O.Vaulina, A.Samarian, O.Petrov, B.James, V.Fortov School of Physics, University of Sydney, Australia Institute for High Energy Densities, Moscow, Russia

2 The experimental data (dc and rf discharge) Vertical and horizontal vortices Oscillation Basic Concepts Modelling and Estimation Simulation Results

3 Outlines Instabilities due to inhomogenaties in plasma. Greater instability of dust structures explained by larger space charge gradient We develop and promote this idea for several years and progress are made on both experimental-theoretical side Here we present overview of obtained results 1. O. S. Vaulina, A. P. Nefedov, O. F. Petrov, and V. E. Fortov, JETP 91, 1063 (2000) 2. O. S. Vaulina, A. A. Samarian, A. P. Nefedov, V. E. Fortov, Phys. Lett. A 289, 240(2001) 3. O. S. Vaulina, A. A. Samarian, A. P. Nefedov, V. E. Fortov, JETP 93, 1789(2001) 4. A.A.Samarian, O.S.Vaulina, W.Tsang, B.W. James Physica Scripta T98, 123 (2002) 5. O.S. Vaulina, A.A. Samarian, O.F. Petrov, B.W. James, V.E. Fortov, JETP 95, (2003) 6. O. S. Vaulina, A. A. Samarian, O. F. Petrov, B. W. James,V. E. Fortov, Dusty Plasmas Focus Issue of New Journal of Physics (2003)

4 Introduction Various self-excited motions are considered in dusty plasma with spatial charge gradient Two basic types of instabilities in systems were studied numerically and analytically. Attention given to vortex motions of dust particles Conditions suitable for instabilities in dusty plasmas are discussed. We showed dust charge gradient is an effective mechanism to excite dust motion. Allows explanation of considerable range of phenomena observed in inhomogeneous laboratory dusty plasma Results of experimental observations of horizontal and vertical vortices in planar capacitive RF discharge are presented

5 Dispersion relations for non-conservative systems Analysis of roots  (k) from equation L( ,k)=0 allows existent region of nontrivial and unstable solution of wave equations to be determined Mathematical models developed for oscillations in non-equilibrium non-linear systems are based on analysis of differential wave equations In these models, there are two basic types of instabilities: Dissipative instability for systems, where dissipation is present (case 1); Dispersion instability, when the dissipation is negligibly small (case 2) We consider a dispersion relation L( ,k)=0 for small perturbations of a stable system G by a harmonic wave with amplitude b: Dispersion relation L( ,k)=0 is linear analogy of differential wave equation of motion. It determines the functional dependency of oscillation frequency  on wave vector k:  = bexp{ikx-i  t}

6 Dispersion relations for non-conservative systems Differential wave equations can be written in functional form as G(ik;i  ;  )b and L( ,k)  det(G)=0 will show whether the model under consideration contains any decay terms When attenuation is present (case 1), L( ,k) will be complex both for stable (  0) states of system. The roots will also be complex (i.e.  =  R +i  I ). And hence: When attenuation is present (case 1), L( ,k) will be complex both for stable (  I 0) states of system. The roots will also be complex (i.e.  =  R +i  I ). And hence:  =bexp{ikx-i  R t}exp{  I t} For  I >0, the solution will increase in time and will be unstable. The point where  I changes sign is the point of bifurcation in the system For case 2, the dispersion relation is a real function. But roots can be a complex conjugate pair:  =  R  i  I. Hence:  =bexp{ikx-i  R t}exp{  I t} and the solution will increase exponentially for any  I  0 For the stable solutions  I =0, harmonic perturbation will propagate dispersively instead of attenuating as in a dissipative system

7 Equation of Motion Lets consider the motion of N p particles with charge, in an electric field, where is the horizontal coordinates in a cylindrically symmetric system. Lets consider the motion of N p particles with charge Z=Z(r,y)=Z oo +  Z(r,y), in an electric field, where r=(x 2 +z 2 ) 1/2 is the horizontal coordinates in a cylindrically symmetric system. y r Z 00 Z 00 +  Z(r,y) r0r0 y0y0

8 Lets consider the motion of N p particles with charge Z=Z(r,y)=Z oo +  Z(r,y), in an electric field, where r=(x 2 +z 2 ) 1/2 is the horizontal coordinates in a cylindrically symmetric system. Taking the pair interaction force F int, the gravitational force m p g, and the Brownian forces F br into account, we get: where l is the interparticle distance, m p 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 force 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 Now is the interparticle potential with screening length D, and e is the electron charge. Also is the total external force 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         D l l yreZ D exp,   ),()(},)({yreZrEjgmyr yEiF 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 yeZrF D   ,)( int  Equation of Motion )()(),(rEjyEiyrE    

9 Equation of motion Assume particle charge Z o = Z oo +  Z(r o,y o ) is in stable state at an extreme point in the dust cloud in the position (r o,y o ) relative to its center. Denote 1 st derivatives of parameters at the point (r o,y o ) as  r =dE e (r)/dr,  y =-dE e (y)/dy  r =  Z(r,y)/  r,  y =  Z(r,y)/  y  r =  E i r (r,y)/  r,  y =  E i y (r,y)/  yand  o =  E i r (r,y)/  y  E i y (r,y)/  r Then the linearized system of equations for the particle deviations can be presented in the form: d 2 r/dt 2 =- fr dr/dt+а 11 r+а 12 y d 2 y/dt 2 =- fr dy/dt+а 22 y+а 21 r where а 11 = -eZ o {  r -  r }/m p, а 12 =  eZ o  o /m p, а 21 =[  eZ o  o + m p g   /Z o ]/m p, а 22 = [-eZ o {  y -  y }+ m p g  y /Z o ]/m p For the case of stationary stable state of the dust particle ( r o =r(t   );  y o =y(t   ); E e (r o )= E i r (r o,  y o ); E e (y o )  E i y (r o,  y o )= m p g/eZ o ) in a position above center of the dust cloud (r o,+y o ) or under it (r o,-y o ) We can obtain a “dispersion relation” L(  )  det(G)=0 from the response of system to a small perturbation  =bexp{-i  t}, which arises in the direction r or y:  4 +(а 11 +а 22 - fr 2 )  2 +(а 11 а 22 -а 12 а 21 )+i fr  {2  2 +а 11 +а 22 }=0 It shows that the small perturbations in system will grow in two cases: Type 1When a restoring force is absent Type 2Near some characteristic resonant frequency  c of the system

10 Condition for Instability An occurrence of Type 1 dissipative instability is determined by the condition: (а 11 а 22 -а 12 а 21 )  0 The equality of the above equation determines a neutral curve of the dissipative instability (  R =0,  I =0). Taking coefficients a ij into account, and assuming that Z o  Z oo >>  Z(r,y), we can obtain: eZ o {(   -   )(  y -  y )-  o 2 } <  o  r g/Z o An occurrence of Type 2 dispersive instability is determined by the condition :  c 2  [4а 12 а 21 +(а 11 - а 22 ) 2 ]/4 fr 2 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 <   =   /2 This 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   

11 Dust Charge Spatial Variation Assuming that drift electron (ion) currents < thermal current, T i  0.03eV and n e  n i, then: = C z aT e = C z aT e Here C z is 2x10 3 (Ar). Thus in the case of Z(r,y)= +  T Z(r,y), where  T Z is the equilibrium dust charge at the point of plasma with the some electron temperatures T e, and  T Z(r,y) is the variation of dust charge due to the  T e, then:  T Z(r,y)/ =  T e (r,y)/T e and  y  / =(  T e /  y)T e -1,    / = (  T e /  )T e -1 If spatial variations  n Z(r,y) of equilibrium dust charge occur due to gradients of concentrations n e(i) in plasma surrounding dust cloud, assuming that conditions in the plasma are close to electroneutral (  n=n i -n e «n e  n i  n and  n Z(r,y)« ), where  n Z(r,y) is the equilibrium dust charge where n e =n i, then  n Z(r,y) is determined by equating the orbit-limited electrons (ions) currents for an isolated spherical particle with equilibrium surface potential ), where  n Z(r,y) is the equilibrium dust charge where n e =n i, then  n Z(r,y) is determined by equating the orbit-limited electrons (ions) currents for an isolated spherical particle with equilibrium surface potential < 0, that is.  n Z( ,y)   where  2000aT e n e /n i =f (r) and T e =f (r) n i(e) TeTe

12 Theory Charge gradient ß Non electrostatic forces (gravity, thermothoretic, ion drag) Keep dust cloud in the region with E  0  eZ p if not F  0 (х  0), thus A  (F non /eZ p ) 2 Efficiency determine by the condition (eZ p /l p ) 2 << F non

13 Vortex in ICP RF discharge 17.5 MHz Pressure from 560 mTorr Input voltage from 500 mV Melamine formaldehyde  m±0.09  m Argon plasma T e ~ 2eV & n e ~ 10 8 cm -3

14 RF discharge 15 MHz Pressure from 10 to 400 mTorr Input power from 15 to 200 W Self-bias voltage from 5 to 80V Melamine formaldehyde μm ± 0.06 μm Argon plasma T e ~ 2 eV, V p =50V & n e ~ 10 9 cm -3 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 length: 5-50 mm). The camcorder is operated at 25 to 100 frames/sec. The video signals are stored on videotapes or are transferred to a computer via a frame-grabber card. The coordinates of particles were measured in each frame and the trajectory of the individual particles were traced out frame by frame Laser beam enters discharge chamber through 40-mm diameter window. Top-view window is used to view horizontal dust-structure. A window mounted on side port in perpendicular direction provides view of vertical cross-section of dust structure. Experiments carried out in 40-cm inner diameter cylindrical stainless steel vacuum vessel with many ports for diagnostic access. Chamber height is 30 cm. Diameters of electrodes are 10 cm for disk and 11.5 cm for ring. Dust particles are illuminated using a He-Ne laser. Experimental Setup

15 Experimental Setup for Vertical Vortex Motion Dust vortex in discharge plasma (superposition of 4 frames) Melamine formaldehyde –2.67 μm (Side view)

16 Experimental Setup for Horizontal Vortex Motion Grounded electrode Dust Vortex Powered electrode Grounded electrode Dust Vortex Pin electrode Grounded electrode Pin electrode Dust Vortex Grounded electrode Pin electrode Dust Vortex Side View Top View Video Images of Dust Vortices in Plasma Discharge

17 Experimental Results

18 Experimental Results Experimental Result vs. Theory

19 Equation of Motion Lets consider the motion of N p particles with charge, in an electric field, where is the horizontal coordinates in a cylindrically symmetric system. Lets consider the motion of N p particles with charge Z=Z(r,y)=Z oo +  Z(r,y), in an electric field, where r=(x 2 +z 2 ) 1/2 is the horizontal coordinates in a cylindrically symmetric system. Taking the pair interaction force F int, the gravitational force m p g, and the Brownian forces F br into account, we get: where l is the interparticle distance, m p 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 force.

20 Results from Simulation

21 Results from Simulation

22 Kinetic Energy Energy gain for two basic types of instabilities: Dissipative instability for systems, where dissipation is present (Type 1); Dispersion instability, when the dissipation is negligibly small (Type 2) The kinetic energy К (i), gained by dust particle after Type 1 instability is: К ( i ) =m p g 2  2 /{8 fr 2 } where  ={А  r /Z oo } determines relative changes of Z(r) within limits of particle trajectory When a=5  m,  =2g/cm 3 and fr  12P (P~0.2Torr), К ( i ) is one order higher than thermal dust energy T o  0.02eV at room temperature for  >10 -3 (  r /Z oo >0.002cm -1, A=0.5cm) Increasing gas pressure up to P=5Torr or decreasing particle radius to a=2  m, К ( i ) /T o >10 for  >10 -2 (  r /Z oo >0.02cm -1, A=0.5cm).

23 Kinetic Energy For Type 2 instability, К (ii) can be estimated with known  c  р  (2e 2 Z(r,y) 2 n p exp(-k){1+k+k 2 /2}/m p ) 1/2 where k=l p /D and Z(r,y)  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)  (aT e ) 2  2 c n /l p where c n =exp(-k){1+k+k 2 /2} and  =А/l p (~0.5 for dust cloud close to solid structure) When a=5  m,  =0.1, k  1-2, l p =500  m, and T e ~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 =c n e 2 2 /4l p

24  -Dependency on Pressure Dependency of the rotation frequency  on pressure for vertical (a) and horizontal (b) vortices w с =   /2= F    /{2m p Z o fr }

25 Conclusion The results of experimental observation of two types of self-excited dust vortex motions (vertical and horizontal) in planar RF discharge are presented First type is the vertical rotations of dust particles in bulk dust clouds Second type is formed in horizontal plane for monolayer structure Induction of these vortices due to development of dissipative instability in the dust cloud with dust charge gradient, which have been provided by extra electrode The presence of additional electrode also produces additional force which, along with the electric forces, will lead to rotation of dust structure in horizontal plane

26 Vertical Component of Particles’ Velocity

27 Number of particles The Effect of Power on Velocity Distribution in Horizontal Plane P= 100W P= 70W P= 30W Velocity Distribution velocity (cm/sec)

28 Velocity distribution

29 Vertical Cross Section P= 120W P= 80W P= 60W P= 30W