Damping of the dust particle Damping of the dust particle oscillations at very low oscillations at very low
neutral pressureneutral pressure
M. Pustylnik,M. Pustylnik, N. Ohno, N. Ohno, S.Takamura, R. SmirnovS.Takamura, R. Smirnov
IntroductionIntroduction
(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:
In the linear approximation the motion of a dust particle trapped in a sheathis described by the harmonic oscillator equation:
,0)(2 20
...
levzzzz 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
levzz
d
d z
EZ
m
e
)(2
0
ca
pepst
4
p – neutral gas pressure, ρ – is the density of the dust particle material, a – is the dust particle radius.
Delayed chargingDelayed chargingDelayed 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:
d
ch
zz
depst mz
ZeE
lev2
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 dragworks. For 2.5 m dust, supposing β/p = 2.3 s-1Pa-1
Zd
x
Zdeq
δZd
Zd
Collisionless sheath model with Collisionless sheath model with bi-Maxwellian electronsbi-Maxwellian electrons
sSEii
plsii
Cnvn
Cmvmi
0
22
22
)(
exp)1(exp
0
0
eSE
c
pl
h
ple
nn
TTnn
Energy and flux conservation for ions:
Boltzman-distributed electrons
Poisson equation
leie Unnez
)0(42
2
z
sheath
presheath
electrode
φ0
Dust particle U
le
Generalized Bohm criterionGeneralized Bohm criterion
n
dvv
vf
mi
i 02
)(1
)()( sSEi
e
Cvnvf
nn
at Φ=Φ0 (Φ = pl- ; Φ0 = pl- 0)
02
exp1
exp0
000
SE
cchh
n
TTTTn
Charging of dustCharging of dust
pli
fi
h
fpl
e
h
c
fpl
e
ce
TaI
Tm
T
Tm
TnaI
1
exp8
exp)1(8
2
02
Equilibrium charge condition – total current equals zero. Electron and ion currents(bi-Maxwellian plasma):
Charging time
f
iech II
a
Experimental setupExperimental setup
U2
U1
R2
R3
R1
Amplifier 100 Hz,
100 sweeps
Ug
Grid
Anode
ProbeFilament
N
S
Ua
Uc
Functiongenerator, constant negative bias,iImpulse to excite vibration (10 ms), syncronized with videocamera
Video imaging parameters:
Frame rate 250 fps
Exposure time 2 ms
Spatial resolution ~13 m/pix
Record duration – 6.55 s
Laser sheet
levitationelectrode
trench
Probe measurements in the bi-Probe measurements in the bi-Maxwellian plasmaMaxwellian plasma
14 15 16 17 18
18.5
19.0
19.5
20.0
5x1014
1x1015
0.30
0.35
0.40
0.45
2.5
3.0
3.5
0.00
0.02
0.04
0.06
Upl [V
]
Ua [V]
n 0 [m
-3]
Tc [
eV
]T
h [e
V]
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
Example of the measurements
10 15 20 25
1E-5
1E-4
1E-3
Ua = 18 V
Ua = 17 V
Ua = 16 V
Ua = 15 V
Ua = 14 V
Ua = 10 V
Ua = 6 V
Ua = 2 V
I e [A
]
U [V]
Probe characteristics
Discharge parameters
Cathode current ~31 mACathode voltage -80 VGrid voltage 18 VAnode voltage varied 0-18 VArgon pressure 0.18 Pa
5 parameters
Dust dynamicsDust dynamics
0 1 2 3 4 5 6 7-300
-200
-100
0
100
200
300
Dis
pla
cem
en
t [m
]
time [s]
Trajectory
0 1 2 3 4 5 6 7
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
log
ari
thm
of
am
plit
ud
e
time [s]
Amplitudes
~-βt
ttAzz lev cosexp0
Pressure variation experimentPressure variation experiment
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91.0
1.1
1.2
1.3
1.45.0x108
1.0x109
0.30
0.35
0.40
2.5
3.0
3.5
0.005
0.010
0.015
0.020
Upl [V
]
p [Pa]
n 0 [c
m-3]
Tc [
eV]
Th
[eV
]
I = 20 mA I = 30 mA I = 40 mA
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-3
-2
-1
0
1
2
3
4
/p [P
a-1s-1
]
p [Pa]
, I = 40 mA (experiment, theory), I = 30 mA (experiment, theory), I = 20 mA (experiment, theory)
instability
Epsteinlaw value
Plasma parameters Damping rate
variation experimentvariation experiment
14 15 16 17 18-1
0
1
2
3
Epstein law value
Exp. Theor., I = 20 mA, p = 0.2 Pa, I = 40 mA, p = 0.4 Pa, I = 40 mA, p = 0.2 Pa, I = 20 mA, p = 0.4 Pa
/p [P
a-1s-1
]
Ua [V]14 15 16 17 18
18.5
19.0
19.5
20.0
5x1014
1x1015
0.30
0.35
0.40
0.45
2.5
3.0
3.5
0.00
0.02
0.04
0.06
Upl [V
]
Ua [V]
n 0 [m
-3]
Tc [
eV
]T
h [e
V]
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
Plasma parameters Damping rate
instability
Calculated map of Calculated map of DCEDCE
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Non-levita table region
Positiv e thr esholdDC E
I = 40 mA
I = 30 mA
I = 20 mA
Non-uniformity of the plasma Non-uniformity of the plasma in the vicinity of the electrodein the vicinity of the electrode
-10 0 10
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
Ele
ctro
n cu
rren
t [A
]
Voltage [V]
~5 cm below the electrode ~5 cm above the electrode
Sheath is governedby several timessmaller than measured
PIC simulation of the sheathPIC 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 Effect of the shape of the ion VDF on the equilibrium VDF on the equilibrium potential of a dust grainpotential of a dust grain
0 1000 2000 30000.0
2.0x1011
4.0x1011
6.0x1011
ion
VD
F [
m-4s-1
]
vi [m/s]
Simulated ion VDF
-4 -3 -2 -1 0
5.0x10-12
1.0x10-11
1.5x10-11
2.0x10-11
2.5x10-11
3.0x10-11
3.5x10-11
4.0x10-11
Cu
rre
nt
[A]
f [V]
Electron Ion (simulated VDF) Ion (cold ion approximation)
Currents
ConclusionsConclusions• 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
