectrum of motion (R 0, I=0) takes place close to resonant frequency c (i.e. when the friction in the system is balanced by incoming potential energy). In general, oscillations with frequency c will develop when dissipation does not destroy the structure of the dispersion solution and does not allow considerable shifts of the neutral curve, where I=0. For amplification of the oscillating solutions, it is necessary that:

fr<c<= /2This formula determines region of dispersion instability. Under condition of synchronized motion of separate particles in dust cloud, solutions similar to waves are possible.In the case of strong dispersion, as a result of development of Type 2 instability, the steady-state motion can represent a harmonic wave with a frequency close to the bifurcation point of the system c

8

Simulation of Results

9

Simulation of Results

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Kinetic Energy

The kinetic energy К(i), gained by dust particle after Type 1 instability is:

К( i )=mpg22/{8fr

2}

where ={Аr/Zoo} determines relative changes of Z(r) within limits of particle trajectory

When a=5m, =2g/cm3 and fr12P (P~0.2Torr), К( i ) is one order higher than thermal dust energy To0.02eV at room temperature for >10-3 (r/Zoo>0.002cm-1, A=0.5cm)

Increasing gas pressure up to P=5Torr or decreasing particle radius to a=2m, К( i )/To >10 for >10-2 (r/Zoo>0.02cm-1, A=0.5cm). This estimation shows that even small variations of dust charge can lead to effective conversion of potential energy from background sources to the kinetic energy of dust motionAs the transport characteristics of a strongly correlated dust system are determined by the dust frequency, for Type 2 instability, К(ii) can be estimated with known c

р(2e2Z(r,y)2npexp(-k){1+k+k2/2}/mp)1/2

where k=lp/D and Z(r,y)<Z> for small charge variations

Assume that resonance frequency c of the steady-stated particle oscillations is close to р. Then kinetic energy К(ii) can be written in the form:

К(ii)5.76 103 (aTe) 22cn/lp

where cn=exp(-k){1+k+k2/2} and =А/lp (~0.5 for dust cloud close to solid structure)

When a=5m, =0.1, k1-2, lp=500m, and Te~1eV, the К(ii)3eV. The maximum kinetic energy (which is not destroying the crystalline dust structure) is reached at =0.5. And К(ii)

lim=cne2<Z>2/4lp

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RFRF discharge 15 MHz discharge 15 MHz

Pressure from 10 to 400 mTorr Pressure from 10 to 400 mTorr

Input power from 15 to 200 WInput power from 15 to 200 W

Self-bias voltage from 5 to 180VSelf-bias voltage from 5 to 180V

Carbon (C) particles diameter ~ 1 μmCarbon (C) particles diameter ~ 1 μm

Melamine formaldehyde - 2.79 μm ± 0.06 μm Melamine formaldehyde - 2.79 μm ± 0.06 μm

Melamine formaldehyde - 6.13 μm ± 0.10 μmMelamine formaldehyde - 6.13 μm ± 0.10 μm

Argon plasma TArgon plasma Te e ~ 2 eV, V~ 2 eV, Vp p =50V &=50V & nne e ~ 10~ 1099 cm cm--

3 3

The laser beam enters the discharge chamber The laser beam enters the discharge chamber through a 40-mm diameter window. through a 40-mm diameter window. A window mounted on a side port in a A window mounted on a side port in a perpendicular direction provides a view of the perpendicular direction provides a view of the vertical cross-section of the dust structure. vertical cross-section of the dust structure. In addition, we use the top-view window to In addition, we use the top-view window to view the horizontal dust-structure. view the horizontal dust-structure.

Experimental Setup

The experiments were carried out in a The experiments were carried out in a 40-cm inner diameter cylindrical 40-cm inner diameter cylindrical stainless steel vacuum vessel with stainless steel vacuum vessel with many ports for diagnostic access. many ports for diagnostic access. The chamber height is 30 cm. The The chamber height is 30 cm. The diameters of electrodes are 10 cm for diameters of electrodes are 10 cm for the disk and 11.5 cm for the ringthe disk and 11.5 cm for the ring The The dust particles suspended in the plasma dust particles suspended in the plasma are illuminated using a Helium-Neon are illuminated using a Helium-Neon laser. laser.

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Back ViewBack ViewExperimental Setup

Images of the illuminated dust cloud are obtained using a charged-coupled Images of the illuminated dust cloud are obtained using a charged-coupled device (CCD) camera with a 60mm micro lens and a digital camcorder (focal device (CCD) camera with a 60mm micro lens and a digital camcorder (focal length: 5-50 mm). length: 5-50 mm). The camcorder is operated at 25 to 100 frames/sec.The camcorder is operated at 25 to 100 frames/sec.

The video signals The video signals are stored on are stored on videotapes or are videotapes or are transferred to a transferred to a computer via a computer via a frame-grabber cardframe-grabber card. . The coordinates of The coordinates of particles were particles were measured in each measured in each frame and the frame and the trajectory of the trajectory of the individual particles individual particles were traced out were traced out frame by frameframe by frame

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Experimental Setup for Vertical Vortex Motion

Dust vortex in discharge plasma (superposition of 4 frames)

Melamine formaldehyde –2.67 μm(Side view)

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Experimental Setup for Horizontal Vortex Motion

When When aamm, where , where aa is dust radius and is dust radius and mm is the free-path length of molecules in a gas, is the free-path length of molecules in a gas,

the frictional frequency the frictional frequency fr,fr, can be written in the free-molecular approximation: can be written in the free-molecular approximation:

frfr CCvvPP//aa

where where is dust density, is dust density, PP is gas pressure, and is gas pressure, and CCvv is a constant determined by is a constant determined by

background gas (e.g. background gas (e.g. CCv v 600 ( 600 (NeNe)) and and 820 (820 (ArAr) at room temperature ~ 300K)) at room temperature ~ 300K)

Powered electrode

Groundedelectrode

DustVortex

Pin electrode

Side View Top View

Groundedelectrode

Pin electrode

Dust Vortex

15

Vortex Motion

Assuming that drift electron (ion) currents < thermal current, Assuming that drift electron (ion) currents < thermal current, TTii0.03eV and0.03eV and n neennii, then:, then:

<Z><Z> = = CCzzaaTTee

HereHere CCzz is is 2x102x1033 ( (ArAr). ). Thus in the case of Thus in the case of ZZ((r,yr,y)=<)=<ZZ>+>+TTZZ((r,yr,y), where ), where TTZZ is the is the

equilibrium dust charge at the point of plasma with the some electron temperatures equilibrium dust charge at the point of plasma with the some electron temperatures TTee, and , and

TTZZ((r,yr,y) is the variation of dust charge due to the ) is the variation of dust charge due to the TTee, then:, then:

T T ZZ((r,yr,y)/<)/<ZZ> = > = TTee((r,yr,y)/)/TTee

andandyy//<Z><Z>==((TTee//yy))TTee-1-1, , //<Z> <Z> = = ((TTee//))TTee

-1-1

Pin Electrode

(Top view)

Video Images of Dust Vortices in Plasma Discharge

16

Illustration of Dust Vortex Motion

If spatial variations If spatial variations n n ZZ((r,yr,y) ) of of equilibrium equilibrium dust charge dust charge occur due to gradients of occur due to gradients of

concentrations concentrations nne(i)e(i) in plasma surrounding dust cloud, assuming that conditions in the in plasma surrounding dust cloud, assuming that conditions in the

plasma are close to electroneutral (plasma are close to electroneutral (nn==nnii--nnee«n«neenniinn and and nnZZ((r,yr,y))«<Z>)«<Z>), where , where nnZ(r,y)Z(r,y)

is the equilibrium dust charge where is the equilibrium dust charge where nnee==nnii, then , then nnZZ((r,yr,y) ) is determined is determined by equating the by equating the

orbit-limited electrons (ions) currents for an isolated spherical particle with equilibrium orbit-limited electrons (ions) currents for an isolated spherical particle with equilibrium surface potential < 0, that is.surface potential < 0, that is.

n n ZZ((,y,y))-- -0.26-0.26

where <where <ZZ>>20002000aaTTee

Z n

n e Z aTe

( / )1 2

Zn

n

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Vortex Movie

18

Theoretical Analysis of Vortex Motion

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120 140 160 180 200

Pressure, mTorr

pg/{Zov fr }

=12 mm-1

0

10

20

30

40

50

60

0 20 40 60 80 100 120 140 160 180 200

Pressure, mTorr

U=40 V

Ft

p/{2mdZov fr }=320 mm-1

a) b)

Dependency of the rotation frequency on pressure for vertical (a) and horizontal (b) vortices

wс = /2= F /{2mpZofr}

19

Conclusion

• The results of experimental observation of twoThe results of experimental observation of two types of types of self-excited dust vortex motions (vertical and horizontal) self-excited dust vortex motions (vertical and horizontal) in planar in planar RF RF discharge are presenteddischarge are presented

• The fThe first irst type type is the vertical rotations of is the vertical rotations of dust dust particles in particles in bulk dust cloudsbulk dust clouds

• The second type of dust The second type of dust vorvorttexex isis formed in the horizontal formed in the horizontal plane for monolayer structure of particlesplane for monolayer structure of particles

• We attribute the inducWe attribute the inductiontion of these vortices with the of these vortices with the developdevelopmentment of dissipative instability in the dust cloud of dissipative instability in the dust cloud with the dust charge gradient, which have been provided with the dust charge gradient, which have been provided by extra electrode. by extra electrode. The presence of additional electrode The presence of additional electrode also produces the additional force which, along with the also produces the additional force which, along with the electric forces, will lead to the rotation of dust structure in electric forces, will lead to the rotation of dust structure in horizontal planehorizontal plane