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

2 Introduction Various self-excited motions are considered in dusty plasma with spatial charge gradient Two basic types of instabilities in these systems were studied numerically and analytically. The basic attention is given to the cases of vortex motions of dust particles Conditions suitable for forming of considered instabilities in discharge dusty plasmas are discussed. It was shown that dust charge gradient is an effective mechanism to excite the dust motion, which allows explanation of considerable range of phenomena observed in the inhomogeneous laboratory dusty plasma Conditions suitable for forming of considered instabilities in discharge dusty plasmas are discussed. It was shown that dust charge gradient is an effective mechanism to excite the dust motion, which allows explanation of considerable range of phenomena observed in the inhomogeneous laboratory dusty plasma The results of experimental observations of the horizontal and vertical vortices in the planar capacitive RF discharge are presented

3 Dispersion relations for non-conservative systems Analysis of the roots  (k), of equation L( ,k)=0 allows the existent region of nontrivial and unstable solutions of the wave equations to be determined Mathematical models developed for study of 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 the linear analogy of differential wave equation of motion. It determines the functional dependency of oscillation frequency  on wave vector k:  = bexp{ikx-i  t}

4 Dispersion relations for non-conservative systems The 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

5 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  Dispersion relations for non-conservative systems )()(),(rEjyEiyrE    

6 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

7 Equation of motion 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   

8 Simulation of Results

9

10 Kinetic Energy 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). 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 motion As 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  (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

11 RF discharge 15 MHz Pressure from 10 to 400 mTorr Input power from 15 to 200 W Self-bias voltage from 5 to 180V Carbon (C) particles diameter ~ 1 μm Melamine formaldehyde μm ± 0.06 μm Melamine formaldehyde μm ± 0.10 μm Argon plasma T e ~ 2 eV, V p =50V & n e ~ 10 9 cm -3 The laser beam enters the discharge chamber through a 40-mm diameter window. A window mounted on a side port in a perpendicular direction provides a view of the vertical cross-section of the dust structure. In addition, we use the top-view window to view the horizontal dust-structure. Experimental Setup The experiments were carried out in a 40-cm inner diameter cylindrical stainless steel vacuum vessel with many ports for diagnostic access. The chamber height is 30 cm. The diameters of electrodes are 10 cm for the disk and 11.5 cm for the ring The dust particles suspended in the plasma are illuminated using a Helium-Neon laser.

12 Back View Experimental Setup 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: 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

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

14 Experimental Setup for Horizontal Vortex Motion When a  m, where a is dust radius and m is the free-path length of molecules in a gas, the frictional frequency fr, can be written in the free-molecular approximation: fr  C v P/a  fr  C v P/a  where  is dust density, P is gas pressure, and C v is a constant determined by background gas (e.g. C v  600 (Ne) and 820 (Ar) at room temperature ~ 300K) Powered electrode Grounded electrode Dust Vortex Pin electrode Side View Top View Grounded electrode Pin electrode Dust Vortex

15 Vortex Motion 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 Pin Electrode (Top view) Video Images of Dust Vortices in Plasma Discharge

16 Illustration of Dust Vortex Motion 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

17 Vortex Movie

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

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