Damping of the oscillations of dust particles levitating
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Transcript Damping of the oscillations of dust particles levitating
Damping of the dust particle
oscillations at very low
neutral pressure
M. Pustylnik, N. Ohno,
S.Takamura, R. Smirnov
Introduction
In the linear approximation the motion of a dust particle trapped in a sheath
is described by the harmonic oscillator equation:
..
.
z 2 z 02 ( z zlev ) 0,
where z is the vertical coordinate, β is the damping rate and ω0 is the
eigenfrequency. If the dust particle is balanced against gravity by the
electrostatic force only
e (EZd )
md z z zlev
2
0
(Zd, md – dust particle charge and mass, E – local electric field). Usually it is accepted
that oscillations of the dust particles are damped by the neutral drag. Damping rate
is given by the Epstein formula:
epst
4 p
ca
p – neutral gas pressure, ρ – is the density of the dust particle material, a – is the
dust particle radius.
Delayed charging
Zd
Delayed charging is the effect,
associated with the finite charging
time of a dust particle. It has been
shown that this effect leads to the
modification of the damping factor:
Zdeq
Zd
δZd
epst
Z d
eE
z
ch
z zlev
2md
x
ch – is the characteristic charging, i.e. time, required to compensate small
deviation of the dust particle charge from its equilibrium value.
Convinient representation of damping factor – β/p. β/p is constant if only Epstein drag
works. For 2.5 mm dust, supposing 1.44, β/p = 2.3 s-1Pa-1
Collisionless sheath model with
bi-Maxwellian electrons
Energy and flux conservation for ions:
mi v i2
mi Cs2
0 pl
2
2
ni vi nSE Cs
Boltzman-distributed electrons
Poisson equation
(0) U le
sheath
electrode
Ule
2
4ene ni
2
z
Dust
particle
z
presheath
φ0
pl
pl
(1 ) exp
ne n0 exp
T
T
h
c
nSE ne (0 )
Generalized Bohm criterion
1
mi
0
f i (v )
n
dv
2
v
at
Φ=Φ0 (Φ = pl- ; Φ0 = pl- 0)
n ne
f i (v) nSE (v Cs )
0 1
0 nSE
n0 exp
exp
0
2
T
T
T
T
c
0
h
c
h
Charging of dust
Equilibrium charge condition – total current equals zero. Electron and ion currents
(bi-Maxwellian plasma):
8 Tc
pl f
I e a n0
(1 ) exp
Tc
me
2
f
I i a 1
T
i
pl
2
Charging time
a
ch
I e I i
f
pl f
8 Th
exp
me
Th
Experimental setup
S
Video imaging
parameters:
Laser sheet
Ua
Frame rate 250 fps
Anode
Exposure time 2 ms
N
R1
Filament
Probe
Amplifier
100 Hz,
100 sweeps
Uc
Spatial resolution ~13
mm/pix
U1
levitation
electrode
Grid
trench
Ug
Record duration –
6.55 s
R2
R3
U2
Function
generator,
constant negative bias,
iImpulse to excite vibration
(10 ms), syncronized with
videocamera
Probe measurements in the biMaxwellian plasma
Example of the measurements
Probe characteristics
I = 40 mA, p = 0.4 Pa
I = 20 mA, p = 0.4 Pa
I = 40 mA, p = 0.2 Pa
I = 20 mA, p = 0.2 Pa
1E-3
10
15
20
25
U [V]
3.5
3.0
2.5
0.45
0.40
0.35
0.30
15
1x10
-3
Discharge parameters
5 parameters
0.00
Th [eV]
1E-5
0.04
0.02
Tc [eV]
1E-4
n0 [m ]
Ua = 18 V
Ua = 17 V
Ua = 16 V
Ua = 15 V
Ua = 14 V
Ua = 10 V
Ua = 6 V
Ua = 2 V
Cathode current ~31 mA
Cathode voltage -80 V
Grid voltage 18 V
Anode voltage varied 0-18 V
Argon pressure 0.18 Pa
14
5x10
20.0
Upl [V]
Ie [A]
0.06
19.5
19.0
18.5
14
15
16
Ua [V]
17
18
Dust dynamics
Trajectory
300
Displacement [mm]
200
100
0
-100
-200
-300
0
1
2
3
4
5
6
7
time [s]
z zlev A0 expt cost
Amplitudes
3.2
3.0
logarithm of amplitude
2.8
~-βt
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
0
1
2
3
time [s]
4
5
6
7
Pressure variation experiment
Plasma parameters
I = 20 mA
I = 30 mA
I = 40 mA
0.020
0.015
Damping rate
,
,
,
0.010
4
Th [eV]
0.005
3.5
3
3.0
2
-1 -1
0.35
0.30
-3
Epstein
law value
0.40
/p [Pa s ]
Tc [eV]
2.5
n0 [cm ]
I = 40 mA (experiment, theory)
I = 30 mA (experiment, theory)
I = 20 mA (experiment, theory)
9
1.0x10
1
0
instability
-1
8
5.0x10
1.4
-2
Upl [V]
1.3
1.2
-3
1.1
1.0
0.0
0.1
0.2
0.3
0.4
0.5
p [Pa]
0.6
0.7
0.8
0.9
0.0
0.1
0.2
0.3
0.4
0.5
p [Pa]
0.6
0.7
0.8
0.9
variation experiment
Plasma parameters
Damping rate
I = 40 mA, p = 0.4 Pa
I = 20 mA, p = 0.4 Pa
I = 40 mA, p = 0.2 Pa
I = 20 mA, p = 0.2 Pa
0.06
0.04
0.02
Th [eV]
0.00
3
3.5
E ps te in law
v alu e
3.0
2.5
E xp . Th eor .
,
I
,
I
,
I
,
I
=
=
=
=
20
40
40
20
mA,
mA,
mA,
mA,
p
p
p
p
2
0.40
/p [Pa s ]
0.35
-1 -1
Tc [eV]
0.45
0.30
-3
n0 [m ]
15
1x10
1
14
5x10
0
Upl [V]
20.0
19.5
instability
19.0
-1
18.5
14
14
15
16
Ua [V]
17
18
15
16
U a [V]
17
18
=
=
=
=
0.2
0.4
0.2
0.4
Pa
Pa
Pa
Pa
Calculated map of DCE
Positiv e DC E thr eshold
14
13
12
I = 40 mA
11
10
9
I = 30 mA
8
7
6
I = 20 mA
5
4
3
2
1
0
-1
Non-levitatable region
Electron current [A]
Non-uniformity of the plasma
in the vicinity of the electrode
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
~5 c m belo w the ele ctro de
~5 c m abov e the ele c tro de
-10
0
Voltag e [V ]
10
Sheath is governed
by several times
smaller than measured
PIC simulation of the sheath
• Bi-Maxwellian electrons
• Ions are injected as Maxwellian with
the room temperature
• Elastic and charge-exchange
collisions for ions are taken into
account
• Plasma particles penetrate through
the electrode with the probability
0.88
• Length of the simulated domain 2 cm
Effect of the shape of the ion
VDF on the equilibrium
potential of a dust grain
Simulated ion VDF
6.0x10
Currents
11
-11
4.0x10
-11
Electron
Ion (simulated VDF)
Ion (cold ion approximation)
-11
4.0x10
2.0x10
3.0x10
11
Current [A]
-4 -1
ion VDF [m s ]
3.5x10
11
-11
2.5x10
-11
2.0x10
-11
1.5x10
-11
1.0x10
-12
5.0x10
0.0
0
1000
vi [m/s]
2000
3000
-4
-3
-2
f [V]
-1
0
Conclusions
• Large deviations of the damping rate from the
value, predicted by the Epstein neutral drag
formula are observed
• The deviation appears at low pressure and is
larger at lower values of
• At comparatively lower plasma density the
damping rate is smaller than the Epstein value
and transition to instability is clearly observed.
• At higher plasma density damping rate is higher
than the Epstein value
• Qualitative agreement between the theoretical
calculations and experimental measurements is
acieved