Transcript r,y - School of Physics

O.Vaulina, A.Samarian, O.Petrov, B.James, V.Fortov
School of Physics, University of Sydney, Australia
Institute for High Energy Densities, Moscow, Russia
The experimental data (dc and rf discharge)
Vertical and horizontal vortices
Oscillation
Basic Concepts
Modelling and Estimation
Simulation Results
2
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)
3
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
4
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}
5
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
(I<0) and for unstable (I >0) states of system. The roots will also be
complex (i.e. =R+iI). And hence:
=bexp{ikx-iRt}exp{It}
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Rt}exp{It}
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
6
Equation of Motion
Lets consider the motion of Np particles with charge Z=Z(r,y)=Zoo+Z(r,y), in



an electric field E (r , y)  i E ( y)  j E (r ) , where r=(x2+z2)1/2 is the horizontal
coordinates in a cylindrically symmetric system.
y
y0
Z00
r0
drD
Fint (r )  eZ (  , y )
dr
Z00+Z(r,y)
7
Equation of Motion
Lets consider themotionvof Np particles
with charge Z=Z(r,y)=Zoo+Z(r,y), in

an electric field E(r, y)  i E( y)  jE(r) , 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: 
 



lk  l j
d 2lk
d lk
 Fint (l )    
mp


m


F

F

p fr
br
ext
l  l k l j
j
dt
dt 2
lk  l j
where l is the interparticle distance, mp is the particle mass and fr is the
friction frequency
eZ(r, y)  l 
Now   l exp D ÷
is the interparticle
potential with screening
length D, and e



v
is the electron charge. Also Fext  i {E ( y)eZ (r , y ) m p g}  j E (r )eZ (r , y) is the total
external force
D
So total external force and interparticle interaction Fint (r )   eZ ( , y )
are
r
dependent on the
the particle’s
particle’scoordinate.
coordinate. When the curl of these forces  0, the
system
can curl
do positive
work
to compensate
the dissipative
of energy.
It
When the
of these
forces
 0, the system
can dolosses
positive
work to
means
that infinitesimal
perturbations
to thermal
or other
in
compensate
the dissipative
losses ofdue
energy.
It means
thatfluctuations
infinitesimal
the
system can grow
perturbations
due to thermal or other fluctuations in the system can grow
D
8
Equation of motion
Assume particle charge Zo = Zoo + Z(ro,yo) is in stable state at an extreme point in the
dust cloud in the position (ro,yo) relative to its center.
Denote 1st derivatives of parameters at the point (ro,yo) as
r=dEe(r)/dr, y=-dEe(y)/dy
r=Z(r,y)/r, y=Z(r,y)/y
r=Eir(r,y)/r, y=Eiy(r,y)/y
and
o=Eir(r,y)/y Eiy(r,y)/r
Then the linearized system of equations for the particle deviations can be presented in
the form:
d2r/dt2=-frdr/dt+а11r+а12y
d2y/dt2 =-frdy/dt+а22y+а21r
where
а11= -eZo{r-r}/mp , а12= eZoo/mp ,
а21=[eZoo + mpg/Zo]/mp, а22= [-eZo{y-y}+ mpgy/Zo]/mp
For the case of stationary stable state of the dust particle ( ro=r(t ); yo=y(t );
Ee(ro)= Eir (ro , yo); Ee(yo) Eiy(ro,yo)= mpg/eZo) in a position above center of the dust
cloud (ro,+yo) or under it (ro,-yo)
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+(а11+а22-fr2)2+(а11а22-а12а21)+ifr{22+а11+а22}=0
It shows that the small perturbations in system will grow in two cases:
Type 1
When a restoring force is absent
Type 2
Near some characteristic resonant frequency c of the system
9
Condition for Instability
An occurrence of Type 1 dissipative instability is determined by the condition:
(а11а22-а12а21)0
The 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 ZoZoo>>Z(r,y), we can obtain:
eZo{( -)(y-y)-o 2} <or g/Zo
An occurrence of Type 2 dispersive instability is determined by the condition :
c2[4а12а21+(а11 - а22)2]/4fr2
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  
10
Dust Charge Spatial Variation
ne/ni=f
ni(e)
Te
(r) and Te=f (r)
Assuming that drift electron (ion) currents < thermal current, Ti0.03eV and neni, then:
<Z> = CzaTe
Here Cz is 2x103 (Ar). Thus in the case of Z(r,y)=<Z>+TZ(r,y), where TZ is the
equilibrium dust charge at the point of plasma with the some electron temperatures Te,
and TZ(r,y) is the variation of dust charge due to the Te, then:
T Z(r,y)/<Z> = Te(r,y)/Te
and
y/<Z>=(Te/y)Te-1, /<Z> = (Te/)Te-1
If spatial variations n Z(r,y) of equilibrium dust charge occur due to gradients of
concentrations ne(i) in plasma surrounding dust cloud, assuming that conditions in the
plasma are close to electroneutral (n=ni-ne«nenin and nZ(r,y)«<Z>), where nZ(r,y) is
the equilibrium dust charge where ne=ni, then nZ(r,y) is determined by equating the
orbit-limited electrons (ions) currents for an isolated spherical particle with equilibrium
surface potential < 0, that is.
 0.26 Z n
Z n
n Z(,y)  n(1  e 2 Z / aT ) 
n
e
where <Z>2000aTe
11
Theory
Charge gradient ß
Non electrostatic forces (gravity, thermothoretic, ion drag)
Keep dust cloud in the region with E  0  eZp
if not F  0 (х  0),
thus A  (Fnon/eZp)2
Efficiency determine by the condition (eZp/lp)2 << Fnon
12
Vortex in ICP
1mm
RF discharge 17.5 MHz
Pressure from 560 mTorr
Input voltage from 500 mV
13
Melamine formaldehyde 6.21m±0.09m
Argon plasma Te~ 2eV & ne ~
108cm-3
Experimental Setup
Experiments
out in 40-cm inner
Images of carried
the illuminated
dust
diameter
stainless
steel
cloud are cylindrical
obtained using
a chargedvacuum
many
portswith
for
coupled vessel
device with
(CCD)
camera
diagnostic
a 60mm access.
micro lens and a digital
camcorder (focal length: 5-50 mm).
Chamber
height isis30operated
cm. Diameters
The camcorder
at 25 of
to
electrodes
are 10 cm for disk and 11.5
100 frames/sec.
cm for ring. Dust particles
illuminated using a He-Ne laser.
Pressure from 10 to 400 mTorr
Input power from 15 to 200 W
Probe Inlet
Self-bias voltage from 5 to 80V
Melamine formaldehyde - 2.79 μm ± 0.06 μm
Argon plasma Te ~ 2 eV,
Oil Diffusion
Pump
14
Particle
Dispenser
Laser
Probe Inlet
Argon Gas
Inlet
Confining Ring
Electrode
Oil Diffusion
Pump
Top Ground
Electrode
Confining Ring
Electrode
Vp =50V
& ne ~
Top Ground
Electrode
are
Observation
Window
RF discharge 15 MHz
Observation
Window
109 cm-3
Particle
Dispenser
The beam
videoenters
signals
arechamber
storedthrough
on
Laser
discharge
Laser
40-mm
diameter
videotapes
or window.
are transferred to a
computerwindow
viaArgon
a frame-grabber
card.
Top-view
is used to view
horizontal
Gas
Inlet
dust-structure.
The coordinates of particles were
A
window inmounted
side and
port thein
measured
each on
frame
perpendicular
direction
provides
view
trajectory of the
individual
particles
wereof
vertical
cross-section
dust structure.
traced out
frame byofframe
Experimental Setup for Vertical Vortex Motion
Dust vortex in discharge plasma
(superposition of 4 frames)
Melamine formaldehyde –2.67 μm
(Side view)
15
Experimental Setup for Horizontal Vortex Motion
Grounded
electrode
Grounded
electrode
Pin electrode
Dust
Vortex
Grounded
Grounded
electrode
electrode
Dust
Vortex
Dust
Vortex
Dust
Vortex
Powered electrode
Side View
electrode
PinPin
electrode
Top View
Video Images of Dust Vortices in Plasma Discharge
16
Experimental Results
Grounded Electrode
4cm
Powered Electrode
8cm
11cm
17
Experimental Results
Experimental Result
vs. Theory
Experimental Result
1,2
potencial, [V]
50
( r )()
1,0
40
0,8
30
0,6
20
0,4
10
0,2
0
0
1
2
3
distance, [cm]
18
4
5
r/R
0,0
0,0
0,2
0,4
0,6
0,8
1,0
Equation of Motion
Lets consider the motion of Np particles with charge Z=Z(r,y)=Zoo+Z(r,y), in



an electric field E (r , y)  i E ( y)  j E (r ) , 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:

d l
m p 2k   Fint (l )l  l l
k
j
dt
j
2
 


lk  l j
dlk 

m


F

F
 
p fr
br
ext
dt
lk  l j
where l is the interparticle distance, mp is the particle mass and fr is the
friction frequency.
NowD 
eZ (r , y )
l
exp( ) is the interparticle potential with screening length D,
l
D
and e is the electron charge.



Also Fext  i {E ( y)eZ (r , y)  m p g}  j E (r )eZ (r , y) is the total external force.
19
Results from Simulation
20
Results from Simulation
21
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 )=mpg22/{8fr2}
where ={Аr/Zoo} determines relative changes of Z(r) within limits of
particle trajectory
When a=5m, =2g/cm3 and fr12P (P~0.2Torr), К( i ) is one order higher
than thermal dust energy To0.02eV 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=2m, К( i )/To >10 for >10-2 (r/Zoo>0.02cm-1, A=0.5cm).
22
Kinetic Energy
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) 22cn/lp
where cn=exp(-k){1+k+k2/2} and =А/lp (~0.5 for dust cloud close to solid
structure)
When a=5m, =0.1, k1-2, lp=500m, 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
23
-Dependency on Pressure
wс =  /2= F /{2mpZofr}
Dependency of the rotation frequency  on pressure
for vertical (a) and horizontal (b) vortices
a)
10
U=40 V
b)
60
9
8

50
7

pg/{Zov fr }
6
=12 mm-1
5
40
Ftp/{2mdZov fr }
30
=320 mm-1
4
20
3
2
10
1
0
0
0
20
40
60
80
100
120
Pressure, mTorr
24
140
160
180
200
0
20
40
60
80
100
120
Pressure, mTorr
140
160
180
200
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
25
Vertical Component of Particles’ Velocity
The Effect of Power on z-component of
the Velocity of Particle
P= 30W
P= 60W
P= 80W
P= 120W
26
Velocity Distribution
Number of particles
The Effect of Power on Velocity
Distribution in Horizontal Plane
70W
P= 30W
100W
P=
velocity (cm/sec)
27
Velocity distribution
Spatial Velocity Distribution
15cm/s
Velocity Distribution Function
8cm/s
3cm/s
P= 70W
0cm/s
P= 100W
P= 30W
28
Vertical Cross Section
P= 120W
P= 80W
P= 60W
P= 30W
29