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PARALLEL PLATE CAPACITOR SIMULATION AND
TRANSVERSE OSCILLATION OF VERTICAL DUSTY
PLASMA CHAINS IN A GLASS BOX
by
Raymond Fowler
A Thesis Submitted to the Faculty of
The Wilkes Honors College
in Partial Fulfilment of the Requirements for the Degree of
Bachelor of the Arts in Liberal Arts and Sciences
with a Concentration in Mathematics and Physics
Wilkes Honors College of
Florida Atlantic University
Jupiter, Florida
May 2014
PARALLEL PLATE CAPACITOR SIMULATION AND
TRANSVERSE OSCILLATION OF VERTICAL DUSTY PLASMA
CHAINS IN A GLASS BOX
by
Raymond Fowler
This thesis was prepared under the direction of the candidate’s thesis advisors,
Dr. Terje Hoim and Dr. Andrew Johnson, and has been approved by the members
of his supervisory committee. It was submitted to the faculty of The Honors
College and was accepted in partial fulfillment of the requirements for the degree
of Bachelor of Arts in Liberal Arts and Sciences.
SUPERVISORY COMMITTEE:
Dr. Terje Hoim
Dr. Andrew Johnson
Dr. Meredith Blue
Dean, Harriet L. Wilkes Honors College
Date
ii
Acknowledgements
I would like to thank my advisor, Dr. Terje Hoim, for reviewing many copies
of my thesis. She has been an encouraging and tireless advisor, always pushing
me forward in my work. She has also supervised several independent studies for
me and has lent me books so that I could learn whatever mathematics I wished to
know. I could not have asked for a better advisor. I would also like to thank my
second reader and advisor, Dr. Andrew Johnson, who has taken much time out of
his busy schedule to advise students and to teach upper division physics classes in
the absence of a physics professor. I further thank Dr. Meredith Blue for being
the third reader of this thesis.
Furthermore, I thank the CASPER research group of Baylor University. The
unpublished paper I co-wrote with Brandon Harris, Dr. Lorin Matthews, and Dr.
Truell Hyde at Baylor’s REU program has been incorporated into this thesis, and
the rest of the thesis results from work I later did with Brandon Harris at long
distance. I also appreciate the time Brandon took to answer all my research-related
questions once the research was officially finished.
Finally, this thesis could not have been written without the NSF funding the
research at the REU program (grant number Phy-1002637).
iii
Abstract
Author:
Raymond Fowler
Title:
PARALLEL PLATE CAPACITOR SIMULATION AND
TRANSVERSE OSCILLATION OF VERTICAL DUSTY
PLASMA CHAINS IN A GLASS BOX
Institution:
Harriet L. Wilkes Honors College, Florida Atlantic
University
Thesis Advisor:
Dr. Terje Hoim and Dr. Andrew Johnson
Concentration:
Mathematics and Physics
Year:
2014
Dust particles in plasma can arrange themselves into a vertical configuration
when placed inside a glass box. What really occurs in the glass box is still unknown
and is difficult to find out by direct methods of measurement. In this thesis, the
dust chains are investigated through probe oscillations and a Matlab simulation
that uses a parallel plate capacitor to model an electric field. Parameters of the
plasma and dispersion relations for the transverse oscillation are extracted from
the chain’s motion. For both longitudinal and transverse oscillatory motion, the
simulation is used to find the charge on all the dust in the chains and to simulate
the chain’s motion. The plasma parameter values are expected within experimental error. The dispersion relations show that more than one wave motion
propagates through the chain. The charge found is much higher than expected,
and so the simulated motion may not be very accurate.
iv
Contents
List of Figures
vii
List of Tables
viii
1 Introduction
1
2 Physics Background and Literature Review
4
3 Mathematical Background
3.1 Introduction . . . . . . .
3.2 Gravity . . . . . . . . .
3.3 Electric Force . . . . . .
3.4 Neutral Drag . . . . . .
3.5 Interparticle Repulsion .
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4 Experimental Work
4.1 Introduction . . . . . . . . .
4.2 Materials and Equipment . .
4.3 Method . . . . . . . . . . .
4.3.1 Plasma Parameters .
4.3.2 Dispersion Relations
4.3.3 Model . . . . . . . .
4.4 Results . . . . . . . . . . . .
4.4.1 Plasma Parameters .
4.4.2 Dispersion Relations
4.4.3 Model . . . . . . . .
4.5 Discussion . . . . . . . . . .
4.5.1 Plasma Parameters .
4.5.2 Dispersion Relations
4.5.3 Model . . . . . . . .
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5 Conclusion
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12
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Appendices
A Antisymmetric Mode Particle Vertical Motion Code
47
B Antisymmetric Mode Calculation of Charge Code
52
v
C Antisymmetric Mode Particle Horizontal Motion Code
55
Bibliography
60
vi
List of Figures
2.1
Vertical Dust Chain . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
Underdamped Data vs Probe’s peak-to-peak amplitude .
Particle’s amplitude vs Probe’s amplitude . . . . . . . .
Particle’s distance from Probe’s tip vs Time . . . . . . .
Underdamped curve fit . . . . . . . . . . . . . . . . . . .
Particles’ heights vs DC bias . . . . . . . . . . . . . . . .
Sheath edge height vs System power . . . . . . . . . . .
Dispersion relation 4-particle chain . . . . . . . . . . . .
Dispersion relation 5-particle chain . . . . . . . . . . . .
Particle 1, antisymmetric mode without perturbation . .
Particle 1, standing mode without perturbation . . . . .
Particle 1, transverse motion without perturbation . . . .
Close-up view of oscillations between 6.5 sec and 10.5 sec
Particle 1, antisymmetric mode with perturbation . . . .
Particle 1, standing mode with perturbation . . . . . . .
Particle 1, transverse motion with perturbation . . . . .
Particle 1, antisymmetric mode with oscillatory force . .
Particle 1, standing mode with oscillatory force . . . . .
Particle 1, symmetric mode with oscillatory force . . . .
Particle 1, transverse motion with oscillatory force . . . .
Electric field for antisymmetric mode . . . . . . . . . . .
Electric field for standing mode . . . . . . . . . . . . . .
Electric field for transverse motion case . . . . . . . . . .
vii
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6
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List of Tables
4.1
4.2
Parameters of Chains for each Mode . . . . . . . . . . . . . . . . . 16
Results of Capacitor Model . . . . . . . . . . . . . . . . . . . . . . 31
viii
Chapter 1
Introduction
Plasma is considered the fourth state of matter. When you heat up a liquid,
it becomes a gas, and when you heat up a gas, it becomes a plasma. Heating
up the gas causes the electrons in many of the gas molecules to dissociate and to
freely move around in the plasma. Hence, plasma consists of neutral gas molecules
and freely floating electrons and ions. Introducing micron-sized particles into the
plasma turns it into a dusty plasma. Dusty plasmas are found in many places,
from the semiconductor industry to nuclear fusion devices to dust clouds in outer
space. They can also be found in the everyday phenomenon of an ordinary flame;
the soot providing the dust for the plasma. The dust in the plasma can produce
novel properties as the dust gives the dusty plasma characteristics of both a liquid
and a solid [1]. These properties include dust levitation in the plasma and dust
acoustic waves. Dust crystal structures, which consist of dust particles held in
place by interparticle repulsive electrostatic forces, can also form, and plasma
parameters like damping can change. Further, if illuminated by a laser, the dust
can be seen with the naked eye and recorded on camera, so one can easily track the
dust motion in the plasma. This allows for the observation of phase transitions;
for example, we can watch dust crystal structures “melt.” They go from a “solid”
configuration of unmoving dust “frozen” in a crystal pattern to a “liquid” form of
more freely moving dust particles. These can then change to a “gas” of chaotic
and quickly moving dust particles. Although dusty plasmas are fascinating for
scientists, the dust in the plasmas behaves as a contaminant in industry and in
1
fusion devices. Hence, a double motivation exists to study the properties of dusty
plasmas, of which there is still much unknown.
Some interesting configurations to study are one-dimensional dust structures
that form inside a glass box and are vertically aligned with the bottom of the
box, i.e., parallel to the box’s sides and perpendicular to the bottom of the box.
These structures, called “vertical dust chains,” are investigated in this thesis.
Little is known about what occurs inside the glass box to bring about these chains
and sustain them, and little is known about the physics of the dust in these
chains. Furthermore, it can be difficult to physically measure plasma parameters
without significantly disturbing the plasma [2]. The approach taken in this thesis
to investigate both the physics of the vertical dust chains and the plasma conditions
that produce them is twofold: longitudinal and transverse waves are propagated
in these chains, and a simulation of the motion of the dust particles in those waves
is made. Oscillations can give some insight into the forces involved that push the
chain back to equilibrium, and a simulation can use experimentally determined
parameters to predict the charge on the dust in the chains. To find the charge on
the dust particles, the electric field for the lower particles in the chain is found by
using a parallel plate capacitor model.
The transverse waves are used to find two important parameters. One parameter is the resonance frequency of a single, oscillating particle, which is then used
to calculate the damping coefficient for the plasma at particular settings. The
other parameter is a dispersion relation, which is found by calculating and fitting
a curve to the data resulting from the transverse oscillation. By changing the DC
bias on the lower electrode and the system power put into the plasma (also called
the “plasma power”), two plasma properties are found: the height of the chain
above the lower electrode for varying DC biases, and the sheath edge for varying
plasma powers. The sheath edge at each plasma power is found by dropping a
2
single, small dust particle and observing its levitation height. The simulation is
used to find the charge of the particles in a vertical chain by finding the values
of charge needed for the particles in the chain to levitate at the experimentally
observed equilibrium positions (the Debye length that the dust needs is found by
using a dispersion relation). The motion of the dust is then modeled by summing
the forces on the particles and using kinematics equations.
This thesis is organized as follows. The basic physics and laboratory set up is
explained in Chapter 2, giving the technical details useful for understanding the
experiments and the simulation. In Chapter 3, the most important mathematical
equations used in the research are presented and explained. Then, in Chapter
4, a brief but exact description of the experiments run and the simulation made
is given. Results of the experiments and simulation are reported in Section 4.4,
followed by a brief discussion of those results in Section 4.5. The conclusion of the
thesis is given in Chapter 5.
3
Chapter 2
Physics Background and
Literature Review
Plasmas are gases that have been heated to the point where the electrons
dissociate from their atoms or molecules and move around in the gas freely. Having
lost the negatively charged electrons, the molecules or atoms become positively
charged ions. Hence, plasmas are called “ionized gases.” However, there may
still be plenty of neutral gas molecules remaining, and the way the electrons and
ions move in the plasma tend to make the plasma neutral overall, which is called
“quasi-neutrality.” When neutrally charged particles of micron- or nanometer-sized
condensed matter are introduced into a plasma, the plasma is called a “dusty
plasma,” the “dust” referring to those particles of condensed matter. However,
these neutrally charged particles do not remain neutral for long. The electrons
moving freely in the gas collide with and stick to these particles, giving them a
negative charge. If the dust is light enough and the charge strong enough, the
dust above a charged surface can levitate against the force of gravity.
A few other forces besides gravity are involved in a dusty plasma, and the
plasma itself can give rise to other forces. When a plasma comes into contact with
a surface, the surface absorbs some of the electrons and ions from the plasma,
creating a non-neutral region of space: an electric field [2]. There is also a Yukawa
interparticle repulsion between dust particles. The Yukawa repulsion is an ordinary electrostatic repulsion screened by the ions in the plasma. The ions tend
4
to surround the negatively charged dust, partially cancelling electric forces interacting with the dust. This screening of the charge is measured in terms of what
is called the Debye length, which will be described more precisely in the section
that explains the Yukawa force equation. There is also neutral drag, resulting
from collisions of the dust with the neutral particles, and ion drag, resulting from
collisions of the dust with ions [3].
For research purposes, plasmas are ignited in a vacuum chamber that is part of
a carefully controlled set of apparatus called a “cell.” The particular cell used in
this paper is a radio-frequency (rf) powered GEC reference cell, where the plasma
is ignited and powered by a radio signal [4]. Originally designed to study the dust
contamination problem in the semiconductor industry, this cell is a standardized
one that can form dust crystals above its lower electrode quite easily [3]. This cell
contains two electrodes inside of its vacuum chamber: a grounded upper electrode
and a negatively charged lower electrode. The charge on the lower electrode can
be fixed and set by a current called a “DC bias.” It is also possible to ignite and
power the plasma by a direct current (dc) discharge, but using an rf cell has a
few main advantages. The semiconductor industry uses rf powered plasmas for
silicon wafer etching [1], the rapidly oscillating rf field allows the dust to levitate
throughout the sheath region instead of only at its edge [5], and the rf field allows
for a negative charge to develop at the lower electrode [3].
The standard procedure that has been used for studying and forming vertical
dust chains with the GEC reference cell is as follows. The plasma is first ignited
between the two electrodes, and then dust of micrometer diameter is dropped into
the plasma. The dust is dropped from a container located near the top of the
vacuum chamber, called a “dust dropper.” As the dust falls, it picks up a negative
charge from the plasma and then levitates above the lower electrode in the sheath
[6]. In order to horizontally confine the dust, a portion of the lower electrode is
5
cut a little lower than the rest of the electrode. The lower, cut out area creates a
potential well that traps the dust [5]. When horizontally confined, the interaction
of the dust particles with each other can form crystallized structures: the dust
particles levitate above the electrode so as to form a crystal-like pattern. This
pattern is called a “Coulomb crystal” because the force that holds the crystal
together is the electrostatic force. Coulomb crystals can be 2D or 3D, and these
crystals have been studied in various places including [7–9]. A general overview of
Coulomb crystals can be found in [5].
In an earlier study by Kong, it was found that if a small glass box was placed
on the lower electrode inside the cut out area, the dust in the crystal structures
could be rearranged into a “vertical chain.” As the plasma power was gradually
lowered, the dust rearranged itself from a chaotic state into a stable state; one
particle was right under another in a straight, vertical line as shown in Figure
2.1 [6]. The glass box further confines the dust because the walls of the glass
Figure 2.1: An actual vertical dust chain with five particles.
box quickly become negatively charged by the moving electrons inside the plasma,
providing a repulsive force to the dust [6]. How exactly the box permits the dust
to form vertical dust chains is currently a matter that is somewhat unknown.
6
A possible cause for the formation of vertical dust chains in the box is the ion
wakefield. The ion wakefield results from ion streaming: the positively charged
ions in the plasma flowing towards the negatively charged lower electrode. When
there is a negatively charged object (like dust) in the way of the ions’ path to the
lower electrode, the ions are deflected inward, creating a positive potential in front
of the negatively charged object. The positive potential creates a small attractive
force between two negatively charged objects. That attractive force is one reason
why the wakefield may be responsible for the chain formation.
As part of studying these vertical chains, waves in these chains have previously
been studied by using a probe with a nanometer diameter to generate longitudinal
waves in the chains [10]. The probe has a metallic tip that can be given an electric
potential. By oscillating the potential between negative and positive, a wave can
be generated in the dust chains as the dust particles are alternately attracted to
and then repelled by the potential on the probe’s tip. For the longitudinal waves,
the probe is placed directly above the dust chain to cause the top particle to be
forced into the particle below, which in turn is repelled by the top particle and
forced into the particle below that one. This then creates a longitudinal oscillation.
To create the transverse waves, the same experimental setup and probe is used as
in [10], but the probe is placed to the side of the chain. The top particle is kicked
to the side, which disturbs the particle below it and creates a transverse oscillation
through the chain.
7
Chapter 3
Mathematical Background
3.1
Introduction
The mathematical equations for the physics in this paper are primarily used
to represent and calculate forces. There are a few main forces, and each force
will be described separately. There are also some other equations that are used,
and they will be described in the sections most related to them. In explaining the
equations, consistency will be attempted so that each variable only needs to be
explained once. The sign conventions followed have “up” in the positive direction
and “to the right” in the positive direction, and repulsive forces are taken to be
positive.
3.2
Gravity
The gravitational force is the usual one associated with Newton’s Second law:
Fg = md g, where md is the dust mass and g is the acceleration due to gravity.
This is one of the two most important forces for our chains. Because the chain is
modeled as one-dimensional, this force always pulls the dust particles downwards
towards the lower electrode. This is the strongest force that the particles must
balance against in order to levitate. Other equations derived from Newton’s Second law are the kinematic equations. The kinematic equation that is used most
8
frequently in the chain model is
1
y = y0 + v0 t + at2 ,
2
(3.1)
where t is time, a is the acceleration of the dust particle, v0 is the initial velocity
of the particle, and y0 is the initial position.
3.3
Electric Force
The electric force is the usual Fe = qE, where q is the charge on a dust
particle and E is the electric field at the location of the dust particle. This is
the second most important force for the particles. The electric field points into
the lower electrode, so the negatively charged particles are always repelled upward
by this force from the lower electrode. This is the strongest of the forces that
push the particles against gravity. There are other sources of electric force besides
the lower electrode. However, the forces from these other sources are calculated
without directly using this equation.
3.4
Neutral Drag
The neutral drag force is given by Fd = −βmd v, where β is the damping coefficient of the plasma, and v is the speed of the particle. As previously mentioned,
this force arises from random collisions of the dust with neutral gas particles in
the plasma. Because of the negative sign, this force always pushes in the opposite
direction of the particle motion, e.g., if a particle moves upward, the drag force
pushes downward.
9
3.5
Interparticle Repulsion
The interparticle repulsion force is a screened Coulomb force and is often called
a Yukawa force. As mentioned in an earlier section, the repulsion of dust particles
is screened by positive ions being attracted to the negatively charged dust and
surrounding them. Coulomb’s law is given by
F =
1 q 1 q2
,
4π0 r2
(3.2)
where 0 is the permittivity in a vacuum, q1 is the charge of one dust particle
affected by the force, q2 is the charge of the other dust particle affected by the
force, and r is the distance between the two particles. The Yukawa force is derived
from Poisson’s equation, but another way to look at the force is as follows. We can
model the screened nature of the force by putting a multiplier into Coulomb’s law
that causes the force to die off quickly with distance. The distance from the particle
can be characterized in terms of a specific distance called the Debye length, λD .
The multiplier will then need to depend both on distance and the Debye length.
− λr
In fact, the multiplier, e
D
, accomplishes these goals. The multiplier clearly goes
to zero as the distance from the particle increases, and it decreases to zero in
terms of the Debye length. Each multiple of the Debye length causes the factor to
become one more power of 1e , and thus, for each Debye-length distance from the
particle, the force falls off by 1e . This type of multiplier can be found in many other
equations in physics. For example, a similar factor is used in electromagnetism
for determining the attenuation of waves in materials, using a skin depth in the
multiplier instead of the Debye length.
10
Incorporating the multiplier above, the interparticle repulsion force then becomes
F =
1 q1 q2 − λr
e D.
4π0 r2
One further modification needs to be made. It is actually the Coulomb potential, not the Coulomb force, that is directly multiplied by the screening factor.
The Coulomb potential with this screening factor is given by
φ=
1 q1 q2 − λr
e D.
4π0 r
Note that the Coulomb potential decreases with
1
r
(3.3)
instead of
1
r2
as in (3.2).
Taking the negative of the derivative of (3.3) with respect to r will then produce
the Yukawa force that was being sought:
F =
1
1
+
r λD
1 q1 q2 − λr
e D.
4π0 r
(3.4)
Notice that a screening length of infinity means the screening occurs an infinite
distance away, and so there is no screening at all. When the Debye length goes to
infinity, the original Coulomb force law (3.2) is recovered because the exponential
factor goes to 1 and the quantity
1
λD
goes to zero in the limit. Notice also that
the sign of the force (3.4) depends on the signs of the charges. Because the dust
particles are negatively charged, the force between any two will be positive and
hence repulsive.
11
Chapter 4
Experimental Work
4.1
Introduction
This chapter describes the experiments that were run and the data obtained.
A discussion of the results follows. Each experiment is described in a separate
subsection. In these subsections, the oscillation experiments are discussed first
and then the model and its related experiments follow. Before describing the
experiments, the materials and equipment used in the experiments are explained.
4.2
Materials and Equipment
An rf powered GEC reference cell [4] is used for the experiments to ignite
and study dusty plasmas. In the cell’s vacuum chamber, the spacing between the
powered lower and grounded upper electrodes is 1.9 cm, and both electrodes are
8 cm in diameter. The probe is a position adjustable Zyvex S100 NanoEffecter
probe, and it is used to oscillate Melamine Formeldahyde (MF) particles levitated
in a vertical chain. The particles in the chain are oscillated in an open-ended
glass box that is flush with the lower electrode. In order to observe the particles,
they are illuminated by a laser and viewed with a high-speed camera. The chains
are formed by dropping particles from a dust dropper at approximately 10 W rf
plasma power, and the power is then slowly decreased as in [6] or [10]. According
to the manufacturer, the spherical particles have diameter 8.89 µm and density
12
1.514 g/cm3 . The glass box is 12 mm tall with identical 2 mm thick walls. The
inner separation between the walls is 10.5 mm, so the outer length of the walls is
10.5 mm + 2 mm + 2 mm = 14.5 mm. The probe tip is 50 nm in radius. The
probe is attached above the upper electrode, and it is positioned vertically as in
[6] or [10].
4.3
4.3.1
Method
Plasma Parameters
Resonance Frequencies
To calculate resonance frequencies, a chain is formed, and then the particles are
removed by adjusting the power until only one particle remains. The experimental
parameters are the following: 2.22 W system power, 80 mTorr neutral gas pressure,
-24 V natural DC bias, and 0 V probe bias. The probe is positioned inside the box
in a corner, and it drives the particle by oscillating its tip’s potential with a square
wave at 1 Hz with amplitudes going from 15 V to 70 V peak-to-peak. The particle
is pushed horizontally by the wave and then oscillates around an equilibrium point
before being pushed by the next crest or trough. The motion of the particle is
observed and recorded at 60 frames/second by a camera above the box. The
particle levitates near the center of the box and alternately moves horizontally
away from and towards the probe. An underdamped harmonic oscillator curve is
fit to the particle’s natural oscillatory motion, and the damping coefficient and
the damped frequency are found from the curve fit [10]. Resonance frequencies
are then calculated from the fitted damping coefficients and damped frequencies
13
using the following equation
ω0 =
p
ω2 + β 2,
where ω0 is the resonance frequency, β is the damping coefficient, and ω is the
damped frequency. The resulting resonance frequencies are then averaged together.
DC Bias
In this experiment, the height above the lower electrode for each particle in
a chain is measured as the DC bias is varied. The experimental conditions are
2.42 W and 80 mTorr with a probe bias of 0 V. The waves have an amplitude of
30 V peak-to-peak and a frequency of 7 Hz.
Sheath Edge
Following Samarian’s technique [11], the sheath edge is found by observing
where a single MF particle, 0.46 µm in diameter, levitates in the plasma for various
powers. The parameters are 80 mTorr with a natural DC bias of -25.5 V.
4.3.2
Dispersion Relations
The dispersion relations are found by fitting sine waves to the transverse oscillatory motion of two particles in a chain. The fitted curves are used to calculate
the phase difference between the motions of the two particles, and the phase difference is used to find the wavenumbers. These calculations are then done for
various frequencies. Two dispersion relations are found for two different chains.
Chain 1. The probe is placed outside the box and moved downwards so
that the tip is 1.51 mm down from the box’s top and 2.74 mm from the box’s
14
wall. The parameters used are: 2.14 W system power, 0.00421 W reverse power,
-21.5 V natural DC bias, 79 mTorr neutral gas pressure, and 0 V probe bias. A
4-particle chain is formed in the box and driven by a 50 V peak-to-peak sinusoidal
probe potential in frequencies from 1 Hz to 10 Hz. This creates a transverse wave
through the chain. The average distance of the top particle from the third particle
(the third particle down from the box’s top) and the frequencies from 4 Hz to
10 Hz are used to calculate a dispersion relation. Nearest neighbors are not used
in calculating the phase difference because their phase differences are too small.
The theoretical dispersion curve is modified from [12] to suit a vertical chain.
The dispersion relation is given by
s
ω(k) =
ω02
4Q2
− r
e λD
−
3
4π0 md r
r
kr
2
1+
sin
,
λD
2
(4.1)
where md is the mass of the dust (5.52 x 10−13 kg), λD is the Debye length
(700 micrometers [13]), ω0 is the resonance frequency (50.8 rad/s to fit the data),
k is the wavenumber, r is the mean distance between the particles throughout all
frequencies (6.956 x 10−4 m), and Q is the charge on all three particles (50590e to
fit the data, where e is the charge of an electron).
Chain 2. The probe’s tip is placed 4.79 mm above the box and is put
0.77061 mm from the box’s wall. The parameters are: 2.16 W system power,
0.00134 W reverse power, -19.9 V natural DC bias, 81 mTorr neutral gas pressure,
and 0 V probe bias. A 5-particle chain is formed and transverse waves are driven
through the chain by a sine wave of 40 V amplitude at frequencies of 5.0 Hz to
7.5 Hz. These frequencies and the average distance between the top particle and
the next one underneath it are used to calculate a dispersion relation. The second
particle from the top was more accurate in finding the phase difference because
the lower particles had less exact sinusoidal motion. These parameters produced
15
an excellent transverse wave through the chain. However, no theoretical curve has
been fit to the data.
4.3.3
Model
To find the charge on each of the particles in a chain, a force balance is created
for the particles when they are in their equilibrium positions. Because the particles
are not moving, the only forces that will be considered are the gravitational, the
interparticle repulsion, and the total electric force from the lower electrode and
glass box walls. The electric force is found by a free fall experiment.
Once the charges are calculated, the motions of the particles in the chain are
simulated in both the horizontal and vertical directions. There are four different
“modes” of oscillation: longitudinal, symmetric, standing, and transverse. The
data recorded for these modes occurred for different plasma parameters, and the
particles in the chains for each mode varied. Hence, each mode must be treated
as a separate situation with different plasma parameters, and so there are four
different force balances to calculate. The parameters relevant to the chains for
each mode are shown below in Table 4.1.
Table 4.1: Parameters of Chains for each Mode
Mode
Probe Frequency
(Hz)
Antisymmetric
6
Symmetric
9
Standing
9
Transverse
7
Probe Amplitude
(V peak-to-peak)
5
14
5
40
Plasma Power DC Bias
(W)
(V)
2.37
-21
3.01
-20
3.94
-21
2.42
-24
Pressure
(mTorr)
43
80
150
80
Free Fall Experiment
A single particle is dropped into the glass box as in [14]. The particle’s motion
is observed by a camera, and a line is fit to its motion to find the particle’s
16
acceleration. The forces acting on the particle are the gravitational force, the
electric force, and the neutral drag. Using the observed particle acceleration,
Newton’s Second Law is used to solve for the electric force Fe according to
md z̈ = Qd E − βmd ż − md g,
where z is the height above the lower electrode, Qd E = Fe , and the dot above a
variable indicates a time derivative.
Force Balance
Once the forces are calculated, a system of equations is generated by setting
the sum of all the forces on each particle equal to zero. Calculating the force
balance takes some effort, and so it will be described in three parts. The following
naming convention will be used for particles in a chain. The particle that is highest
above the lower electrode, also called the top particle, is called “Particle 1.” The
particles in the chain are then labeled in increasing increments, e.g., “Particle 2”
for the next highest and “Particle 3” for the one just under “Particle 2.” For a
chain with n particles, the particle nearest to the lower electrode is “Particle n,”
which is also called the bottom particle. The order of the system of equations
will be listed in the same manner, with the top equation being the force sum for
“Particle 1” and the bottom equation being for “Particle n”.
In order to see how the charge is calculated, it is useful to look at the case of
n = 3 particles. For simplicity, let G = Fe − Fg . Then the system of equations
17
reads:
F12 (q1 , q2 , λD1 ) + G = 0
F23 (q2 , q3 , λD2 ) − F12 (q1 , q2 , λD2 ) + G = 0
(4.2)
−F23 (q2 , q3 , λD3 ) + G = 0,
where the dependence of the interparticle repulsion forces Fij on charge and Debye
length has been made explicit. This system of equations cannot be solved because
there are six unknowns (q1 , q2 , q3 , λD1 , λD2 , λD3 ) and only three equations. The
average Debye lengths for the longitudinal motion are found experimentally by
fitting the theoretical dispersion relation curve (4.1) to the experimentally found
dispersion relations. Each average Debye length for the longitudinal modes is found
to have the same value: 1000 micrometers. Because the transverse motion occurs
under plasma conditions that overlap the plasma conditions of the longitudinal
motions, its average Debye length is also set to 1000 micrometers in the model.
Since these are average Debye lengths, they can be assumed to hold for each
particle in the chain. For each “mode,” the average Debye length can then be
plugged into the system of equations (4.2) to eliminate the dependence of the
forces on Debye lengths.
The new system of equations is:
F12 (q1 , q2 ) + G = 0
F23 (q2 , q3 ) − F12 (q1 , q2 ) + G = 0
(4.3)
−F23 (q2 , q3 ) + G = 0.
Starting with the bottom equation and solving upwards produces F23 = G and
F12 = 2G. However, the top equation shows that F12 = −G, which means that
the only consistent solution requires everything to be equal to zero. This is clearly
18
not the case. In order to avoid this situation, a different electric field is chosen for
the second-to-bottom particle down to the lower electrode. The chosen field is one
that linearly decreases to the lower electrode because a linear field has been found
by experiments without a glass box [15]. To create this sort of linear field, the field
at the lower electrode must be known. To find the field at the lower electrode, the
two electrodes are modeled as a parallel plate capacitor. Although the electrodes
are circular and finite, this modeling is worth attempting. The electric field on
the lower electrode is then given by
Ef =
σ
,
20
where σ denotes the charge density on the lower electrode. The charge density on
the lower electrode is not known, so it must be found indirectly. To do this, first
find the charge Q on the lower electrode by Q = CV , where C is the electrode’s
capacitance and V is the potential difference between the lower electrode and the
upper electrode. Charge density is then given by
σ=
Q
CV
=
,
A
A
where A is the area of the lower electrode. However, for a parallel plate capacitor,
C=
0 A
,
d
where d is the distance between the plates. Therefore,
σ=
CV
0 A V
0 V
=
=
.
A
d A
d
19
Hence, the electric field is given by
Ef =
σ
(0 V ) /d
V
=
= .
20
20
2d
Finally, recall that these particles are levitating in the sheath of the lower
electrode. The sheath is created by the plasma interacting with the lower electrode
surface, losing electrons to the surface and creating a space of positive charge.
There is also a sheath at the upper electrode. Hence, the sheaths create two
capacitors: there is one capacitor formed between the upper electrode and the
plasma, and there is another one between the lower electrode and the plasma.
Because there is much controversy over the sheath, the location of its edge, and
its definition, the other surface of the capacitor formed with the lower electrode is
simply placed midway between the upper and lower electrode, i.e., at the midpoint
of the plasma. This means that d = 19 mm/2 = 9.5 mm, and V is the potential
difference between the plasma and the lower electrode. Recalling that the lower
electrode has a DC bias that determines its potential, this means V is the potential
of the DC bias with respect to the potential of the plasma. The plasma potential
is then found using the same method as in [16].
To form a linearly decreasing electric force, the following coordinate system
is used. The y coordinate is the height above the lower electrode with hi being
the height of the ith particle above the lower electrode. Hence, the electrode is at
y = 0, and Particle 2 is at y = h2 . Because it is the field, not the force, which
linearly decreases to the lower electrode, the field for Particle 2 must be calculated
by dividing the electric force at Particle 2 by the charge of Particle 2. The linear
equation of the electric field in the point slope form is E − Ef = m(y − 0), where
20
the slope of the line can be found by
4y
m=
=
4x
Fe
q2
− Ef
h2
=
Fe
Ef
−
.
q2 h2
h2
(4.4)
To get the force, the electric field E must be multiplied by the charge of the
particle. Hence, the force (on the ith particle) is given by
0
Fe = qi (mhi + Ef ) .
(4.5)
0
Plugging the force Fe into the system of equations (4.3) and then defining
0
0
G = Fe − Fg for ease of writing gives
F12 (q1 , q2 ) + G = 0
F23 (q2 , q3 ) − F12 (q1 , q2 ) + G = 0
(4.6)
0
−F23 (q2 , q3 ) + G (q2 , q3 ) = 0.
Solving from the top gives F23 = −2G as before. Writing out the force in terms
of charge results in
F23 = q2 q3 K23 ,
where K23 is the rest of the Yukawa force
K23 =
1
1
+
r23 λD
r
1
− 23
e λD .
4π0 r23
The charge of the second particle can then be solved in terms of the charge of
the third particle by setting the equations equal to each other to get
q2 = −
2G
.
q3 K23
21
(4.7)
0
0
0
By the third equation in (4.6), G = Fe − Fg = F23 = −2G, so Fe = −2G + Fg .
0
Plugging in the linear equation (4.5) for Fe results in
0
Fe = q3 (mh3 + Ef ) = −2G + Fg .
(4.8)
0
Note that because Fe is the electric force at the bottom particle’s height, the
height and charge of the third particle are used in (4.8). Also note that (4.8)
cannot be solved for the charge of the third particle yet because the slope, m,
depends on the charge of Particle 2, as seen in (4.4). Using the expression (4.7) of
q2 in terms of q3 gets rid of this dependency. This results in
Ef
Fe
Fe
m=
−
=
q2 h2
h2
h2
q3 K23
Ef
.
−
−
2G
h2
The equation (4.8) can now be solved for q3 , giving
q3
Ef
Fe K23
q3 −
−
2Gh2
h2
h3 + Ef
= −2G + Fg .
Moving the right hand side over to the left hand side and multiplying through
by q3 gives a quadratic equation in q3 :
Aq32 + Bq3 + C = 0,
e K23
where A = − F2Gh
h3 , B = Ef −
2
Ef
h,
h2 3
and C = 2G − Fg = 2Fe − 3Fg .
The charge of the third particle can then be found by the quadratic formula.
The charges of the rest of the particles can be found by back-substituting each
previously found charge into the system of equations. This method for calculating
22
the bottom particle of a chain generalizes to a chain with n particles as
Aqn2 + Bqn + C = 0,
with A =
Fe Kn−1,n
h ,
(n−1)(Fg −Fe )hn−1 n
B = Ef 1 −
hn
hn−1
, and C = (n−1)Fe −nFg . The full
code for calculating the charge in the antisymmetric case is given in Appendix B.
Simulation of Longitudinal Motion
Once the force balance is found, the motion of the particles in each chain
is simulated in three ways. The first way is to let the simulation run without
touching the particles in order to see if the force balance is stable. The second
way is to displace the top particle from equilibrium and observe the resulting
motion. The third way is to add an oscillatory force. In the case of perturbing
the chain by displacing the top particle, the top particle is moved 1 mm above its
starting equilibrium position. In the case of the oscillations, the oscillatory force
is a sinusoid as follows:
−
q
2.5 × 10−3 sin(2π · 9t).
0.003
(4.9)
The intention of this equation is to mimic the force of the probe. The oscillatory
part is the potential, while the fraction results from taking the negative gradient
and multiplying it by q. The 0.003 denotes the fact that the probe tip was intended
to be placed 3 mm from the particles, but the equation is clearly mistaken because
the force should be divided by the square of the probe tip’s distance from the
particles. Further, the equation is too idealized because the tip is actually a
different distance from each particle. However, the goal was merely to put a
working oscillatory force into the Matlab script to find preliminary results.
23
The longitudinal (vertical) motion is calculated by the kinematics equation
(3.1) in a loop. For each time step, the forces on the particles are calculated. The
particles’ accelerations are then found by setting the sum of the forces on them
equal to their mass times acceleration. The accelerations are then plugged into the
kinematics equation (3.1), and the positions of the particles are recorded. Each
case of wave motion is run for 10 sec, except the transverse motion case; it is run
for 50 sec in both the equilibrium and perturbed motion cases and for 10 sec in
the oscillatory force case. The full code for calculating the longitudinal motion in
the antisymmetric case is given in Appendix A.
Simulation of Transverse Motion
Because of a lack of data, little was done to simulate the transverse (horizontal)
motion of the particles. However, a working script was made for this situation. In
the script, we neglect all vertical motion, and we assume the horizontal electric
force from the box walls acts like a spring. Hence, the electric force is modeled by
Hooke’s law, F = −kx, where k is the spring constant and x is the displacement
from equilibrium. This model is assumed to work because the particles are motionless in their initial positions but should be repelled by the box walls once they
leave their initial positions. The full code for calculating the transverse motion in
the antisymmetric case is given in Appendix C.
24
4.4
4.4.1
Results
Plasma Parameters
Resonance Frequencies
The average resonance frequency was found to be 40.19 rad/s (6.40 Hz). The
average damping coefficient β was 8.79 Hz. The results of the underdamped fit are
shown in Figures 4.1-4.4. Figure 4.1 shows the data for the damping coefficient,
the damped frequency, and the resonance frequency. These values are plotted
against the peak-to-peak amplitude of the probe. Lines are fit to the data and are
also shown on the figure.
Figure 4.1: Underdamped Data vs Probe’s peak-to-peak amplitude
Figure 4.2 shows the particle’s peak-to-peak amplitude versus the probe’s peakto-peak amplitude applied to the particle, and the figure has a fit to the points
that follow a linear trend. A vertical dashed line separates the data that does
25
follow a linear trend from the outliers that do not follow a linear trend. The peakto-peak amplitude of the particle’s motion was measured between the first crest
and trough in the particle’s fitted underdamped motion. The particle’s amplitude
was measured for each corresponding probe amplitude.
Figure 4.2: Particle’s amplitude vs Probe’s amplitude
Figure 4.3 shows the distance of the moving particle from the probe’s tip. The
particle is periodically kicked by the wave from the probe and briefly oscillates
in an underdamped fashion. Recall that the probe perturbs the particle with a
square wave, so the underdamped oscillations between crests and troughs are not
driven but natural.
Figure 4.4 shows one of the underdamped curves fitted to the motion of the
particle. As seen in Figure 4.3, there can be more than one underdamped motion
in the course of a single experiment for the particle. That is the reason for multiple
data points at many of the amplitude values in Figure 4.2.
26
Figure 4.3: Particle’s distance from Probe’s tip vs Time
DC Bias
The particles’ heights in a chain are plotted against a changing DC bias in
Figure 4.5. The particles tend to move slowly downwards as the DC bias becomes
more positive.
Figure 4.4: Underdamped curve fit
27
Figure 4.5: Particles’ heights vs DC bias
Figure 4.6: Sheath edge height vs System power
28
Sheath Edge
The height of the sheath edge from the lower electrode is plotted for varying
system powers in Figure 4.6. The curve exhibits a maximum near 3 W and may
approach a height of around 8.5 mm as the system power is increased.
4.4.2
Dispersion Relations
Chain 1. In Figure 4.7, the data points start at the arrow labeled “start” and
follow the arrows. The dashed quadratic curve fits the first four points. The solid
curve is the theoretically predicted curve (4.1) that fits the next two points. The
dotted line connects the last three points.
Figure 4.7: Dispersion relation 4-particle chain
29
The line and the quadratic are there to suppose that there are different wave
motions that require different dispersion theory at those wavenumbers; these
curves show there are definite trends in their regions.
Figure 4.8: Dispersion relation 5-particle chain
Chain 2. No theoretical curve has been fit for Figure 4.8 yet. There was a
lack of time to fit the horizontal dispersion relation (4.1) to the data.
30
4.4.3
Model
The charges calculated were fairly high for all the modes and are shown in
Table 4.2. The symmetric mode did not have a stable configuration for all particles
and had a negative electric force. All the other modes had a positive electric force,
as seen in Table 4.2. The charge could not be calculated for the symmetric mode,
so the charge shown in the table below for that mode was calculated by ignoring
the bottom, 4th particle, treating it as a free parameter. The charges required for
the other particles to balance, i.e., the charges that made their force-sums zero,
were calculated by a self-consistent approach. That is, a value of charge for a
particle was placed into the model and the resulting forces calculated. The charge
was then changed to make the forces smaller. This was repeated in an algorithmic
manner for each particle until the force sum was as small as desired.
Table 4.2: Results of Capacitor Model
Mode
Particle 1
Particle 2
Particle 3 Particle 4
Particle 5
Antisymmetric
Symmetric
Standing
Transverse
-90490e
-168759e
-784104e
-399616e
-359604e
-150001e
-44707e
-21208e
-376649e
-255994e
-305780e
-546960e
N/A
N/A
-229471e
-618585e
N/A
-208943e
-33551e
-36670e
Electric Force
(N)
2.2050e-13
-1.4358e-12
2.2144e-12
7.1849e-13
The results of the model for the motions of the particles are in the following
Figures 4.9-4.19. For the simulation of their motion without perturbation, the
particles oscillate in a damped fashion until they settle on an equilibrium position
(Figures 4.9-4.10). Because the symmetric mode had no force balance, it had no
stable motion for this case, and so it is not shown.
The particles for the transverse motion settings exhibit different behavior from
the other motions. The transverse particles oscillate in a jagged fashion and are
still for a while, but they eventually jump to and settle in a new equilibrium
31
position as seen in Figure 4.11.
Figure 4.9: Particle 1, antisymmetric mode without perturbation
Figure 4.10: Particle 1, standing mode without perturbation
In order to see the jagged motion better, Figure 4.12 shows a close-up view of
the transverse motion.
32
Figure 4.11: Particle 1, transverse motion without perturbation
Figure 4.12: Close-up view of oscillations between 6.5 sec and 10.5
sec
33
Figure 4.13: Particle 1, antisymmetric mode with perturbation
For the simulation of the particles’ motions resulting from displacing the top
particle, the motions are qualitatively the same for all settings (see Figures 4.134.15). For each setting, the particles oscillate until they reach their original equilibrium positions prior to perturbation. As before, the symmetric mode had no
stable motion for this situation and so is not shown.
34
Figure 4.14: Particle 1, standing mode with perturbation
Figure 4.15: Particle 1, transverse motion with perturbation
For the simulation of the particles’ motions with an oscillatory force, they
oscillate essentially sinusoidally (see Figures 4.16-4.17).
35
Figure 4.16: Particle 1, antisymmetric mode with oscillatory force
Figure 4.17: Particle 1, standing mode with oscillatory force
36
Despite not having charges that permitted a force balance, the particles in the
symmetric case also oscillated essentially sinusoidally as shown in Figure 4.18.
Figure 4.18: Particle 1, symmetric mode with oscillatory force
As in all the previous figures, the particles’ motions in the transverse case differ
from the rest. The particles oscillate, but they oscillate in a jagged fashion rather
than as a sine wave (Figure 4.19).
37
Figure 4.19: Particle 1, transverse motion with oscillatory force
Figure 4.20: Electric field for antisymmetric mode
38
The electric fields for all but the symmetric case are shown in Figures 4.20-4.22.
The electric field is found by dividing the force on each particle by the calculated
charges of the particles. Because no calculated charge for the symmetric case could
be found, the electric field for that mode could not be found.
Figure 4.21: Electric field for standing mode
Figure 4.22: Electric field for transverse motion case
39
4.5
4.5.1
Discussion
Plasma Parameters
Resonance Frequencies
The frequencies and resonance frequencies resulting from the fitted curves are
nearly constant (Figure 4.1). The average resonance frequency makes sense because the best transverse waves through the chain were found between 5 Hz and
7 Hz. The average value of 8.79 Hz for β is close to the expected value of 8 Hz.
However, β shows some decrease with increasing amplitude; although, it is expected to be constant. This could be due to a smaller camera resolution that did
not capture the more quickly moving particle at those probe amplitudes. However, the graph of the particle’s oscillation amplitude (Figure 4.2) suggests that
at higher amplitudes the assumption of underdamped oscillation no longer holds
because the particle’s oscillation amplitude no longer increases with increasing
probe amplitude. The probe at high potentials is probably changing the plasma
conditions by absorbing many electrons. There could also be higher order terms
at those amplitudes from the horizontal confinement. Ignoring the data at those
higher amplitudes gives an average β of 9.22 Hz, which is higher than expected
but close enough to 8 Hz to be within experimental error. Because the resonance
frequency experienced little change with increasing amplitude, the value for it can
be trusted even at high peak-to-peak probe amplitudes.
DC Bias
Figure 4.5 shows that negative DC biases move the chain upwards while positive
DC biases move the chain downwards—just as expected. The bump in the data
that occurs where the chain loses a particle probably happened because arcing
40
occurred while collecting the data. The data was taken at -24 V and then taken
in steps towards -30 V. Arcing occurred at -30 V and a particle was lost, so the
plasma conditions were probably different for the last measurements going to 10 V.
Future experiments can fill the large data gap so that a trend can be seen better.
Sheath Edge
The sheath edge increases, reaches a maximum, and then decreases to some
equilibrium value as power increases (Figure 4.6). A fourth order polynomial is
the smallest degree that fits the sheath edge plot. Notice that, by comparing the
Figure 4.6 to the plasma powers in Table 4.1, the sheath edge height confirms
the use of 9.5 mm in the parallel plate capacitor model. Despite the sheath edge
pressure being at 80 mTorr, all the modes fit nicely with this plot except for the
symmetric mode, which was at 80 mTorr.
4.5.2
Dispersion Relations
Chain 1. The theoretical curve predicts a resonance frequency greater than
the one found by fitting the underdamped curve to the square wave oscillations
(40.19 rad/s). Taking data at a higher frame rate could produce higher resolution
for fitting the curve. The theoretical curve also does not take into account density,
charge, Debye length, and electron temperature change in the vertical direction.
These could lower the predicted curve enough so that it would match with the
earlier found resonance frequency.
Future work could find fits for the other parts of the dispersion relation. The
variance in the dispersion relation suggests that there are three different wave
motions going on through the chain as the frequency changes. Also, the imaginary
parts of the dispersion relation could be examined.
41
Chain 2. The dispersion relation for the 5-particle chain (Figure 4.8) is similar
to that of the 4-particle chain at lower frequencies (Figure 4.7), which confirms
the existence of this dispersion relation.
4.5.3
Model
The electric force found by free fall was negative in the case of the symmetric
mode. This was unexpected and is the reason why no force balance can be found
for particles in that mode. The negative electric force does not appear to result
from poor calculation of the data in the experiment; more than one attempt was
made to recalculate the electric force from the data but the result was still negative
and about the same magnitude.
All the vertical motions for each mode are fairly expected, but the transverse
mode produced different results in all cases. The jagged oscillations and the temporary stability that jumps to a final jagged stability are very different behaviors
from the other modes. It cannot be a result of plasma conditions or the number
of particles because the standing mode had roughly the same conditions and the
exact same number of particles. Note also that the other modes produced similar
results to each other.
The horizontal motions resulted in the particles springing back and forth to
infinity, so further work needs to be done on it. It is possible that the instability
results from the value of the spring constant being wrong. An oscillation experiment should be performed in order to determine the spring constant accurately
for each mode.
The model produced higher charges than expected. One possible reason for
this is that the parallel plate capacitor model is not accurate enough. There may
be a stronger electric field in the sheath region of the plasma, or there may be
some other way in which the electric field behaves in a manner very different from
42
a parallel plate capacitor. The electrodes are circular and finite instead of planar
and infinite, and this is a probable cause for some of the inaccuracies. There may
also be edge effects, or there may be irregularities introduced by the fact that
there is plasma between the “plates” of the capacitor. Further work is needed to
see whether other factors may allow for an accurate calculation of charges. Once
the charges are accurately calculated, the electric field at each of the particles’
positions can be accurately calculated. Because of this inaccuracy, no attempt
was made to make the oscillatory force more realistic.
43
Chapter 5
Conclusion
There is much to learn about vertical dust chains and their formation. The
exact charge on the dust particles in the chains is still unknown, and it is wondered
whether the ion wakefield is responsible for the formation of these chains. However,
it can be difficult studying these chains because direct, physical measurements can
significantly disturb the plasma. Our approach in this thesis extracts information
about the chains by perturbing them with oscillations and by simulating their
motion.
The measurement of the resonance frequencies shows that the resonance frequency stays mostly constant. The measurement of the amplitude of particle
oscillation shows that the underdamped fit to the particle oscillation no longer
holds for probe amplitudes greater than 40 V peak-to-peak. Despite this, the resonance frequency is mostly constant in that region of probe amplitude. Unlike the
resonance frequency, the damping coefficient does change with the probe’s amplitude, so the coefficient data for amplitudes on the probe above 40 V peak-to-peak
might not be accurate. However, if that data is ignored, the average damping
coefficient is larger than expected.
The DC bias moved the chain in the expected manner. A negative DC bias
moved the chain upwards, and a positive DC bias moved the chain downwards.
The sheath edge changed its height with power, and the change in the edge’s
height with respect to power could be fit by a fourth order polynomial. Despite
being found at a different pressure from most of the modes, the sheath edge height
44
also confirmed the use of 9.5 mm in the capacitor model for all but the symmetric
mode. However, more work should be done to check the quantitative accuracy of
these measurements.
The dispersion relations show that there are at least three kinds of wave motion
going through the chain and that the theoretical curve fits the transverse motion
after it starts damping with increased frequency. The theoretical curve could
be greatly improved with more data and by taking into account more variables.
Theoretical curves still need to be found for the other wave motions.
The model shows that while reasonable results for motion can be found by this
method of free fall (except for the symmetric mode) and simulation, the parallel
plate capacitor method is not accurate enough to estimate the charge on the dust
particles. Because the electric force on the particles above the bottom one is
the same (and by experiment, is the same for the bottom one too), the charge
distribution does not follow any particular pattern. One would expect the charges
to decrease from top to bottom. Instead, the charges tend to alternate between
extremely high and extremely low values.
It is possible that the electric force is not actually constant and that the free
fall experiment needs to be run again more carefully. However, even if that is the
case, it would seem that the electric field needs to be stronger in order to decrease
the charge on the particles in the chain. The experimentally calculated electric
forces are smaller than the gravitational force on the particles.
It is also possible that some other electric field better models the situation.
One possible alternative is an unusual electric field that may be explained by a
5th order polynomial as in [17]. If the particles are in a particular region where
the electric field curves as a polynomial, the results for charge might be different,
and the negative electric force might be explainable.
One other potential inaccuracy in the model is that the charge is assumed to be
45
constant on all the particles. It is possible that the particles change charge quickly
when moving, and this might result in a less accurate model. Until a more accurate
model is developed, the question of whether the ion wakefield is responsible for
the chain formation cannot be answered by this simulation method.
46
Appendix A
Antisymmetric Mode Particle
Vertical Motion Code
%
%
%
%
equilibrium charges found by Raymond Fowler
numerical framework by Raymond Fowler
reduction in step size and addition of drag by Brandon Harris
flags and reporting established by Brandon Harris
% this program plots particles in vertical chain
% using: gravity, electric field, particle repulsion
% initally assuming infinite transverse potential
clear all; close all;
load('ResultsPositionsYvsT.mat')
% report step results? (1 for yes, 0 for no)
report = 0;
mode = 1;
beta = 8.65;
% average terminal velocity (in m/s)
% termv = (1.1640 + 0.9411257 + 1.1586)/3;
epsilon0 = 8.85e-12;
linewidthsize = 2.0;
% debye length
debye2(1) = (1000)*1e-6;
debye2(2) = (1000)*1e-6;
debye2(3) = (1000)*1e-6;
% meters per pixel
scale = 0.000011988011988;
% box top
top = 1001;
% mass of dust
diameter = 8.89e-6;
density = 1500; % kg/mˆ3
volume = 4/3*3.1415926535*(diameter/2)ˆ3; % mˆ3
dustmass = density*volume; % kilograms
47
% force of gravity
gforce = -dustmass*9.81;
% charge on dust
echarge = 1.6e-19;
%Linear E field Balance (From CalcQAnti.m)
q(1) = -90490*echarge;
q(2) = -359604*echarge;
q(3) = -376649*echarge;
boxdomainidx = (1:top)';
boxdomain = linspace(1.1988011988e-05,12*1e-3,top)';
sizeboxdomain = size(boxdomain,1);
onesboxdomain = ones(sizeboxdomain,1);
% import workspace file for chain particle equilibrium positions
load('AntiSymPositions.mat')
%Different scale for the june 28 transverse chain, and it was ...
never erased.
%So instead, the scalejune28 was set to 1.
scalejune28 = 1;
partpos = partpos*1e-3; %partpos in mm; convert to m
partposscale = partpos*scalejune28;
% partposscale(1) = partposscale(1) + 1e-3; %Use to start top ...
particle at non-equil position
% calculate separations between actual particles in chain
for j = 1:length(q)-1
sep(j) = partposscale(j)-partposscale(j+1);
end
% plot forces
figure;
hold on;
% plot gravity force
plot(boxdomain*1000,gforce*onesboxdomain,'Color',[0 0.6 0],'...
LineWidth',linewidthsize);
% Get ready to plot the electric force
Eforce = zeros(sizeboxdomain-1,length(q));
domainstep = boxdomain(2)-boxdomain(1);
%Plasma Potential
V = 38;
Ef = -V/(2*9.5e-3); %Electric field on lower electrode
m = ((eforce(mode)/q(length(q)-1)) - Ef)/partposscale(length(q)-1)...
; %Slope
for j = 1:length(q)
clear('EforceNew');
EforceNew = eforce(mode)*onesboxdomain;
48
if j == length(q)
EforceNew = q(length(q))*(m*partposscale(length(q)) + Ef)*...
onesboxdomain;
end
plot((boxdomain(1:sizeboxdomain)+domainstep/2)*1000,EforceNew,...
'k','LineWidth',linewidthsize);
Eforce(boxdomainidx(1:sizeboxdomain),j) = EforceNew;
end
% plot the interparticle repulsion force
% top repulsion only up
j = 1;
repulsionF(j) = (1/debye2(j)+1/sep(j)).*(1./(4*pi*epsilon0*sep(j))...
)*q(j)*q(j+1).*...
exp(-sep(j)/debye2(j));
plot(boxdomain*1000,repulsionF(j)*onesboxdomain,'c','LineWidth',...
linewidthsize);
for j = 2:length(q)-1
uprepulsionF(j) = (1/debye2(j)+1/sep(j)).*(1./(4*pi*epsilon0*...
sep(j)))*q(j)*q(j+1).*...
exp(-sep(j)/debye2(j));
downrepulsionF(j) = (1/debye2(j)+1/sep(j-1)).*(1./(4*pi*...
epsilon0*sep(j-1)))*q(j-1)*q(j).*...
exp(-sep(j-1)/debye2(j));
repulsionF(j) = uprepulsionF(j)-downrepulsionF(j);
plot(boxdomain*1000,repulsionF(j)*onesboxdomain,'c','LineWidth...
',linewidthsize);
end
% bottom repulsion down
j = length(q);
repulsionF(j) = -(1/debye2(j)+1/sep(j-1)).*(1./(4*pi*epsilon0*sep(...
j-1)))*q(j-1)*q(j).*...
exp(-sep(j-1)/debye2(j));
plot(boxdomain*1000,repulsionF(j)*onesboxdomain,'c','LineWidth',...
linewidthsize);
% plot sum of forces
for j=1:length(q)
forcesum(j) = Eforce(sizeboxdomain,j)+gforce+repulsionF(j);
plot((boxdomain(1:(sizeboxdomain))+domainstep/2)*1000,...
Eforce(:,j)+gforce+repulsionF(j),'r','LineWidth',...
linewidthsize);
end
forcesum
% plot where particles are from data
range = get(gca,'Ylim');
rangesize = size(range,2);
onesrange = ones(rangesize,1);
for j = 1:length(q)
49
plot(partposscale(j)*onesrange*1000,range,'b--','LineWidth',...
linewidthsize)
end
axis([6.0 12 range(1) range(2)]);
xlabel('Distance from lower electrode (mm)','FontName','Arial','...
FontSize',14);
ylabel('Force (N)','FontName','Arial','FontSize',14);
hold off;
v = zeros(length(q),1);
y = partposscale;
step = 100000;
handle = figure;
yMatrix = y;
stepamount = 0.0001;
t = 0:stepamount:step*stepamount;
dt = stepamount;
for i = 1:(step + 1)
yMatrix(:,i) = y;
a = [(forcesum/dustmass)-beta*v']';
y = y + v*dt + 0.5*a*dtˆ2;
if report
y
end
v = v + a*dt;
Eforce(:,length(q)) = q(length(q))*(m*y(length(q)) + Ef);
% calculate separations between actual particles in chain
for j = 1:length(q)-1
sep(j) = y(j)-y(j+1);
if report
sep
end
end
% Calculate the interparticle repulsion force
% top repulsion only up
j = 1;
repulsionF(j) = (1/debye2(j)+1/sep(j)).*(1./(4*pi*epsilon0*sep...
(j)))*q(j)*q(j+1).*...
exp(-sep(j)/debye2(j));
for j = 2:length(q)-1
uprepulsionF(j) = (1/debye2(j)+1/sep(j)).*(1./(4*pi*...
epsilon0*sep(j)))*q(j)*q(j+1).*...
exp(-sep(j)/debye2(j));
50
downrepulsionF(j) = (1/debye2(j)+1/sep(j-1)).*(1./(4*pi*...
epsilon0*sep(j-1)))*q(j-1)*q(j).*...
exp(-sep(j-1)/debye2(j));
repulsionF(j) = uprepulsionF(j)-downrepulsionF(j);
end
% bottom repulsion down
j = length(q);
repulsionF(j) = -(1/debye2(j)+1/sep(j-1)).*(1./(4*pi*epsilon0*...
sep(j-1)))*q(j-1)*q(j).*...
exp(-sep(j-1)/debye2(j));
oscil = -q.*(2.5e-3*sin(2*pi*9*t(i))/0.003); %Oscilatory force
% Calculate sum of forces
%
for j=1:length(q)
forcesum(j) = Eforce(sizeboxdomain,j)+gforce+repulsionF(j)...
+ oscil(j);
forcesum(j) = Eforce(sizeboxdomain,j)+gforce+repulsionF(...
j); %Wtihout oscilatory force
end
if report
forcesum
v
a
end
% plot where particles are from data
range = get(gca,'Ylim');
rangesize = size(range,2);
onesrange = ones(rangesize,1);
clf(handle);
hold on;
if i == (step + 1)
% plot where particles are from data
range = get(gca,'Ylim');
rangesize = size(range,2);
onesrange = ones(rangesize,1);
for j = 1:(length(q))
%Use with linear Eforce.
plot(partpos(j)*scalejune28*onesrange*1000,range,'k.-'...
,'LineWidth',linewidthsize)
end
end
hold off;
if ~mod(i,step/100)
fprintf('Step Number: %d %s\n',i,datestr(now));
end
end
51
Appendix B
Antisymmetric Mode Calculation
of Charge Code
% this program calculates the charge on each particle in a ...
vertical chain
% given the forces required to balance them and using a nearest ...
neighbor
% Yukawa interparticle repulsion force.
clear all; close all;
load('ResultsPositionsYvsT.mat')
mode = 1;
beta = 8.65;
epsilon0 = 8.85e-12;
linewidthsize = 2.0;
% debye length
debye2(1) = (1000)*1e-6;
debye2(2) = (1000)*1e-6;
debye2(3) = (1000)*1e-6;
% meters per pixel
scale = 0.000011988011988;
% box top
top = 1001;
% mass of dust
diameter = 8.89e-6;
density = 1500; % kg/mˆ3
volume = 4/3*3.1415926535*(diameter/2)ˆ3; % mˆ3
dustmass = density*volume; % kilograms
% force of gravity
gforce = dustmass*9.81;
% charge on dust
echarge = 1.6e-19;
q(1) = -21000*echarge;
q(2) = -70000*echarge;
q(3) = -76000*echarge;
52
boxdomainidx = (1:top)';
boxdomain = linspace(1.1988011988e-05,12*1e-3,top)';
sizeboxdomain = size(boxdomain,1);
onesboxdomain = ones(sizeboxdomain,1);
% import workspace file for chain particle equilibrium positions
load('AntiSymPositions.mat')
%Different scale for the june 28 transverse chain, and it was ...
never erased.
%So instead, the scalejune28 was set to 1.
scalejune28 = 1;
partpos = partpos*1e-3; %partpos in mm; convert to m
partposscale = partpos*scalejune28;
% calculate separations between actual particles in chain
for j = 1:length(q)-1
sep(j) = partposscale(j)-partposscale(j+1);
end
%electric force
Eforce = zeros(sizeboxdomain-1,length(q));
domainstep = boxdomain(2)-boxdomain(1);
%Plasma Potential
V = 38;
Efield = -V/(2*9.5e-3);
m = ((eforce(mode)/q(length(q)-1)) - Efield)/partposscale(length(q...
)-1); %Slope
for j = 1:length(q)
clear('EforceNew');
EforceNew = eforce(mode)*onesboxdomain;
Eforce(boxdomainidx(1:sizeboxdomain),j) = EforceNew;
end
j = length(q);
K = (1/debye2(j)+1/sep(j-1)).*(1./(4*pi*epsilon0*sep(j-1))).*exp(-...
sep(j-1)/debye2(j));
a = (eforce(mode)*partposscale(length(q))*K)/((j-1)*partposscale(...
length(q)-1)*(gforce - eforce(mode)));
b = Efield - Efield*(partposscale(length(q))/partposscale(length(q...
)-1));
c = -(j*gforce - (j-1)*eforce(mode));
qpos(j)
qneg(j)
qpos(j)
qneg(j)
=
=
=
=
(-b + sqrt(bˆ2 - 4*a*c))/(2*a);
(-b - sqrt(bˆ2 - 4*a*c))/(2*a);
round(qpos(j)/echarge)*echarge;
round(qneg(j)/echarge)*echarge;
for j = length(q)-1:-1:1
qpos(j) = (j*(gforce - eforce(mode)))/(qpos(j+1)*(1/debye2(j)...
+1/sep(j)).*(1./(4*pi*epsilon0*sep(j))).*exp(-sep(j)/debye2(j))...
);
53
qpos(j) = round(qpos(j)/echarge)*echarge;
end
for j = length(q)-1:-1:1
qneg(j) = (j*(gforce - eforce(mode)))/(qneg(j+1)*(1/debye2(j)...
+1/sep(j)).*(1./(4*pi*epsilon0*sep(j))).*exp(-sep(j)/debye2(j))...
);
qneg(j) = round(qneg(j)/echarge)*echarge;
end
q2pos = round(qpos/echarge);
q2neg = round(qneg/echarge);
54
Appendix C
Antisymmetric Mode Particle
Horizontal Motion Code
%
%
%
%
equilibrium charges found by Raymond Fowler
numerical framework by Raymond Fowler
reduction in step size and addition of drag by Brandon Harris
flags and reporting established by Brandon Harris
% this program plots particles in vertical chain
% using: gravity, electric field, particle repulsion
% initally assuming infinite transverse potential
clear all; close all;
load('ResultsPositionsYvsT.mat')
% report step results? (1 for yes, 0 for no)
report = 0;
mode = 1;
beta = 8.65;
% average terminal velocity (in m/s)
% termv = (1.1640 + 0.9411257 + 1.1586)/3;
epsilon0 = 8.85e-12;
linewidthsize = 2.0;
% debye length
debye2(1) = (1000)*1e-6;
debye2(2) = (1000)*1e-6;
debye2(3) = (1000)*1e-6;
% meters per pixel
scale = 0.000011988011988;
% box top
top = 1001;
% mass of dust
diameter = 8.89e-6;
density = 1500; % kg/mˆ3
volume = 4/3*3.1415926535*(diameter/2)ˆ3; % mˆ3
dustmass = density*volume; % kilograms
55
% force of gravity
gforce = -dustmass*9.81;
% charge of electron
echarge = 1.6e-19;
% charge on dust
q(1) = -295822*echarge;
q(2) = -110000*echarge;
q(3) = -1231313*echarge;
boxdomainidx = (1:top)';
boxdomain = linspace(1.1988011988e-05,12*1e-3,top)';
sizeboxdomain = size(boxdomain,1);
onesboxdomain = ones(sizeboxdomain,1);
% import workspace file for chain particle equilibrium positions
load('AntiSymPositions.mat')
%Different scale for the june 28 transverse chain, and it was ...
never erased.
%So instead, the scalejune28 was set to 1.
scalejune28 = 1;
partpos = partpos*1e-3; %partpos in mm; convert to m
partposscale = partpos*scalejune28;
% partposscale(1) = partposscale(1) + 1e-3; %Use to start top ...
particle at non-equil position
% calculate vertical separations between actual particles in chain
for j = 1:length(q)-1
vsep(j) = partposscale(j)-partposscale(j+1);
end
% starting horizontal separations from equilibrium
hsep = zeros(1,length(q));
% Distance between actual particles in chain
sep = sqrt(vsep.ˆ2 + hsep.ˆ2);
% Multiplicative factor needed to get horizontal component of ...
force
for j = 1:length(q)-1
hcomp(j) = (hsep(j) - hsep(j+1))./sep;
end
% plot forces
figure;
hold on;
% plot gravity force
plot(boxdomain*1000,gforce*onesboxdomain,'Color',[0 0.6 0],'...
LineWidth',linewidthsize);
% plot the electric force
56
Eforce = zeros(sizeboxdomain-1,length(q));
domainstep = boxdomain(2)-boxdomain(1);
for j = 1:length(q)
clear('EforceNew');
k=5; %"Spring" constant
EforceNew = -k*hsep(j)*onesboxdomain;
plot((boxdomain(1:sizeboxdomain)+domainstep/2)*1000,EforceNew,...
'k','LineWidth',linewidthsize);
Eforce(boxdomainidx(1:sizeboxdomain),j) = EforceNew;
end
% plot the interparticle repulsion force
% top repulsion only up
j = 1;
repulsionF(j) = hcomp(j)*(1/debye2(j)+1/sep(j)).*(1./(4*pi*...
epsilon0*sep(j)))*q(j)*q(j+1).*...
exp(-sep(j)/debye2(j));
plot(boxdomain*1000,repulsionF(j)*onesboxdomain,'c','LineWidth',...
linewidthsize);
for j = 2:length(q)-1
uprepulsionF(j) = hcomp(j)*(1/debye2(j)+1/sep(j)).*(1./(4*pi*...
epsilon0*sep(j)))*q(j)*q(j+1).*...
exp(-sep(j)/debye2(j));
downrepulsionF(j) = hcomp(j-1)*(1/debye2(j)+1/sep(j-1)).*(1....
/(4*pi*epsilon0*sep(j-1)))*q(j-1)*q(j).*...
exp(-sep(j-1)/debye2(j));
repulsionF(j) = uprepulsionF(j)-downrepulsionF(j);
plot(boxdomain*1000,repulsionF(j)*onesboxdomain,'c','LineWidth...
',linewidthsize);
end
% bottom repulsion down
j = length(q);
repulsionF(j) = -hcomp(j-1)*(1/debye2(j)+1/sep(j-1)).*(1./(4*pi*...
epsilon0*sep(j-1)))*q(j-1)*q(j).*...
exp(-sep(j-1)/debye2(j));
plot(boxdomain*1000,repulsionF(j)*onesboxdomain,'c','LineWidth',...
linewidthsize);
% plot sum of forces
for j=1:length(q)
forcesum(j) = Eforce(sizeboxdomain,j)+gforce+repulsionF(j);
plot((boxdomain(1:(sizeboxdomain))+domainstep/2)*1000,...
Eforce(:,j)+repulsionF(j),'r','LineWidth',linewidthsize);
end
forcesum
% plot where particles are from data
range = get(gca,'Ylim');
rangesize = size(range,2);
onesrange = ones(rangesize,1);
for j = 1:length(q)
57
plot(partposscale(j)*onesrange*1000,range,'b--','LineWidth',...
linewidthsize)
end
axis([6.0 12 range(1) range(2)]);
xlabel('Distance from lower electrode (mm)','FontName','Arial','...
FontSize',14);
ylabel('Force (N)','FontName','Arial','FontSize',14);
hold off;
v = zeros(length(q),1);
y = partposscale;
step = 100000;
handle = figure;
yMatrix = y;
stepamount = 0.0001;
t = 0:stepamount:step*stepamount;
dt = stepamount;
for i = 1:(step + 1)
yMatrix(:,i) = y;
a = [ (forcesum(1:2)/dustmass)-beta*v(1:2)' 0 ]';
y = y + v*dt + 0.5*a*dtˆ2;
if report
y
end
v = v + a*dt;
% calculate separations between actual particles in chain
for j = 1:length(q)-1
sep(j) = y(j)-y(j+1);
if report
sep
end
end
% plot the interparticle repulsion force
% top repulsion only up
j = 1;
repulsionF(j) = (1/debye2(j)+1/sep(j)).*(1./(4*pi*epsilon0*sep...
(j)))*q(j)*q(j+1).*...
exp(-sep(j)/debye2(j));
for j = 2:length(q)-1
uprepulsionF(j) = (1/debye2(j)+1/sep(j)).*(1./(4*pi*...
epsilon0*sep(j)))*q(j)*q(j+1).*...
exp(-sep(j)/debye2(j));
downrepulsionF(j) = (1/debye2(j)+1/sep(j-1)).*(1./(4*pi*...
epsilon0*sep(j-1)))*q(j-1)*q(j).*...
exp(-sep(j-1)/debye2(j));
repulsionF(j) = uprepulsionF(j)-downrepulsionF(j);
end
58
% bottom repulsion down
j = length(q);
repulsionF(j) = -(1/debye2(j)+1/sep(j-1)).*(1./(4*pi*epsilon0*...
sep(j-1)))*q(j-1)*q(j).*...
exp(-sep(j-1)/debye2(j));
% plot sum of forces
for j=1:length(q)
forcesum(j) = Eforce(sizeboxdomain,j)+gforce+repulsionF(j)...
;
end
if report
forcesum(1:2)
v
a
end
% plot where particles are from data
range = get(gca,'Ylim');
rangesize = size(range,2);
onesrange = ones(rangesize,1);
clf(handle);
hold on;
if i == (step + 1)
% plot where particles are from data
range = get(gca,'Ylim');
rangesize = size(range,2);
onesrange = ones(rangesize,1);
for j = 1:(length(q)-1)
plot(partpos(j)*scalejune28*onesrange*1000,range,'k.-'...
,'LineWidth',linewidthsize)
end
end
hold off;
if ~mod(i,step/100)
fprintf('Step Number: %d %s\n',i,datestr(now));
end
end
59
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