Electromagnetic Propulsion
ii
The thesis of Jeffrey D. Contri was reviewed and approved* by the following:
Michael M. Micci
Amy Pritchett
*Signatures are on file in the Graduate School.
iii
Abstract
A parallel molecular dynamics code was developed to simulate ensembles of water molecules in
the presence of electric and magnetic fields to showcase the Abraham effect. Capitalizing on this
phenomenon would allow the exploration of a new method of electric rocket propulsion without
the requirement to ionize propellant – an outstanding impediment of the technology – by saving
on power and raising thruster efficiency.
Perfect cube ensembles ranging from 8 to 1000 molecules were investigated to find the smallest
size that gave consistent, favorable behavior across 10 different initializations with random 3D
orientations and velocities based on a Maxwell-Boltzmann distribution. It was sought that no
matter the initialization, the average normalized z-velocity profile when the electric field was
present remained higher than the profile when the electric field was absent, successfully
demonstrating the Abraham effect.
An 85 kV/m sinusoidal electric field was applied in the x-direction at a frequency of 75 GHz
while a 2.5 T static magnetic field was applied in the y-direction. Over a period of 250 ps, it was
found that the 512-molecule ensemble was the smallest size to produce the desired velocity
profile comparisons across all initializations. Further optimizing the electric field to 15 GHz
provided results with an average normalized Δ of 23.87 mm/s/pulse and an average normalized
z-acceleration of approximately 3.58 × 108 m/s2.
For an assumed acceleration length of 20 cm, this would provide an of approximately 1220 s,
which is comparable with current electric propulsion methods. Further raising the magnetic field
strength an order of magnitude would theoretically raise the acceleration by an order of
magnitude, rendering an of about 3860 s, far surpassing that of current Hall thrusters and ion
engines.
Should this revolutionary method of propulsion be experimentally verified, power spent on
ionization would be eliminated and the green propellant would allow for an “island-hopping”
type of space exploration by visiting a given water-bearing celestial body, refueling, and
progressing to the next.
1.2.1 Solid ................................................................................................................................ 2
1.2.2 Liquid ............................................................................................................................. 2
1.2.3 Electric ............................................................................................................................ 3
1.3 Motivation ............................................................................................................................. 3
1.4.1 The Abraham Effect ....................................................................................................... 5
1.4.2 Force Derivation ............................................................................................................. 6
1.4.2.3 Walker & Walker .................................................................................................... 8
1.4.4 Investigation of Propellants ............................................................................................ 9
1.4.4.1 Rydberg Atoms/Molecules ..................................................................................... 9
1.4.4.2 Alkali Hydrides ..................................................................................................... 10
2.1 Water Molecule ................................................................................................................... 14
2.2.2 Paschen Curve .............................................................................................................. 16
2.3.1 Electric Field ................................................................................................................ 18
2.3.2 Magnetic Field .............................................................................................................. 19
2.4.3 No Vibrational Modes .................................................................................................. 20
2.4.4 No Spontaneous Ionization........................................................................................... 21
3.1 Introduction ......................................................................................................................... 23
3.3 Molecular Dynamics ........................................................................................................... 30
4.5 Refinement of the Electric Field Frequency ....................................................................... 41
CHAPTER 5: EXPERIMENTAL PROSPECTS.......................................................................... 42
Appendix B: Center of Mass Calculation ..................................................................................... 48
Appendix C: Dipole Calculation ................................................................................................... 49
Appendix D: Newton-Raphson Iteration of van der Waal’s Equation of State ............................ 50
Appendix E: Dissociation of Water .............................................................................................. 51
Appendix F: Random Polarity Distribution .................................................................................. 55
Appendix G: Random Initial Velocity Distribution ...................................................................... 65
Appendix H: The Arcana2 Cluster ............................................................................................... 70
Appendix I: MPI_BROADCAST() .............................................................................................. 71
Appendix J: MPI_ALLGATHER() .............................................................................................. 72
Appendix L: Sandstorm ................................................................................................................ 74
List of Figures
Figure 1: Electric thruster efficiency where the only loss mechanism is ionization energy ........... 5
Figure 2: Collection of randomly oriented dipoles with no net polarization .................................. 6
Figure 3: Implied dynamics of a collection of randomly oriented dipoles when an electric field is
present ....................................................................................................................................... 6
Figure 4: Illustration for Lorentz force derivation on rotating dipole molecules [6] ...................... 7
Figure 5: Dipole representation in Penfield and Haus force derivation [6] .................................... 8
Figure 6: Illustration of how the Abraham force affects the dynamics of a water molecule ........ 11
Figure 7: SPC/E water molecule model illustration [20] .............................................................. 15
Figure 8: Breakdown electric field strength vs. total pressure [23] .............................................. 17
Figure 9: Phase diagram of water [24] .......................................................................................... 18
Figure 10: Four orientations to determine the optimal frequency [25] ......................................... 19
Figure 11: Initial arrangement of a 1000-molecule ensemble ...................................................... 20
Figure 12: Illustration of EII and SEE phenomena occurring between DC-biased parallel plate
electrodes [28] ......................................................................................................................... 21
Figure 14: Flow chart of Sandstorm ............................................................................................. 24
Figure 15: Initial orientation of molecules before any random rotations were applied ................ 25
Figure 16: Possible orientation of a molecule after the 1st rotation about the z-axis ................... 26
Figure 17: Possible orientation of a water molecule after the 2nd rotation about the new y-axis 26
Figure 18: Possible orientation of a water molecule after the 3rd rotation about the new x-axis 27
Figure 19: Illustration of a Maxwell-Boltzmann distribution of gas particle speeds [31] ............ 29
Figure 20: Lennard-Jones potentials of H2O interactions ............................................................. 32
Figure 21: Illustration of the velocity Verlet algorithm ................................................................ 33
Figure 22: Illustration of how the MPI Broadcast command communicates data [40] ................ 35
Figure 23: Illustration of how the MPI Allgather command communicates data [41] ................. 36
Figure 24: Average normalized z-velocity of 125 molecules with and without an electric field
(Conditions: Ex=85 kV/m @ 75 GHz; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -2) ...... 38
Figure 25: Average normalized z-acceleration of 125 molecules with and without an electric
field (Conditions: Ex=85 kV/m @ 75 GHz; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -2)
................................................................................................................................................. 39
Figure 26: Average normalized z-velocity of 512 molecules with and without an electric field
(Conditions: Ex=85 kV/m @ 75 GHz; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -4) ...... 40
Figure 27: Average normalized z-acceleration of 512 molecules with and without an electric
field (Conditions: Ex=85 kV/m @ 75 GHz; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -4)
................................................................................................................................................. 40
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Figure 28: Comparison of average normalized z-velocity profiles across different frequencies for
512 molecules (Conditions: Ey=85 kV/m; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -4) 41
Figure 29: Illustration of the design parameters of a rectangular waveguide [43] ....................... 42
Figure 30: A rocket and its propellant at rest ................................................................................ 46
Figure 31: A rocket after an infinitesimal time of firing .............................................................. 46
Figure 32: Diagram of a water molecule ...................................................................................... 48
Figure 33: Equilibrium constants for the dissociation of H2O as a function of temperature [Hill]
................................................................................................................................................. 52
Figure 35: Dipole distribution for 8 molecules ............................................................................. 56
Figure 36: Dipole distribution for 27 molecules ........................................................................... 57
Figure 37: Dipole distribution for 64 molecules ........................................................................... 58
Figure 38: Dipole distribution for 125 molecules ......................................................................... 59
Figure 39: Dipole distribution for 216 molecules ......................................................................... 60
Figure 40: Dipole distribution for 343 molecules ......................................................................... 61
Figure 41: Dipole distribution for 512 molecules ......................................................................... 62
Figure 42: Dipole distribution for 729 molecules ......................................................................... 63
Figure 43: Dipole distribution for 1000 molecules ....................................................................... 64
Figure 44: Speed distribution of 8 molecules at 300 K ................................................................ 65
Figure 45: Speed distribution of 27 molecules at 300 K .............................................................. 66
Figure 46: Speed distribution of 64 molecules at 300 K .............................................................. 66
Figure 47: Speed distribution of 125 molecules at 300 K ............................................................ 67
Figure 48: Speed distribution of 216 molecules at 300 K ............................................................ 67
Figure 49: Speed distribution of 343 molecules at 300 K ............................................................ 68
Figure 50: Speed distribution of 512 molecules at 300 K ............................................................ 68
Figure 51: Speed distribution of 729 molecules at 300 K ............................................................ 69
Figure 52: Speed distribution of 1000 molecules at 300 K .......................................................... 69
Figure 53: Illustration of how the MPI Braodcast command communicates data [41] ................ 71
Figure 54: Illustration of how the MPI Allgather command communicates data [41] ................. 72
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Table 1: Single ionization energies of various propellants [1] ....................................................... 4
Table 2: Experimental characteristics of the three lightest alkali hydrides [16]........................... 10
Table 3: SPC/E water molecule model characteristics [19].......................................................... 14
Table 4: Van der Waals equation of state constants [21].............................................................. 16
Table 5: Collection of positive & negative polarity components for all ensemble sizes .............. 28
Table 6: Table of theoretical and computational equations that define a gas’ distribution [32] .. 29
Table 7: Computation speeds and respective errors across all ensemble sizes............................. 30
Table 8: Lennard-Jones parameters for interactions between the atoms of water molecules ....... 31
Table 9: Summary of simulation parameters ................................................................................ 36
Table 10: Equilibrium constants for the dissociation of H2O as a function of temperature ......... 53
Table 11: Table of standard commercially available waveguides [43] ........................................ 73
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Nomenclature
Symbols
coefficient [ m
s2 ] ; [cm]; [\]
Pa⋅m ]
√1 = skewness measure [\]
constant [T]; [
fraction coefficient [m]; [m]; [\]
= mole fraction coefficient [\]
= electric field [ V
= force [N]
= equilibrium constant [atm]
= principal quantum number; waveguide mode, number of
moles [\]; [\]; [\]
= charge [C]
= coordinate direction [\]
= coordinate direction [\]
= Lennard-Jones potential well depth [ N
m ]
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= specific volume [ m3
collision area [Å]; [m2]
= rotation angle [rad]
= rotation angle [rad]
= rotational frequency [ rad
= speed of light 3 × 108 m
s
= unit of electric charge 1.6022 × 10−19 C
0 = acceleration due to Earth’s gravity 9.81 m
s2
K
mol
kmol ⋅ K
m
= The ratio of a circle’s circumference to its diameter 3.14159265 …
Subscripts
L = Lorentz
LJ = Lennard-Jones
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Acknowledgments
First and foremost, I would like to thank Dr. Micci for his continued support, guidance, and
patience over the last several years as a research advisor. His hands-off, yet informative &
supportive, approach to advising has allowed me to venture down different avenues, bringing me
to conclusions on my own, helping me to grow both academically and rhetorically. Across all
internships and other research endeavors, I have cherished this project the greatest and endless
thanks are given to him for allowing me to inherit the endeavor.
I would also like to extend significant gratitude to Kirk Heller, the department’s Systems
Administrator, for his assistance in IT support, from setting up software to anything &
everything related to working on the cluster. My growth in parallel computing and Linux is
largely in part because of you.
Lastly, I would like to express how grateful I am to Kerstyn – my roommate, fellow researcher,
and closest confidant – as well as to my mom, dad, and sister. Without your love and support, I
most certainly would not have made it through grad school and be where or who I am today.
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Epigraph
“We set sail on this new sea because there is new knowledge to be gained, and new rights to be
won, and they must be won and used for the progress of all people. For space science, like
nuclear science and all technology, has no conscience of its own. Whether it will become a force
for good or ill depends on man, and only if the United States occupies a position of pre-eminence
can we help decide whether this new ocean will be a sea of peace or a new terrifying theater of
war. I do not say that we should or will go unprotected against the hostile misuse of space any
more than we go unprotected against the hostile use of land or sea, but I do say that space can be
explored and mastered without feeding the fires of war, without repeating the mistakes that man
has made in extending his writ around this globe of ours.
There is no strife, no prejudice, no national conflict in outer space as yet. Its hazards are hostile
to us all. Its conquest deserves the best of all mankind, and its opportunity for peaceful
cooperation may never come again. But why, some say, the Moon? Why choose this as our goal?
And they may well ask why climb the highest mountain? Why, 35 years ago, fly the Atlantic?
Why does Rice play Texas?
We choose to go to the Moon. We choose to go to the Moon in this decade and do the other
things, not because they are easy, but because they are hard, because that goal will serve to
organize and measure the best of our energies and skills, because that challenge is one that we
are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the
others too.”
CHAPTER 1: INTRODUCTION
1.1 Rocket Equation
Rocket propulsion is achieved based on the conservation of momentum, the product of mass and
velocity. For a collision in an isolated system, the total momentum of two objects before the
collision, is equal to the total momentum of the two objects after the collision. That is, the
momentum lost by one object is equal to the momentum gained by the other. The transfer of
momentum within a rocket is due to the propellant particles colliding with the head of the
combustion chamber, the diverging section of the nozzle, and any collisions with the
chamber/nozzle walls that have an axial component.
The rocket equation relates a spacecraft’s change in velocity to its mass before and after the
propellant is burned.
) (1.1)
The logarithmic relationship between the initial and final masses is why such large rockets are
needed to transport such relatively small payloads to space. Staging a rocket helps overcome this
logarithmic relationship. However, after about 5 or 6 stages, the complexity of adding another
stage begins to outweigh the advantage that that stage’s increase in Δ would provide. The Δ
(phonetically pronounced as “delta-v”) a spacecraft achieves is also largely dependent on the
engine’s specific impulse, a measure of how effectively the propellant is being used to generate
thrust. Specific impulse is the thrust divided by the weight flow of the propellant or – in its
reduced form – the equivalent velocity divided by the acceleration due to Earth’s gravity.
=
0 (1.2)
Given a propellant mass, a spacecraft whose thruster has a higher will be able to achieve a
greater Δ. A derivation of Equation 1.1 can be found in Appendix A. Different types of rocket
propulsion have different specific impulses. There are three primary methods of propulsion: solid
rocket motors, liquid engines, and electric thrusters.
2
1.2.1 Solid
In solid propulsion, a fuel and an oxidizer are pre-mixed into a solid form. These propellants can
be heterogeneous – in which the fuel and oxidizer are on different molecules – or homogeneous
– in which they are on the same molecule. The propellant is molded so that the motor has a
specific cross section, which determines the thrust and chamber pressure profiles. Once ignited,
the solid rocket motor burns until completion. Some engineering ingenuity can be applied to the
nozzle that allows for throttling but most solid motors do not incorporate this. Typical values
are in the 100-300 s range. This type of propulsion is often used for launch vehicles because they
are cheap and easy to make, while still providing a large amount of thrust.
1.2.2 Liquid
As the name suggests, liquid propulsion involves the fuel and oxidizer in a liquid state being
injected into the combustion chamber where they are ignited. Liquid propulsion engines that use
one liquid are monopropellant engines, those that use two are bipropellant engines, and those that
use three – which are rare – are tripropellant engines.
Monopropellant thrusters usually use some variation of hydrazine which is simply ignited in the
combustion chamber. They are usually shorter in length, ranging from 10 cm to about 1 m and
are primarily used for reactive control systems (RCS) to reorient spacecraft in orbit or
rendezvous and proximity operations (RPO). Cold gas thrusters are also used for these purposes
and involve relieving the pressure of a gas, allowing it to flow out a de Laval nozzle.
Bipropellant engines vary in size, being as small as monopropellant engines all the way up to the
F-1 engine, which stands at about 5.5 m with an exit diameter of 3.7 m and provided 7.8 MN of
thrust to the first stage of the Saturn V rocket. There are also hypergolic propellants in which no
ignition is required; once the fuel and oxidizer reach the combustion chamber, they combust on
contact. These are rarely used on missions because of this very reason, adding complexity to the
handling, storage, and transportation.
The specific impulse of liquid rocket engines is typically no greater than 350-400 s and can be
used for every stage in a rocket from launch, to orbit changing, to interplanetary travel. With
liquid and solid propellants, the equivalent velocity is defined as
= + ( − )
. (1.3)
where is the mass flow rate, is the exit area of the de Laval nozzle, is the propellant’s
exit velocity, and and are the exit and ambient pressures, respectively. It takes this form
and is not simply the same as the exit velocity because they are operational in the atmosphere
3
and always involve a de Laval nozzle. For many forms of electric propulsion, the equivalent
velocity is simply assumed to be the exit velocity since there is no nozzle.
1.2.3 Electric
With electric propulsion, the propellant is excited by electrical means through three different
practices: electrothermal, electrostatic, and electromagnetic. In electrothermal propulsion, a gas
is heated via an induction coil or microwave, and accelerated out a de Laval nozzle. These
thrusters are around 10 cm in length and generally operate with an of around 500 s. Because
of their size, their use is limited to RCS. Electrostatic thrusters first ionize propellant and then
accelerate the ions with an electric field. Gridded ion engines and Hall thrusters are common
electrostatic engines, which are only used for orbit raising and interplanetary travel. Since Hall
thrusters utilize a magnet in their operation, there is some debate whether they are electrostatic or
electromagnetic. However, their magnet is only used to hold electrons in place, and is not used to
accelerate the propellant, making it of the electrostatic type. Electromagnetic thrusters accelerate
ionized propellant with an electric field as well as a magnetic field. Some common technologies
are magnetoplasmadynamic (MPD) thrusters and field-reversed configuration (FRC) thrusters,
which can vary in size. Electrostatic and electromagnetic specific impulses are highest at around
1000-1500 s, with some experimental gridded ion engines reaching as high as 6000 s.
The equivalent velocity in electric propulsion is assumed to be equal to the exit velocity.
= (1.4)
Flow rates for these thrusters are on the order of milligrams, whereas exit velocities are on the
order of tens of kilometers per second. This provides for miniscule thrust, many orders of
magnitude lower than with solid and liquid propulsion. Noble gases are often used for
electrostatic and electromagnetic propulsion because they are easy to handle, being inert. Xenon
is predominantly used in the industry because of its high molecular weight. However, recently
SpaceX has started using krypton for its ion thrusters for the Starlink mission.
In electric propulsion, the transfer of momentum is due to the charged particles pushing off on
the electric and/or magnetic fields, which gives a reacting force to the spacecraft. The main
difference between chemical and electric propulsion is that chemical is energy limited – having
received the combustion power from breaking the chemical bonds of the propellant’s molecules
– and electric is power limited. The higher the power, the higher the electric and magnetic fields
will be, creating a greater acceleration.
1.3 Motivation
The major loss mechanism of electrostatic and electromagnetic thrusters is the requirement to
ionize the propellant. Power is supplied to the propellant to ionize it, creating a net charge for
acceleration. Once it exits the engine, the spacecraft becomes negatively charged requiring the
4
firing of electrons to the plasma beam to neutralize it and the spacecraft. If this act were
neglected, the spacecraft would experience beam stalling, where it would attract the plasma back
towards itself, effectively canceling the thrust it just produced. Because of recombination and
radiation, the ionization energy factor is one-to-two orders of magnitude above the single
ionization energy for the propellant being used, essentially wasting energy and decreasing
thruster efficiency. This irrecoverable ion production energy cost is why electric thruster
efficiency increases with specific impulse; the ion production energy becomes a lower fraction of
the total thruster energy input as the exhaust kinetic energy increases.
The thruster efficiency is defined as the ratio of the thrust power to the sum of thrust power and
power losses.
=
+ (1.5)
Focusing on the ionization of the propellant as the only loss mechanism, implementing a
common practice factor of 10 to the ion production energy to account for recombination, and
reducing the equation to focus on the energy input into the propellant, the efficiency becomes
=
. (1.6)
Table 1 shows a collection of commonly used propellants for electric propulsion and their
ionization energies, with Figure 1 illustrating how thruster efficiency increases with molecular
weight and, more importantly, .
Table 1: Single ionization energies of various propellants [1]
Element Mass,
5
Figure 1: Electric thruster efficiency where the only loss mechanism is ionization energy
It can be seen that the loss in efficiency due to the lost ionization energy can be substantial, even
at high specific impulses. Revolutionary gains in electric thruster efficiency could be obtained if
the need to ionize the propellant could be avoided, which could be accomplished by using both
electric and magnetic fields to accelerate the propellant.
1.4 Concept and Literature Review
1.4.1 The Abraham Effect
The Abraham effect describes how a polar molecule can experience a force in a given direction
due to a time-varying polarization in a magnetic field and/or a polarization in a time-varying
magnetic field without ionization occurring [2]. It is expressed as
=
= ∑
(1.8)
and represents each atom on the molecule [3]. Figure 2 shows a collection of polar molecules
that would hypothetically have a net zero polarization outside the influence of an electric field
due to random molecular orientation.
6
Figure 2: Collection of randomly oriented dipoles with no net polarization
If an electric field were to be applied, the dipole moments of the molecules would attempt to
align themselves with the electric field lines, which can be seen in Figure 3.
Figure 3: Implied dynamics of a collection of randomly oriented dipoles when an electric field is present
The movement of these dipoles in the presence of a perpendicular magnetic field would generate
a force perpendicular to both the electric and magnetic field.
1.4.2 Force Derivation
1.4.2.1 Cox
Cox first proposed this acceleration scheme for Shuttle propulsion in 1980 with atmospheric
gases [4]. He derived the Lorentz body forces as follows with a supporting illustration in Figure
4 [5].
7
Figure 4: Illustration for Lorentz force derivation on rotating dipole molecules [6]
The Lorentz force is given by
= × (1.9)
where the velocity of each charge is the product of the rotational frequency and the radius.
=
× (1.10)
If the dipole moment is defined as the magnitude of the charge distance times the separation
= , (1.11)
=
Therefore, the sum of the forces on two charges is
= × , (1.13)
=
× (1.14)
1.4.2.2 Penfield and Haus
Penfield and Haus developed an earlier derivation of forces acting on dipolar gases, but their
derivation differs from Cox’s in that they accounted for any given motion of the dipole [6, 7]. As
seen in Figure 5, the position of the negative end of the dipole is given by and the dipole is
acted on by both electric and magnetic fields which are functions of position.
8
Figure 5: Dipole representation in Penfield and Haus force derivation [6]
The force acting on the negative end is
= − ( ) − × ( ) (1.15)
and the force on the positive end, at the point + and traveling with the velocity + /, is
= ( + ) + [ +
] × ( + ) . (1.16)
Thus the net force on the dipole is
= [ ( + ) − ( )] + [ × ( + ) − × ( )] +
× ( + ) (1.17)
and when taken to the first order, the expression becomes
= ⋅ ∇ + × ( ⋅ ∇ ) +
× . (1.18)
The last term represents the time rate of change of the dipole moment with the magnetic field
and, in spatially uniform electric and magnetic fields, only the final term remains giving the
same result as derived by Cox [4, 6].
=
1.4.2.3 Walker & Walker
Walker and Walker and Walker et al. experimentally measured this force in solid dielectric
crystals within 4% of the Penfield and Haus derivation [7, 8, 9]. The experiment performed
involved an alternating electric field inducing an alternating radial polarization. The resultant
force produced an alternating torque on the crystal when a steady axial magnetic field was
applied. The rotation was measured by optical means. Walker and Walker, and Walker et al.
were searching for the force described in Equation 1.7. However, they only found a force due to
the first term, namely the Abraham force.
=
9
Even though they showed that Abraham’s derivation was in error, they proved that the force
exists and can be experimentally measured.
1.4.3 Furthering the Concept with Acceleration Analysis
Using Newton’s 2nd Law of Motion in conjunction with Equation 1.14 expresses the acceleration
as
Δ (1.21)
if the dipole’s rotation is perfectly perpendicular with the magnetic field. Thus, a high
acceleration results from a high dipole moment-to-mass ratio, high magnetic field strength, and a
high electric field frequency. The dipole acceleration must exceed a minimum value in order to
produce a comparable exhaust velocity and specific impulse. Basic kinematics find the
acceleration of a given exhaust velocity, with an assumed initial velocity of zero, to be
= e
2 (1.22)
To make this acceleration scheme viable, a specific impulse of 1000 s is desired. With a
conservative acceleration length of 20 cm, the required acceleration is 2.4 × 108 m/s2. Since the
solid crystal that was experimentally measured by Walker & Walker possessed a dipole moment
larger than that of any gas suitable for propulsion, other dipoles were investigated [6, 8].
1.4.4 Investigation of Propellants
1.4.4.1 Rydberg Atoms/Molecules
Cox first suggested the use of Rydberg atoms/molecules in his 1981 paper in order to obtain
larger dipole moments [4]. A Rydberg atom or molecule is one whose valence electron is excited
to a high energy level just below the ionization limit [10]. The principal quantum number, , is
generally greater than 10 and the lifetime of such an excited state is extraordinarily long, usually
greater than 100 μs. They are typically created by electron collision excitation or tuned laser
absoprtion. The quantum mechanics of hydrogen-like atoms, which are examples of Rydberg
atoms, can be used to obtain the permanent dipole moment given by
= 3( − 1)
where is the net nuclear charge [11].
Zimmerman et al. published experimental evidence for the existence of these large permanent
dipole moments for the alkali metals at MIT [12]. However, there are a few reasons why
choosing Rydberg atoms/molecules for an alternating polarization thruster is not optimal. The
field ionization potential for Rydberg atoms is proportional to −4, whereas the force on the
dipole increases with 2. Therefore, the maximum acceleration that can be achieved is an inverse
square relation to the electronic excitation.
10
Rydberg states are also easily destroyed by the absorption of electromagnetic radiation, resultant
radiation, or by collisions with other particles. In order to avoid ionizations due to collisions,
operation at cryogenic temperature would be required. Additionally, Rydberg atoms/molecules
may also be photoionized by background infrared and submillimeter radiation due to their
decreased ionization potential. For example, Rydberg sodium atoms have been photoionized at
300 K background radiation, and in order to prevent this unintential photoionization, thermal
shielding was required [13]. The culmination of these cases makes Rydberg atoms/molecules a
poor choice for an alternating-polarization thruster.
1.4.4.2 Alkali Hydrides
Alkali hydride molecules – ionic and non-covalent bonds between a single hydrogen atom and an
alkali metal – have significantly larger dipole moments than mixed alkali diatomics [14]. Since
the molecule is diatomic and includes the lightest element, their dipole moment-to-mass ratios
are highest amongst naturally occurring molecules, making them ideal for an alternating-
polarization acceleration scheme.
Paunescu first looked at lithium hydride (LiH), which has the highest naturally occurring dipole
moment-to-mass ratio at 1.492 D/amu. He developed a molecular dynamics simulation to
observe the motion of one molecule in the presence of a static magnetic field and a varying
electric field. The dipole vector was initialized along the y-axis in the same direction of the
magnetic field, while the electric field varied in the x-direction. Under a static magnetic field of
25 T and a “saw tooth” electric field of 20 kV/m at a frequency of 1 GHz, a center-of-mass
velocity of 78.6 mm/s was achieved after 8 ns [15]. Assuming an acceleration length of 20 cm
with an average acceleration of 9,825 km/s2, an exhaust velocity of 1982 m/s and an of 202 s
could be achieved, making it a competitor to some forms of electrothermal propulsion.
Sodium hydride (NaH) and potassium hydride (KH) are the next two heaviest alkali hydrides.
The bond lengths were taken from the Computational Chemistry Comparison and Benchmark
DataBase and the dipole moments were calculated via the QC-Lab tool on NanoHub using the
TZVP basis set [16]. The characteristics and, more importantly, the dipole moment-to-mass
ratios, are tabulated in Table 2.
Table 2: Experimental characteristics of the three lightest alkali hydrides [16]
Molecule Charge [C] Bond

[

]
11
Additionally, as one goes farther down Group 1, the energy needed to ionize the molecule
decreases, defeating the purpose of the acceleration technique. While alkali hydrides show
promise through successful calculations and simulations, they are pyrophoric, meaning they
spontaneously combust when in contact with oxygen, and therefore are to be avoided for
propulsion purposes. Fortunately, there is a molecule that is safe, abundant on Earth and in the
solar system, green, and has a high dipole moment-to-mass ratio that is even comparable to that
of most alkali hydrides: water.
1.4.4.3 Water
The water molecule has an experimental dipole of 1.85 D and a moment-to-mass ratio of 0.103
D/amu. Paunescu originally began his investigation by simplifying the molecule to a two-particle
system. The configuration, bond length, masses, and charges of the two particles were adapted to
maintain the same polar moment as that of an actual water molecule. Under the same electric and
magnetic field conditions as with the lithium hydride molecule, the of the hypothetical water
molecule reached 5 mm/s after 8 ns [15]. Figure 6 shows a detailed illustration of how the
electric and magnetic fields affect the dynamics of a water molecule.
Figure 6: Illustration of how the Abraham force affects the dynamics of a water molecule
It was mentioned earlier in the introduction how most current electric propulsion systems operate
with xenon. Unfortunately, xenon is extremely expensive because of its scarcity. NASA’s Dawn
mission, which explored Vesta and Ceres over an 11-year period, used 425 kg of xenon gas for
its ion engine [17]. At approximately $1200/kg, NASA would have spent $510k on the
12
propellant alone and while the propellant cost is a small fraction of a space mission, there is still
substantial room for savings [18].
Water, on the other hand, is one of the most abundant resources on Earth. Even after being
distilled and purified, the production and handling costs would pale in comparison to that of
xenon. It is also found in various forms throughout the solar system. Therefore, a spacecraft
could theoretically travel to Mars, refuel, travel to an asteroid that contains ice, gather and melt
it, and move on to other bodies in our solar system that harbor ice, providing for an island-
hopping type of space exploration.
1.5 Thesis Objective
The objectives of this thesis are to create a molecular dynamics (MD) simulation that models a
large ensemble of water molecules with realistic conditions and phenomena, optimizing the state
parameters and electric field characteristics to show that the Abraham effect causes a collective
increase in average velocity in the z-direction. These realistic conditions include initializing the
molecules with random orientations, as well as random speeds based on a Maxwell-Boltzmann
distribution. Additionally, the pressure at which the molecules receive their initial spacing will
be determined from the Paschen curve, which will also determine the electric field strength.
Another goal is to find the smallest ensemble size of which the Abraham effect is apparent,
regardless of the initial conditions. This serves to minimize the computational resources required
for an extended simulation to produce an animation in which the acceleration in the z-direction is
apparent, a tertiary goal of the research.
Successful completion of these goals allows the calculation of parameters to accurately compare
it to current forms of electric propulsion and provide the necessary conditions to set up an
experiment to physically verify the phenomenon.
The following chapter presents the setup of the simulation where the water molecule model is
detailed as well as the rationale for the state parameters chosen. How the electric and magnetic
field parameters were determined was also included. Finally, this chapter provides the
assumptions taken, their rationale, and how they were included in the study.
The next chapter addresses the code itself with a brief introductory flowchart followed by how
the random initial conditions and molecular dynamics algorithms are implemented. Chapter 3
ends with the boundary conditions and manner in which the code is parallelized, before
providing a summary of the simulation conditions.
The results are laid out in Chapter 4 with results followed by a short passage featuring the first
steps to be taken for an experimental setup in Chapter 5. This thesis ends with a conclusion
13
where defining parameters are calculated from the results and compared with current
technologies, along with aspirations for future work.
14
2.1 Water Molecule
The SPC/E model of the water molecule was utilized because of its simplicity [19, 20]. The
atoms are modeled as point charges and point masses, with the magnitude of the partial negative
charge of the oxygen atom evenly countered and distributed by the magnitude of the partial
positive charges of the hydrogen atoms. Additionally, the bonds are modeled as rigid rotors, so
vibrational and rovibrational modes are not accounted for, drastically reducing the dynamical
complexity and, thus, computational cost. Table 3 displays the characteristics of the chosen
model with respect to the illustration in Figure 7.
Table 3: SPC/E water molecule model characteristics [19]
Hydrogen (H) atoms (1, 1) Mass: 1.6739 × 10−27 kg
Charge: 0.4238e
Charge: − 0.8476e
H-O-H bond angle () 109.47°
Effective molecular radius () 3.166 Å
15
Figure 7: SPC/E water molecule model illustration [20]
These characteristics give a dipole moment of 2.35 D, which is higher than the experimental
value, most likely due to the crude approximations. A detailed derivation of the theoretical dipole
moment is shown in Appendix C.
2.2 State Parameters
2.2.1 Equation of State
The Ideal Gas Law is a simple equation that relates the state parameters (pressure, temperature,
and volume) to each other
= (2.1)
where is the number of particles and is the Boltzmann constant. Unfortunately, there are
certain assumptions that go into validating the use of the Ideal Gas Law. They are as follows:
1. Low pressure
3. The number of particle-to-wall collisions are much greater than particle-to-particle
4. All collisions are perfectly elastic
Gases that do not meet all of these criteria will not behave perfectly like an ideal gas.
Additionally, the Ideal Gas Law becomes particularly inaccurate with water vapor because it is a
polar molecule and never fully acts like an ideal gas. Furthermore, whatever dipole molecule is
used for this acceleration scheme, a different equation of state will have to be used to more
accurately simulate and model the dynamics of the system.
Van der Waals’ equation of state [21], sometimes referred to as the Virial Theorem, was utilized
to more accurately set the initial volume of the system and the corresponding molecular
separation.
16
3 − ( + )2 + − = 0 (2.2)
The specific volume is denoted by and is the universal gas constant. This third order relation
also contains two gas-specific constants, and , which can be found in Table 4. It was solved
using the Newton-Raphson iteration method of which Appendix D provides a walk-through of.
Table 4: Van der Waals equation of state constants [21]
Substance [ ⋅
He 3440 0.0234
H2 24.8 0.0266
O2 138 0.0318
CO2 366 0.0429
H2O 580 0.0319
Hg 292 0.0055
2.2.2 Paschen Curve
The Paschen curve relates the RMS electric field strength at which a gas will break down and
ionize (ie., the breakdown electric field) to the gas pressure. Typically, it is plotted with the
product of pressure and the electrode distance as the abscissa (x-axis) and the electric field
strength as the ordinate (y-axis), but for this research’s purposes, a relation between the electric
field and pressure is more convenient. A logarithmic scale is usually utilized to make estimates
easier.
Paschen’s Law most closely explains the Townsend Avalanche concept in which an electron
traveling from the cathode to the anode might collide with a gas particle and ionize it, given it
has enough energy. The curve has the U-shape because of two reasons, but it nevertheless comes
down to the probability that an electron will ionize a gas particle before reaching the cathode
[22].
As the gas pressure is decreased from that at the Paschen minimum, the number density begins to
decrease. If an electron is placed in the electric field, it is less likely to collide with a molecule
before reaching the anode, allowing the breakdown electric field to be higher. As the gas
pressure is increased from that at the Paschen minimum, the number density begins to increase.
If an electron is then placed in the electric field, it may not have enough time to acquire the
necessary kinetic energy to lead to breakdown before colliding with another particle. Therefore,
a higher electric field is permissible. Figure 8 shows the Paschen curve for dry air, H2O and a
mixture of both at 293 K [23].
17
Figure 8: Breakdown electric field strength vs. total pressure [23]
Paschen’s Law is shown in the following equation
=

, (2.3)
=
(2.4)
= 1
. (2.5)
In these two previous equations, is the effective collision area of the molecule and 1 is the
energy required to ionize the neutral particle. The critical frequency at which the physics of
Paschen’s law starts to become irrelevant is
=
√2 (2.6)
where is the collision frequency and is the mass of an electron. Finding the correct
pressure for the simulations was an iterative process. At first, 4,000 Pa was chosen because it
correlated to a high electric field strength, around 100 kV/m. Simulations provided minimal-to-
no Abraham effect because it was found that the Coulomb force between particles greatly
dwarfed that of the Lorentz force. The pressure was then decreased and the electric field strength
was altered accordingly. As the pressure decreased, so did the number density, reducing the
Coulomb interaction greatly, and allowing the Lorentz force to become more prominent. The
lowest pressure of 13.33 Pa (0.1 Torr) was finally chosen to minimize the number density and
maximize the electric field strength.
18
2.2.3 Temperature
The application of a polar propulsion method requires the propellant to be in a gaseous state.
Figure 9 depicts the phase diagram for water [24].
Figure 9: Phase diagram of water [24]
The experimental data provided in Figure 8 were gathered at 293 K. Initial investigations were
conducted at 4,000 Pa and per Figure 9, water is indeed gaseous at these conditions. A
temperature of 300 K was chosen to err on the side of caution and maintain the phase
requirement. As the pressure was decreased to the left side of the Paschen curve, the choice of
temperature remained the same so that as few variables were being changed at a time.
2.3 Electric and Magnetic Fields
2.3.1 Electric Field
The higher the electric field strength, the higher the Lorentz force will be, and thus the greater
the Abraham effect. Given the experimental data in Figure 8, the maximum electric field
associated with a pressure of 13.33 Pa is approximately 85 kV/m. A sinusoidal application with
this maximum value was applied in the x-direction.
Contri found that the optimal frequency was 75 GHz. A single molecule was given four different
initial orientations with the dipole along either the y- or z-axis. The spread of the hydrogen atoms
was then oriented so as to maximize the movement of the dipole as the electric field was applied.
Figure 10 shows the four orientations [25].
19
Figure 10: Four orientations to determine the optimal frequency [25]
As the electric field was applied, the center-of-mass accelerated in the z-direction with the
positive portion of the electric field, and then decelerated with the negative portion. The goal was
to create a plateaued velocity behavior. The frequency was varied from 60 to 100 GHz to find the
best balance between a velocity increase and a plateau, and 75 GHz provided just that.
2.3.2 Magnetic Field
Both Penfield & Haus’ and Walker & Walker’s derivations and experiments only found a force
when the magnetic field was constant. Therefore, a static magnetic field was applied in the y-
direction, per the requirement in Section 1.4.1. Equation 1.21 shows that the acceleration will be
greatest with a high magnetic field. A relatively inexpensive (~$1000) high permanent magnet
has been developed with a high field strength of 2.5 T. It is also much lighter than conventional
equivalent electromagnetic systems by more than an order of magnitude and for these reasons, a
2.5 T magnetic field was chosen for the simulation [26].
20
2.4 Assumptions
Various assumptions went into making the code less computationally expensive, as well as
simplifying the relevant physics, which also reduced computation time.
2.4.1 Initial Cube-like Orientation
An ensemble of molecules was generated to more accurately represent how a real-world scenario
would behave. The molecules were initially configured into a cube, equally spaced from one
another. Figure 1 shows the initial layout of an ensemble of 1000 molecules.
Figure 11: Initial arrangement of a 1000-molecule ensemble
Only the oxygen atoms were plotted due to their close proximity to the center of mass, justified
in Appendix B. Additionally, animations of the ensemble were created and refraining from
plotting the hydrogen atoms and bonds greatly reduced computation time.
2.4.2 No Initial Rotational Velocity
Each atom was given an initial thermal velocity based on a Maxwell-Boltzmann distribution
around a given temperature, but all atoms on a given molecule were given the same velocity.
Ergo, there was only translational and no rotational movement initially. Once they felt the
interactions from other molecules and the electric field was applied – which was after 1 timestep
– they would be free to rotate.
2.4.3 No Vibrational Modes
As mentioned in Section 2.1, the vibrational, and thus rovibrational, modes of the molecules
were not accounted for. While this would provide for a more accurate model, the length scale of
21
the simulation domain was far greater than the deviation in an atom’s position due to vibrations
and so it was not worth the computational cost to include.
2.4.4 No Spontaneous Ionization
The molecules were assumed to never be ionized and no randomly generated electrons were
added to the simulation at any time. Plasma may be generated through four main processes:
electron impact ionization (EII), secondary electron emission (SEE), field emission (FE), and
cosmic radiation (CR) [27]. By staying below the Paschen curve – a criterion for an alternating
polarity acceleration scheme – Townsend Avalanches were avoided and EII could be neglected.
FE is the release of electrons from a cathode commonly into a vacuum and depends on electron
tunneling, only becoming valid for microgaps of small distances on the order of 7 μm and
smaller [27]. Any practical application of this acceleration scheme would involve a waveguide
with a gap many orders of magnitude larger than a micrometer and therefore that method of
electron generation could be neglected. Finally, SEE & CR were disregarded to focus on a
simpler simulation. Additionally, the electronic modes of the molecule are ignored, because the
simulation temperature was not nearly high enough. Even though Figure 12 incorporates a DC
power supply, the illustration of the EII and SEE phenomena is what is of focus.
Figure 12: Illustration of EII and SEE phenomena occurring between DC-biased parallel plate electrodes [28]
2.4.5 Water’s Paschen Curve
Because of the lack of experimental data for water’s Paschen curve and the complexity of
deriving one, the Paschen curve for water is assumed to follow that of dry air in Figure 8.
2.4.6 No Dissociation
The dissociation of a gas is the breaking down of that molecule into its constituents as a direct
effect of temperature on the quantum states of that molecule and the mixture pressure. Equation
2.7 shows the chemical equation as water dissociates into its constituents with through
representing the mole fractions.
22
Figure 13 shows how the mole fractions change as a function of temperature.
Figure 13: Dissociation of H2O at 13.33 Pa
Even at 1500 K, the composition of the gas is still over 99% water, so it can be safely assumed
that at 300 K – the temperature chosen for the simulations – there are no constituents of water
present. A more comprehensive look at the dissociation of water can be found in Appendix E to
further justify the negligence of the phenomenon.
23
3.1 Introduction
The code developed was named Sandstorm as it operates on a swarm of particles, and was
written in FORTRAN 90 using Compaq Visual Fortran as the integrated development
environment (IDE). It was first checked for errors before it was moved to the computer cluster to
be compiled and executed. The data output files were then copied from the cluster to a personal
computer where they were analyzed in MATLAB. The entire parallel code can be found in
Appendix L and Figure 14 shows a flow chart of Sandstorm, highlighting the main sections of
the code that are further explained in this chapter.
24
25
3.2 Initial Conditions
Each molecule was given an initial random orientation and velocity using the ran2 random
number generator [29]. A direction cosine matrix (DCM) was applied to the orientation of each
molecule where the three rotation angles were randomly chosen. The velocity was determined
using the gasDev function which gives a speed by integrating a Maxwell-Boltzmann distribution
of thermal speeds via kinetic theory [29].
3.2.1 DCM
The oxygen atom is illustrated with a red circle and held at the center of each molecule’s
respective coordinate frame. The hydrogen atoms are illustrated with blue circles, whereas the
molecule’s dipole is shown with a green line. An illustration is shown in Figure 15.
Figure 15: Initial orientation of molecules before any random rotations were applied
The first rotation is about the z-axis and ranges from 0 to 2π radians, rotating the molecule
through the xy-plane. Equation 3.1 depicts the rotation matrix about the z-axis where the
trigonometric expressions are abbreviated (ie. ‘cos()’ is ‘’) for simplicity’s sake.
() = [ 0
− 0 0 0 1
] 0 ≤ < 2 (3.1)
An illustration of a molecule’s possible orientation after the first rotation can be seen in Figure
16.
26
Figure 16: Possible orientation of a molecule after the 1st rotation about the z-axis
The second rotation is about the y-axis and ranges from −π/2 to π/2, bringing the molecule
above or below the xy-plane. Equation 3.2 depicts the rotation matrix about the y-axis.
() = [ 0 − 0 1 0 0
] −
2 (3.2)
An illustration of a molecule’s possible orientation after the second rotation can be seen in
Figure 17.
Figure 17: Possible orientation of a water molecule after the 2nd rotation about the new y-axis
The final rotation is about the x-axis and rotates the molecule about its dipole between a range of
0 and π. Equation 3.3 depicts the rotation matrix about the x-axis.
() = [ 1 0 0 0 0 −
] 0 ≤ < (3.3)
An illustration of a molecule’s possible orientation after the third rotation can be seen in Figure
18.
27
Figure 18: Possible orientation of a water molecule after the 3rd rotation about the new x-axis
The DCM is formed from matrix multiplication, multiplying the three rotation matrices from
right to left
DCM = () () () (3.4)
The final form of the DCM matrix can be seen in Equation 3.4.
DCM = [
] (3.5)
This DCM is then applied to each hydrogen’s position vector to give it its initial random
orientation [30].
3.2.2 Randomicity II
This section and the next address the validity and accuracy of the ran2 function. In Ref. [29],
four different random number generators are presented: ran0, ran1, ran2, and ran3. Ran2 is a
combination of the first two generators, invoking an additional shuffle that helps eliminate
sequential correlations deeming the function a “perfect” random number generator by the
authors. They even challenge the reader to find a “statistical test that ran2 fails in a non-trivial
way” and will award the first successor a prize of $1000 [29]. Whether this award is taxed or left
untaxed is unknown.
Once the three random angles were determined for a molecule, its initial dipole vector was
solidified. For a given ensemble size, half of the dipoles should be positive in each coordinate
direction, with the other half being negative. The sign of the components of the collection of
dipoles was tabulated in Table 3.2.
28
Table 5: Collection of positive & negative polarity components for all ensemble sizes
Number of
− 3 5 4
− 11 13 13
− 28 30 30
− 59 63 56
− 100 110 101
− 169 148 159
− 251 263 248
− 356 342 351
− 483 524 480
With an absolutely perfect random number generator, there would be an even split amongst the
number of molecules. Since this is practically unattainable by a computer, there is some
deviation, but for the ran2 function, it is low, staying around or below 5% for intermediate and
larger ensembles, giving great credibility to the use of the generator.
3.2.3 Maxwell-Boltzmann Distribution of Thermal Velocities
The initial speed of a molecule was determined via the ‘gasDev’ function [29]. Each atom on a
given molecule was given a unique, random speed in each direction determined from an
integrated Maxwell-Boltzmann distribution centered around the initial temperature of 300 K.
This was so that the molecule had no initial rotation, an assumption stated in Section 2.3.2 [29].
Equation 3.6 defines the distribution of gas particles’ speeds – where is the temperature and
is the molecule’s mass – and Figure 19 shows the probabilistic distribution with the most
probable, root mean square (RMS), and average speeds represented as such [29].
() = 42 (
Figure 19: Illustration of a Maxwell-Boltzmann distribution of gas particle speeds [31]
The most probable speed is the most likely speed a molecule might have if randomly selected,
thus it is the peak of the curve. The average speed is the arithmetic mean of the speeds, and the
RMS is the measure of the magnitude of the set of speeds, and is used to determine other gas
parameters based on speed. Based on the theoretical expressions’ ratios, they are always related
to each other by a scalar. Table 6 compiles the theoretical and computational equations that
define a gas’ distribution [32]. The computational most probable speed was determined from the
tallest bin of particles. Further explanation along with a detailed look at all the distributions can
be seen in Appendix G.
Table 6: Table of theoretical and computational equations that define a gas’ distribution [32]
Type of Speed Theoretical Computational
√ 3
tallest bin (3.9a, 3.9b )
As with the polarity distributions, the distribution of initial velocities was varied. Table 7 gives a
look at the computed speed values and their errors from the theoretical values which are bolded
in the second row.
30
Table 7: Computation speeds and respective errors across all ensemble sizes
Number of
The percent errors were calculated using the following equation.
= | − |
(3.10)
3.3 Molecular Dynamics
An n-body problem relates to computing the individual motions of a group of bodies interacting
with each other, whether they be molecules, celestial objects, or even billiard balls. Molecular
dynamics is an n-body problem with forces and constraint dynamics specific to the nature of the
problem.
3.3.1.1 Lorentz Force
The SPC/E model of the water molecule assumes the atoms are point charges, with the net
charge evenly distributed along its dipole [20]. Because of this, the Abraham force can be
simplified to the Lorentz force, which describes the behavior of moving charges in electric and
magnetic fields.
This simplification also makes computations in each direction easier.
, = ( − ,) (3.12a)
, = 0 (3.12b)
, = , (3.12c)
3.3.1.2 Coulomb Force
The Coulomb force results from the electrostatic potential between two charged particles. The
point charge assumption allows the potential between atom and to be expressed as
31
(3.13)
where the Coulomb force between two particles is the negative gradient of the potential.
, = −∇ , (3.14)
The Coulomb interaction between two atoms on the same molecule is negated due to the rigidity
of the bonds, effectively making it zero. Thus, the total Coulomb force on particle is
, = ∑ 1
The Lennard-Jones (L-J) potential accounts for interatomic interactions between atoms and .
The −12 term characterizes the strong repulsion at short distances due to the Pauli Exclusion
Principle, whereas the −6 term characterizes the weak, long-range attractive van der Waals
force [33, 34].
, = 4 [(
)
12
− (
)
6
] (3.16)
The Lennard-Jones potential has two parameters specific to the elements the potential is
between. The zero-energy separation distance is and the depth of the potential well is . The
interaction between two unlike atoms requires the use of the Lorentz-Berthelot mixing rules [35]:
= 1
(3.17b)
Table 8 shows appropriate L-J parameters for interactions between the atoms of water molecules.
Table 8: Lennard-Jones parameters for interactions between the atoms of water molecules
Atomic Interaction [Å] []
H-H 2.81 8.6
O-O 2.95 61.6
O-H 2.88 23.0165
Figure 20 gives a better illustration as to how these parameters play a part in the L-J potential.
32
Implementing the Lennard-Jones potential into a molecular dynamics simulation requires
computing the force, done so by taking the negative gradient of the potential.
, = −∇ , (3.18)
There is no L-J force between atoms on the same molecule. Therefore, the total force on atom
is
. (3.19)
Because the atomic separation is raised to a high order, the potential and force rapidly approach
zero. This allows for truncation, greatly reducing the number of computations for each time step.
Common practice sets the cutoff distance, , equal to 2.5 where the L-J potential is only
1.6% of its minimum energy. The following equations show the conditions on how the Lennard-
Jones force is implemented in the code where is the distance where the potential is equal
to −.
, = 0 = (3.20b)
, = ,( ) < < (3.20c)
, = 0 > (3.20d)
33
3.3.1.4 Total Force
The total force on particle , therefore, can be expressed as the sum of the three forces.
, = , + , + , (3.21)
This is essential for computing the acceleration felt by the particle to advance its position and
velocity.
3.3.2 Velocity Verlet
The ‘velocity Verlet’ algorithm variant called RATTLE, is used to integrate the equations of
motion of the water molecules [36, 37, 38]. An illustration of the algorithm can be seen in Figure
21.
Figure 21: Illustration of the velocity Verlet algorithm
It is a two-part algorithm that calculates the acceleration felt by each particle directly for each
part,
. (3.22)
The first advances the position by a full timestep and the velocity by a half time step.
( + Δ) = () + Δ () + Δ2
2 () (3.23a)
(3.23b)
The second occurs after the second set of forces have been calculated and advances the velocity
of the particles by another half time step.
( + Δ) = ( + Δ
2 ) +
Δ
2 ( + Δ) (3.24)
This algorithm determines the positions and velocities at the current time step to (Δ2) with a
minimized round-off error [37].
3.3.3 Constraint Dynamics
As the positions of the atoms are advanced, the bond lengths between the hydrogen atoms and
oxygen atom might shorten or lengthen. The RATTLE algorithm is utilized to retain the proper
bond length within a certain tolerance [37]. If the bond length exceeds the tolerance, most likely
from an improper representation of the physics or too large of a time step, the code writes
34
“Constraint Failure” to a file and no further calculations of the positions and velocities would
occur.
3.3.4 Time Step
Choe and Kim determined the proper time step for molecular dynamics of H2 and CO2 by
analyzing the eigenvalues of the nonlinear dynamics. They found that time steps less than 1.823
fs and 3.808 fs provided stable dynamics for H2 and CO2, respectively [38]. Determining the
maximum time step for this study would have added significant complexity and was outside the
scope of this thesis. Therefore, a time step of 1 fs was chosen to advance the equations of motion
as a conservative measure and for simplicity.
3.4 Boundary Conditions
Periodic boundary conditions (PBC) were used to contain the molecules in a cubic domain.
These conditions allowed for the assumption that the cube was infinitely replicated through
space, thus once all of the atoms on a molecule left the cube, it entered from the opposite side
with the same velocity. This allowed for the total number of molecules to be conserved [33].
Once the length of the cube was determined with Equation 3.25
= 1/3 , (3.25)
it was offset by a certain amount that is half the molecular separation,
=
21/3 (3.26)
to center the evenly spaced molecules with a distance between a molecule on any face of the
cube, and the adjacent face of the domain.
3.5 Parallelization
Message Passing Interface (MPI) is used to parallelize the code and communicate the position
and velocity arrays between all the processors [39]. Each processor had access to the entirety of
the position and velocities arrays. However, a given processor is to only calculate the forces,
positions, and velocities of a certain number of molecules. Once completed, that data was then
communicated between the processors via the Broadcast and Allgather routines. With MPI, the
data communicated between processors need to be scalar or 1-dimensional arrays. Since the
position and velocity arrays used for calculation in the code are 3-dimensional, they are first
converted to 1-dimensional arrays, passed between the processors via the routines, and then
converted back to 3-dimensional arrays so that further calculations can properly take place.
Simulations were run on the Arcana2 cluster in the Aerospace Engineering Department. Details
about Arcana2 can be found in Appendix H.
35
3.5.1 Broadcast
The Broadcast routine is used for sending a set of data from one processor to the rest. The initial
position and velocity arrays are created on the master processor and sent to the remaining
processors so that each processor has access to all the data. Figure 22 shows an illustration of
how the data are communicated and Appendix I gives a deeper insight into this routine.
Figure 22: Illustration of how the MPI Broadcast command communicates data [40]
3.5.2 Allgather
The Allgather routine is – for pedagogical purposes – a combination of the MPI Gather and
Scatter routines. It collects the specified data from all the processors and then distributes the
modified array back to all the processors in one routine. This routine requires that the number of
elements sent from each processor be the same. The Allgatherv routine can account for varying
counts of sent data, but with how the number of molecules is determined, it is unnecessary to use
[39]. In all, the Allgather routine was chosen over the aforementioned routines because of its
simplicity and all-encompassing functional purposes.
Once the time loop starts, the forces, positions, and velocities are calculated as mentioned in the
beginning of the chapter. The updated section of the array on one processor is then sent to every
other processor using the Allgather routine. Because the velocity Verlet has two parts, the routine
is called twice – one after each part – to ensure the data are properly updated. Figure 23 shows an
illustration of how the data are communicated and Appendix J gives a deeper insight into this
routine.
36
Figure 23: Illustration of how the MPI Allgather command communicates data [41]
3.6 Code Summary
Table 9 summarizes the simulation conditions for the results presented in Chapter 4.
Table 9: Summary of simulation parameters
Time Step 1 × 10−15 s
Electric Field Strength (x-direction) 85 kV/m
Electric Field Frequency (x-direction) 75 GHz
Magnetic Field Strength (y-direction) 2.5 T
Pressure 13.33 Pa
Temperature 300 K
4.1 Introduction
The number of molecules simulated varied through all the perfect cubes from 8 to 1000. These
numbers were a result of the cube-like initial orientation. Depending on the number of molecules
they contained, different ensembles were assigned to different classes. Namely, 23 – 43 were
considered small ensembles, 53 – 73 were considered intermediate ensembles, and 83 – 103 were
considered large ensembles.
As mentioned in Section 1.5, the objective was to find the minimum ensemble size, for which
results were consistent across multiple seed values or different initializations. That is, the
average normalized z-velocity of the molecules’ center-of-mass was continuously higher when
the electric field was present, than when it was not. The average z-acceleration would also abide
by this condition.
The average velocity was normalized by taking the initial average velocity of the center-of-mass
of all the molecules and subtracting it from every atom in each respective direction.
, = , − 0, (4.1)
This was conducted so that the initial average z-velocity read zero and an easier comparison of
the magnitude of the velocity increase could be made across simulations regardless of their size
and initialization. The average z-acceleration was then calculated via numerical differentiation in
MATLAB to save on computation time in Sandstorm. The central difference method of order
(2) was used
2 (4.2)
() = ( + Δ) − ( − Δ)
2Δ (4.3)
Investigations began at the smaller ensemble sizes across 10 different initializations. For a
simulation to be considered “favorable,” the velocity profile when the electric field was present
38
needed to be greater than when the electric field was absent, so as to exemplify the Abraham
effect. Furthermore, for a simulation size to be considered the smallest necessary for further
simulations, all the different seed values needed to provide favorable results. For the sake of
computation time, simulations began at 50 ps.
What was found was that the velocity profiles were nearly identical no matter the ensemble size
or initialization. Since there needed to be a distinction between different ensemble sizes and seed
values, the timespan was lengthened to 250 ps.
4.3 Unfavorable Results for Smaller Ensembles
Early investigations at the smaller ensemble sizes provided “unfavorable” results, meaning that
the velocity profile when the electric field was present was not always greater than when the
electric field was absent for the entirety of the simulation or various initializations. Figures 24
and 25 show unfavorable results of the average normalized z-velocity and z-acceleration for a
simulation of 125 molecules with and without the electric field present the entire time, along
with the average dipole components. The dipole components were calculated via Equation 4.4
= ( 1 + 2 − 2 ) (4.4)
The components had to be amplified so that they could be discernable on the electric field
strength scale, and were included to examine their correlation to the velocity profile.
Figure 24: Average normalized z-velocity of 125 molecules with and without an electric field (Conditions: Ex=85
kV/m @ 75 GHz; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -2)
As with the velocity profile, the respective acceleration profile was unfavorable, indicating
ineffectiveness of the electric field.
39
Figure 25: Average normalized z-acceleration of 125 molecules with and without an electric field (Conditions:
Ex=85 kV/m @ 75 GHz; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -2)
Even for the intermediate ensemble sizes, unfavorable simulations occurred for at least one out
of the 10 different initializations. Consistent, favorable results were not found until the larger
ensemble sizes were investigated.
4.4 Favorable Results for Larger Ensembles
An ensemble of 512 molecules proved to be the smallest ensemble size at which favorable
velocity and acceleration profiles were found across all 10 initializations. An ensemble size
containing between 343 (73) and 512 (83) molecules would require considerable amendments to
the initial arrangement algorithm as well as to the parallelization, but it would improve the
computation time due to its smaller size.
Figure 26 shows that, when the electric field is present, the average normalized z-velocity is
always higher than when the electric field is absent, and is ever-increasing in a “step-up” like
fashion. After 250 ps, the velocity reaches about 30 mm/s over almost 19 pulses. This translates
to an average acceleration of 1.2 × 108 m/s2 and a Δ of 1.6 mm/s/pulse.
40
Figure 26: Average normalized z-velocity of 512 molecules with and without an electric field (Conditions: Ex=85
kV/m @ 75 GHz; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -4)
Figure 27 shows that the respective acceleration profile when the electric field is present is also
greater than when it is absent, indicating favorableness. Additionally, the average acceleration
calculated from Figure 27 is on par with the behavior in Figure 26: the acceleration varies from
zero to approximately 2.6 × 108 m/s2 and is close to the 1.2 × 108 m/s2 prediction if averaged
by two.
Figure 27: Average normalized z-acceleration of 512 molecules with and without an electric field (Conditions:
Ex=85 kV/m @ 75 GHz; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -4)
As inferred, the 729- and 1000-molecule ensembles also provided favorable profiles across all
initializations.
41
4.5 Refinement of the Electric Field Frequency
The initial frequency was 75 GHz with rationale explained in Section 2.3.1. After the optimal
simulation was determined, the electric field frequency was then varied to further obtain better
results with a higher z-velocity at the end of the simulation.
The frequency was varied in 15 GHz increments to cover a wider range with fewer simulations
needed. After determining that increasing the frequency provided a smaller Δ per pulse, lower
frequencies were investigated where it was found that the Δ per pulse increased as the
frequency decreased. Figure 28 shows the average normalized z-velocity profiles for frequencies
ranging from 15 GHz to 90 GHz.
Figure 28: Comparison of average normalized z-velocity profiles across different frequencies for 512 molecules
(Conditions: Ey=85 kV/m; By = 2.5 T; T = 300 K; P = 13.33 Pa; Seed = -4)
A frequency of 15 GHz provided the highest Δ per pulse and peak velocity at the end of the
simulation. After 250 ps, the velocity reached about 89.5 mm/s over 3.75 pulses. This translates
to an average acceleration of 3.58 × 108 m/s2 and a Δ of 23.87 mm/s/pulse, almost 1400%
higher than at 75 GHz.
42
CHAPTER 5: EXPERIMENTAL PROSPECTS
Experimentally verifying the results presented by Sandstorm in Chapter 4 would add great
validity to this method of propulsion. The step-up behavior of the velocity profile would be
undistinguishable on a large scale, but obtaining expected exit velocities from the computed
accelerations would showcase the Abraham effect in action. While an experimental procedure
was not the focus of this thesis, nor included in this study, preliminary prospects were made in
investigating waveguides, structures that guide electromagnetic or sound waves with minimal
loss of energy.
An illustration of a rectangular waveguide and the standard parameters can be seen in Figure 29.
Figure 29: Illustration of the design parameters of a rectangular waveguide [43]
The power transmission in megawatts of a waveguide is determined by
= (6.63 × 10−4)√1 − (
) 2
2 (5.1)
where and are in centimeters and the breakdown electric field is in V/cm. Rectangular
waveguides can only propagate TM and TE modes, not the TEM mode. The TE mode is of focus
because it has an electric field in the x-direction and a magnetic field in the y-direction, which is
how the simulation is set up. The TE modes have further specialization and take the
nomenclature of TEmn where m and n are integers. Only the TE10 mode is analyzed here because
it is the dominant mode [43]. The cutoff frequency of any mode is represented as
43
where is the speed of light and reduces to
,10 =
2 (5.3)
for the TE10 mode. It is imperative that electric field frequencies higher than the cutoff frequency
are used, otherwise the power will become complex [43].
Since the waveguide power is independent of the length, the power required to operate a space-
grade thruster would remain the same as the acceleration length increased. Evidently, as the
acceleration length increases, the equivalent velocity – and thus – would also increase. A
table of some commercially available waveguides can be found in Appendix K.
44
6.1 Summary
Current methods of electric propulsion expend a lot of power when ionizing the propellant.
Avoiding this requirement would lead to an increase in thruster efficiency and could be done by
taking advantage of the Abraham force on a dipole molecule in perpendicular electric and
magnetic fields. Simulating an ensemble of water molecules to observe the collective effect of
the Abraham force would provide insight into the validity of this method of propulsion.
One of the goals of this research was to determine the smallest ensemble size that gave average
normalized z-velocity and z-acceleration profiles that were higher with an electric field present
for the entirety of the simulation. The primary goal was to accomplish this with realistic
conditions by implementing random initial orientations and random speeds based on a Maxwell-
Boltmann distribution. Additionally, restrictive conditions on pressure, temperature, and electric
field strength from water’s assumed Paschen curve and phase diagram were implemented.
Consistency was first confirmed across 10 different initializations for an ensemble of 512 (83)
molecules and also for larger ensembles.
The highest velocity achieved was approximately 89.5 mm/s, with an average acceleration of
3.58 × 108 m/s2, and an average Δ of 23.87 mm/s/pulse. Assuming an acceleration length of 20
cm, and utilizing Equations 1.2 and 1.22 an of 1220 s could be reached, rivaling that of
current commercial electric propulsion methods. If the magnetic field were to be raised by an
order of magnitude to 25 T, the acceleration would theoretically increase by an order of
magnitude via Equation 1.21 [44]. With this augmentation, an greater than 3860 s could be
achieved, dwarfing that of current Hall thrusters and ion engines.
6.2 Future Work
While the results presented herein are promising, there are still areas for advancement. Firstly,
conducting much longer simulations – which would showcase the effect of collisions on a longer
scale – would provide a more direct look at an achievable exit velocity. An animation of the
molecules could also be developed to provide a visual representation of the bulk motion in the z-
direction. The RMS speed for water molecules at 300 K is approximately 600 m/s. An assumed
10% increase would allow for a discernable change in the average normalized z-velocity. At
45
23.87 mm/s/pulse, a Δ of 60 m/s would require a 167.6 ns simulation with a 15 GHz electric
field. Due to technical difficulties with the cluster, this was unachievable for this study.
Finally, a physical experiment could be conducted with the optimized and successful parameters.
Upon validation, a new form of electromagnetic propulsion for which power spent on ionization
would be eliminated. Additionally, the green propellant would allow an “island-hopping” type of
space exploration due to water’s abundance in the solar system, revolutionizing the electric
propulsion industry.
Appendix A: Rocket Equation Derivation
Part of a rocket’s mass includes the mass of the propellant.
Figure 30: A rocket and its propellant at rest
After the engine is fired and the propellant is expelled out the rocket at some exit velocity, the
mass of the rocket becomes a function of time as it loses a differential amount of mass and gains
a differential amount of velocity.
Figure 31: A rocket after an infinitesimal time of firing
By conservation of momentum, Equation A.1 is formed.
() ⋅ = ⋅ (A.1)
Since the differential amount of propellant used by the rocket is equal to the amount of mass lost
by the rocket,
= − . (A.2)
Equation A.1 can be simplified and integrated between some initial and final time.
47
∫ v
v


= ∫ − ()



(A.3)
) , (A.4)
where the difference between the initial and final velocity over the course of the burn is referred
to as the “delta-v.”
Δ = − (A.5)
In electric propulsion, the exhaust velocity is assumed to be equal to the equivalent velocity and
thus the rocket equation can be expressed as:
Δ = ln (
Appendix B: Center of Mass Calculation
The center-of-mass is the location that represents the mean position of the matter in a body or
system. Finding this point requires taking a weighted average of the relative position of the
distributed masses.
(B.1)
If the molecule is positioned in a 2D plane with an arbitrary origin placed at the oxygen atom,
Figure 32: Diagram of a water molecule
the center of mass can be calculated as follows:
= 1 1 + + 2 2
1 + + 2
= (1 )(−0.8165 + 0.5774 ) + (16 )(0 ) + (1 )(0.8165 + 0.5774 )
(1 ) + (16 ) + (1 )
= 0.06415
109.47°
Appendix C: Dipole Calculation
The dipole moment of a molecule points from the positive charges to the negative charges and
can only be present if the electronic structure of that molecule is asymmetric. The moment can
be expressed as
= ∑ (C.1)
where represents each atom. Water’s dipole moment can then be expressed as
= 1 1 + + 2 2 . (C.2)
Further computations are as follows:
= (. 4238)(−.8165 , .5774 − .06415 )
+ (−.8476)(0 , −.06415 )
+ (. 4283)(.8165 , .5774 − .06415 )
= 0.489378 ( 1 − 10
1 ) (
3.33564 − 30 ⋅ )
= 2.35
This value is 27% higher than the experimental value, most likely due to the assumption that the
atoms are point charges.
Appendix D: Newton-Raphson Iteration of van der Waal’s
Equation of State The Newton-Raphson method is an iterative root-finding method to solving higher order or
transcendental functions that cannot otherwise be solved analytically [29]. It begins with an
initial guess of the parameter in question. The function, which has to be in the form of () =
0, and the function’s derivative are then evaluated at that guess to find a new value of the desired
parameter.
′() (D.1)