FLUID FLOW THROUGH CARBON NANOTUBES: A NEW MODELING AND
SIMULATION APPROACH
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Michael Avon
August, 2009
FLUID FLOW THROUGH CARBON NANOTUBES: A NEW MODELING AND
SIMULATION APPROACH
Michael Avon
Thesis
Approved:
Accepted:
________________________
Advisor
Dr. Alper Buldum
________________________
Department Chair
Dr. Celal Batur
________________________
Co-Advisor
Dr. S. Graham Kelly
________________________
Dean of the College
Dr. George Haritos
________________________
Faculty Reader
Dr. Fred Choy
________________________
Dean of the Graduate School
Dr. George R. Newkome
________________________
Date
ii
ABSTRACT
In this thesis, the flow of fluids through carbon nanotubes was investigated in
order to get a better understanding of the unique properties and phenomena of
nanofluidics. The previous modeling and simulation efforts were based on diffusion of
atoms or molecules that were thrown to the nanotubes with initial velocities. Here, we
present molecular dynamics simulations of carbon nanotubes that were embedded in
liquid argon. The fluid was pushed through the nanotubes using a moving wall piston of
graphene. Single-walled, double-walled, rigid and relaxed nanotubes in different
diameters were considered. In order to achieve more continuous flow of fluid through the
nanotube, several rounds of pumping were simulated. Pressure difference in different
regions was analyzed.
iii
TABLE OF CONTENTS
Page
LIST OF FIGURES ........................................................................................................vi
CHAPTER
I. INTRODUCTION........................................................................................................1
1.1
Objective of the Thesis .............................................................................1
1.2
Overview of the Thesis .............................................................................2
II. BACKGROUND ........................................................................................................4
2.1
What are Carbon Nanotubes? ...................................................................4
2.2
Nanotube Structure..................................................................................5
2.3
Fluid Flow at the Nanoscale.....................................................................7
2.4
Recent Nanofluidic Research and its Applications ...................................8
2.5
Recent Research on Fluid Flow through Carbon Nanotubes...................13
III. THE METHODS AND THE MODEL.....................................................................17
3.1
Molecular Dynamics..............................................................................17
3.2
Energy Considerations...........................................................................19
3.3
Building a Model...................................................................................26
IV. ARGON FLUID FLOW THROUGH SINGLE-WALLED CARBON
NANOTUBES.........................................................................................................34
4.1
Collecting Data for Analysis..................................................................34
iv
4.2
4.3
Rigid Single-Walled Carbon Nanotubes .................................................37
4.2.1
(10,10) Carbon Nanotube .........................................................37
4.2.2
(20,20) Carbon Nanotube .........................................................44
Relaxed Single-Walled Carbon Nanotubes .............................................48
4.3.1
(10,10) Carbon Nanotube .........................................................48
4.3.2
(20,20) Carbon Nanotube .........................................................53
4.3.3
Repeated Pumping through (20,20) Carbon Nanotube ..............57
V. ARGON FLUID FLOW THROUGH DOUBLE-WALLED CARBON
NANOTUBES..........................................................................................................62
5.1
5.2
Rigid Double-Walled Carbon Nanotubes ................................................62
5.1.1
(10,10)@(15,15) Carbon Nanotube ..........................................62
5.1.2
(15,15)@(20,20) Carbon Nanotube ..........................................66
Relaxed Double-Walled Carbon Nanotubes............................................70
5.2.1
(10,10)@(15,15) Carbon Nanotube ..........................................70
5.2.2
(15,15)@(20,20) Carbon Nanotube ..........................................75
VI. CONCLUSIONS .....................................................................................................82
REFERENCES..............................................................................................................84
APPENDIX...................................................................................................................86
v
LIST OF FIGURES
Figure
Page
2.1 Possible single-walled nanotube configurations ........................................................5
2.2 Schematic of nanotube structure................................................................................6
3.1 Minimum potential energy distance for graphite sheet.............................................20
3.2 Total potential energy of the argon atom / graphite sheet system.............................20
3.3 Argon atom locations above graphite sheet .............................................................21
3.4 Minimum potential energy distance for carbon nanotube ........................................22
3.5 Total potential energy of the carbon nanotube system .............................................22
3.6 Pass-through potential energy of graphite sheet system ...........................................24
3.7 Argon initial placement, end view...........................................................................25
3.8 Argon initial placement, side view ..........................................................................25
3.9 Minimized energy state, end view ...........................................................................26
3.10 Nanotube placed into argon fluid ..........................................................................28
3.11 Graphite wall placed into model............................................................................29
3.12 Identifying argon atoms with different colors........................................................30
3.13 Beginning of MD simulation.................................................................................32
3.14 MD simulation 30% complete...............................................................................32
3.15 MD simulation complete.......................................................................................33
4.1 End view, rigid (10,10) SWNT ...............................................................................38
vi
4.2 Side view, cross-section, rigid (10,10) SWNT.........................................................39
4.3 Argon movement through rigid (10,10) SWNT .......................................................40
4.4 Argon fluid hydrostatic pressure inside rigid (10,10) SWNT ...................................43
4.5 Argon fluid hydrostatic pressure outside rigid (10,10) SWNT .................................44
4.6 End view, rigid (20,20) SWNT ...............................................................................45
4.7 Side view, cross-section, rigid (20,20) SWNT.........................................................46
4.8 Argon movement through rigid (20,20) SWNT .......................................................47
4.9 Argon fluid hydrostatic pressure inside rigid (20,20) SWNT ...................................48
4.10 End view, relaxed (10,10) SWNT .........................................................................49
4.11 Side view, cross-section, relaxed (10,10) SWNT...................................................50
4.12 Argon movement through relaxed (10,10) SWNT .................................................52
4.13 Argon fluid hydrostatic pressure inside relaxed (10,10) SWNT .............................53
4.14 End view, relaxed (20,20) SWNT .........................................................................54
4.15 Side view, cross-section, relaxed (20,20) SWNT...................................................55
4.16 Argon movement through relaxed (20,20) SWNT – 0th Pumping ..........................56
4.17 Argon fluid hydrostatic pressure inside relaxed (20,20) SWNT .............................57
4.18 Argon movement through relaxed (20,20) SWNT – 1st Pumping...........................58
4.19 Argon movement through relaxed (20,20) SWNT -2nd Pumping ...........................59
4.20 Argon fluid average hydrostatic pressure inside relaxed (20,20)
SWNT during three successive pumping cycles, taken at same
step 500 of simulation...........................................................................................60
5.1 End view, rigid (10,10)@(15,15) DWNT ................................................................63
5.2 Side view, cross-section, rigid (10,10)@(15,15) DWNT .........................................64
5.3 Argon movement through rigid (10,10)@(15,15) DWNT........................................65
vii
5.4 Argon fluid hydrostatic pressure inside rigid (10,10)@(15,15) DWNT....................66
5.5 End view, rigid (15,15)@(20,20) DWNT ................................................................67
5.6 Side view, cross-section, rigid (15,15)@(20,20) DWNT .........................................68
5.7 Argon movement through rigid (15,15)@(20,20) DWNT........................................69
5.8 Argon fluid hydrostatic pressure inside rigid (15,15)@(20,20) DWNT....................70
5.9 End view, relaxed (10,10)@(15,15) DWNT ............................................................71
5.10 Side view, cross-section, relaxed (10,10)@(15,15) DWNT ...................................73
5.11 Argon movement through relaxed (10,10)@(15,15) DWNT..................................74
5.12 Argon fluid hydrostatic pressure inside relaxed (10,10)@(15,15)
DWNT..................................................................................................................75
5.13 End view, relaxed (15,15)@(20,20) DWNT ..........................................................76
5.14 Side view, cross-section, relaxed (15,15)@(20,20) DWNT ...................................77
5.15 Argon movement through relaxed (15,15)@(20,20) DWNT..................................78
5.16 Argon fluid hydrostatic pressure inside relaxed (15,15)@(20,20)
DWNT..................................................................................................................79
5.17 Ends of relaxed (10,10)@(15,15) DWNT vs. relaxed (10,10) SWNT
at finish of simulation ...........................................................................................80
5.18 Ends of relaxed (15,15)@(20,20) DWNT vs. relaxed (15,15) SWNT
at finish of simulation ...........................................................................................80
5.19 Midpoint of relaxed (15,15)@(20,20) DWNT at finish of simulation ....................81
viii
CHAPTER I
INTRODUCTION
1.1 Objective of the Thesis
The purpose of this thesis is to investigate the behavior of argon fluid flow
through carbon nanotubes. Carbon nanotubes are atomic-sized hollow cylinders
possessing unique properties. Scientists and engineers have been exploring nanotubes for
the last 30 years in an effort to unlock their potential to revolutionize structural, electrical,
and medical devices. In previous investigations of fluid flow though nanotubes, fluid was
placed at the mouth of the nanotube and given an initial velocity towards the tube like
shooting bullets from a gun. The fluid atoms then slow to a stop. In this paper we instead
push the initially stationary fluid toward the mouth of the nanotube with a moving-wall
graphene piston. This wall continuously applies pressure to the fluid to keep it moving.
The goal was to answer several questions: Can fluid be pushed through a
nanotube? How does nanotube diameter affect the flow? Will the flow change if the
nanotube surface is held rigidly in place or allowed to relax and deform? Can pumping
action act to continuously push the fluid down the length of the nanotube? How does
flow through a single-walled nanotube compare to a double-walled nanotube?
1
1.2 Overview of the Thesis
A simulation was run using computer modeling software and molecular dynamics
tools. In the simulation, argon fluid held at a constant temperature and volume was
pushed through a carbon nanotube by a moving wall graphite piston. Argon was chosen
because it is a commonly used element that is abundant, stable, and resistant to bonding
with other elements. Different configurations and combinations included single- and
double-walled nanotubes, and rigid and relaxed nanotubes. Programs were written to
collect the results and data was tabulated using MATLAB software. Results were
compared and contrasted for argon fluid position, pressure, and configuration.
Chapter II of the thesis gives background on carbon nanotube history, physical
properties, and structure geometry. It also discusses fluid flow at the atomic scale and
recent related research. Nanofluidics is introduced with recent experimental work in areas
of biology, chemistry, and physics. Molecular dynamics simulations involving fluids and
nanostuctures are discussed.
Chapter III starts by explaining the molecular dynamics simulation technique. The
interactions between carbon sheets and carbon nanotubes and argon atoms are
investigated. The steps behind constructing a fluid cell, building a model, and running a
simulation are explained. Reasons are given for why the model was built in a particular
way. Screenshots show how the model looked before, during, and after the simulation
was run.
Chapter IV discusses the results for single-walled nanotubes for different
diameters in both rigid and relaxed configurations. Views of the end and side of the
nanotube are displayed. The arrangement of argon atoms both inside and outside the
2
nanotube is presented. Plots of argon atom movement through and around the tube are
shown at increments of the simulation. Pressure calculations were preformed and plotted
at different increments. The repeated pumping action needed to push the argon fluid
completely through the end of a nanotube is shown in figures. Contour pressure plots are
given for selected nanotube cases.
Chapter V discusses results of double-walled nanotubes for different diameters in
both rigid and relaxed configurations. Chapter VI consists of conclusions and possibilities
for future investigations.
3
CHAPTER II
BACKGROUND
2.1 What are Carbon Nanotubes?
A carbon nanotube is an atomic structure formed of carbon atoms linked as a
molecule into a long, hollow cylinder. The prefix “nano” means extremely small or
microscopic. A nanometer (nm) is one billionth of a meter, or 1/1,000,000,000 m. In
comparison, the diameter of a human hair is about 100,000 nm or 1/10,000 m. Nanotubes
can be thousands of times longer than they are wide. A nanotube can be single-walled, or
can be arranged concentrically into double-walled or multi-walled nanotubes. Nanotubes
have been studied since the 1950s, but it was not until the early 1990s that research
intensified. It was at this time that single-walled carbon nanotubes were discovered along
with the methods to produce them. Nanotubes can be created using a number of methods.
The most widely used methods are arc-discharge, laser ablation, and chemical vapor
deposition.
Nanotubes have novel physical properties. The unique arrangement of the
chemical bonds gives a nanotube its very high strength. Nanotubes are the strongest and
stiffest materials known in terms of tensile strength and elastic modulus, and have a low
density as well. Depending on its structure, a nanotube can conduct electricity like a
metal or a semiconductor. They are expected to be excellent thermal conductors and be
4
able to withstand high temperatures. In the future nanotubes may be used to develop new
technology such as medical implants, military armor, and tiny computers. As more is
understood about how fluid flows through a nanotube, new applications may include drug
delivery, battery storage, and more efficient filters.
2.2 Nanotube Structure
Single-walled carbon nanotubes are similar to rolled-up rectangular strips of
hexagonal graphite sheets. There are only so many ways to roll-up the sheet and connect
the lattice structure. Therefore, single-walled nanotubes come in three configurations:
zigzag, armchair, and chiral.
Figure 2.1 - Possible single-walled nanotube configurations [1]
5
Zigzag tubes have some carbon-carbon (C-C) bonds aligned parallel to the tube
axis. Armchair tubes have some bonds perpendicular to the tube axis. Chiral tubes have a
left- or right-handed screw axis, similar to a DNA helix.
The structure of a single-wall carbon nanotube is specified by the circumferential
vector
r
r
r
C h = n1 ⋅ a1 + n 2 ⋅ a 2
2.1
where a1 and a2 are basis vectors for graphite. Ch is an example of a chiral vector. Each
nanotube is specified by the two integers, n1 and n2. In Figure 2, (n1,n2) = (5,1).
Figure 2.2 - Schematic of nanotube structure
Zigzag tubes are denoted as (n,0), armchair as (n,n), and chiral as (n1,n2).
Armchair nanotubes were used in this research study.
The unit cell length along the nanotube axis is determined by the smallest lattice
vector perpendicular to Ch. This perpendicular vector is
6
r {(n1 + 2n2 ) ⋅ ar1 − (2n1 + n2 ) ⋅ ar2 }
T =
q
2.2
where
(n − n2 ) ≠ mod(3)
N
if 1
q=
(n1 − n 2 ) ≡ mod(3)
3N
2.3
and N is the greatest common divisor of n1 and n2. The larger the value of T, the longer
the nanotube will be.
The nanotube radius is given by
R=
Ch
3d
=
n12 + n 22 + n1n 2
2π
2π
2.4
where d=1.42 angstroms (Å) is the C-C bond length. Therefore, (10,10), (15,15), and
(20,20) nanotubes would have radii of 6.78 Å, 10.18 Å, and 13.57 Å, respectively.
2.3 Fluid Flow at the Nanoscale
Nanotechnology is the design and application of structures on the atomic and
molecular scale. A subfield of nanotechnology is nanofluidics, the study of fluid flow
through and around nano-sized tubes and channels. Fluid flow on the nanoscale is
different than flow on the macroscale due to drastically different length scales. The
character of a flow through a channel or pipe depends on the dimensionless Reynolds
number
Re =
vρd
2.5
η
where v = velocity, ρ = density, d = pipe diameter, and η = viscosity, or the fluid’s
resistance to flow. When the Reynolds number is small the fluid flow is laminar. When
7
the Reynolds number is large, the flow is turbulent. For smaller pipe diameters, viscosity
has a greater affect than velocity or density. This means that liquid that flows freely on
the macroscale flows like honey on the nanoscale. In addition, for small Reynolds
numbers the flow is always laminar and there will be no turbulence. In laminar flow it
can be difficult for two fluids to mix quickly without introducing obstacles or circulation
into the flow stream to promote turbulence and diffusion [2].
For a circular pipe, the flow rate is determined by the Poiseuille equation
∆P =
128µLQ
πd 4
2.6
where ∆P = pressure drop, µ = dynamic viscosity, L = length of the pipe, Q =
volumetric flow rate, and d = pipe diameter. The diameter of the pipe, raised to the forth
power, is the dominating factor. For small diameter pipes the flow rate drops
dramatically, making it difficult to push fluids through nano-sized tubes. These scaling
effects are some of the obstacles which scientists and engineers must overcome to design
and fabricate nanofluidics devices. But there are many promising areas where
nanofluidics may soon become applicable.
2.4 Recent Nanofluidic Research and its Applications
The following gives some recent examples of nanofluidics in the areas of biology,
chemistry, engineering, and others.
Nanofluidics has been applied to soil science to understand how water flows
though plants [3]. The capillaries in plants can have diameters in the nanometer range.
Plants move water through their system using the different energy levels of the water at
8
different locations in the system. Water potential is the potential energy stored in a unit of
water compared to a unit of water at standard temperature and pressure. Water will tend
to move from an area of high water potential to one of lower water potential if acted on
by gravity, diffusion, or pressure. Water potential is what drives plants to pull water at a
higher potential from the ground by their roots to a lower potential in the atmosphere by
their leaves. The knowledge of water flow in plants can be used to predict the flow of
water in similar man-made devices such as labs-on-a-chip.
Lab-on-a-chip devices are another important field where nanofluidics is applied.
A lab-on-a-chip is a piece of glass or silicon a few square millimeters in size that can
perform laboratory functions by itself. An example is a device that tests for traces of lead
contamination in the environment. Lead is toxic to humans and animals even in small
concentrations. Scientists would like to be able to test for lead pollution using a cheap
and reliable device sensitive enough to detect concentrations down to parts per billion.
Researchers created a lead biosensor that used an array of nanocapillaries to control small
volumes of fluid moving between compartments on the sensor [4]. The capillaries consist
of two intersecting channels 50 nm wide, 30 nm deep and 14 nm long. A voltage was
applied to move the lead solution though the capillaries. It was found that the device
could be reused and was sensitive enough to monitor lead in drinking water.
Clean drinking water is in fact another area where nanofluidics is being applied.
In places where clean fresh water is not available, seawater can be desalinated to remove
salts and minerals from the water and make it safe for human consumption and crop
irrigation. But this process is very energy intensive and costly. With demand for water
growing, engineers have tried using nanotubes to purify water [5]. Many desalination
9
plants clean water by the process of reverse osmosis. Reverse osmosis uses pressure to
push dirty water through a semi-permeable membrane. The dissolved salts cannot pass
through the membrane and accumulate on its surface while the cleaned water passes
through to the other side. One possible way to reduce the amount of pressure required for
this process (3 to 6 MPa), and hence the amount of electricity and its cost, is to use a
membrane integrated with carbon nanotubes. The inside of the nanotubes are smooth and
hollow, and are 1 to 2 nanometers in diameter. The membrane filter consists of silicon
nitride film perforated by thousands of these nanotubes. When eighty-nine of the 50 µm
by 50 µm square membranes were laid out in a array (a combined area about the size of a
dime), it was found that a pressure of just 100 kPa, or one atmosphere, was needed to
push water across the film [6]. If a tiny device like that could be enlarged and designed to
block salts and minerals it could substantially lower the amount of pressure and energy
required for desalination in the future.
Nanofluidics is also being applied to area of biofluids. New devices are being
looked at to see if bacteria can be used as tiny motors to pump small volumes of liquids.
In one such device, live E. coli bacteria cells were attached to the sides of a microfluidic
channel [6]. E. coli are cylindrical in shape, 2 to 6 µm long and 0.5 to 0.8 µm in diameter.
The bacterium moves by rotating its long, thin tails called flagella, which rotate at about
600 rpm to create torque to push the cell along. Researchers tethered a harmless stain of
the bacteria in a row along a channel filled with fluid. As the cells moved their flagella
they pushed the fluid down the channel. The researches found that the E. coli should be
able to pump about 0.25 nL of fluid per minute. Delivering precise amounts of fluid is
exactly what future micro and nano-scale devices would require. These mechanical
10
pumps have an advantage over similar pumps that use electrical charges in the fluid, in
that they can work in both conductive and non-conductive fluids. The pump may also
have the ability to heal itself if damaged since it is built from living organisms.
Controlling the direction of flow in a channel is just as important as pumping the
flow. Directing fluid flow in microchannels can be done with a valve that opens and
closes. Researchers have created a valve using a hydrogel that expands when in contact
with an acidic fluid and contracts in the presence of a basic fluid [2]. The gel was placed
at channel intersections like a road block. Exposure to high pH swells the gel and closes
up the channel, while a low pH shrinks the gel and allows fluid to pass. In this way one
network of channels could control two different fluids in two different directions.
Biological science can benefit from nanofluidics in another similar way. Cells
transport ions across their membranes through tiny pores on the cell surface by a nanopumping action. Researchers have reproduced this phenomenon by creating a model
using a sheet of plastic perforated with holes [8]. When an electric field is applied, the
voltage can pump ions from one side of the sheet to the other. The cone-shaped pores had
a base diameter of 50 nm and open-ended tip diameter of 2 nm, produced by shooting
gold atoms through the thin plastic. The sheet was placed in a potassium chloride
solution, and began to pump potassium ions through the pores from the narrow end to the
wide end when the electric field was applied across the sheet. If the pores can be made
smaller on a thinner sheet they would be very similar in size to pores in biological cells.
Nanofluidics not only involves flow though nanotubes and nanochannels but also
how fluid interacts at the surface of nanotubes. Researchers have investigated how a
water droplet would interact with the ends of nanotubes. In one experiment, a “forest” of
11
vertically aligned carbon nanotubes was grown on a substrate [9]. The scientists looked to
see if a droplet of water falling onto the canopy of the nanotube forest would be absorbed
or not. It was found that ordinary carbon nanotubes were hydrophilic and water was
absorbed into the voids between the tubes like rain on a forest of trees. But when a
uniform coating of the polymer Teflon was deposited onto the top of the nanotubes, it
was found that the nanotubes became hydrophobic a repelled the water droplet. This
same effect can be seen in nature when a water droplet falls onto a leaf and rolls right off.
In fact a high-speed camera observed that the Teflon coating made a falling droplet
bounce back off the nanotube surface like a trampoline. These coated nanotubes could be
used to create microscopic surface coatings that shed water and resist contamination.
This nanosurface science is being applied in the study of stiction in engineering
design. Stiction, or static friction, is the force required to cause one body in contact with
another to begin to move. Stiction can become a big problem for nano-sized devices and
structures. At the nano-scale, electrostatic forces are relatively strong and can tend to glue
two nearby surfaces together. Two adhered surfaces can be difficult to separate without
damaging their fragile nanostructure. One way to prevent stiction between nano-surfaces
is to apply a coating of carbon nanotubes, much like non-stick coating on cookware.
These nanotubes are applied to a nanosurface in a “forest” arrangement sticking upwards
from the surface. In one example, Latex microbeads were applied to the nanotube coating
and found to evenly disperse and could easily be moved across the nanotube-coated
surface with a microlever. These results were more favorable than those observed with a
surface coated with a smooth diamond-like carbon [10].
12
Nanofluidics is emerging as a new important area of nanoscience. Its applications
in nanotechnology will become important building blocks for future nanomachines and
devices. Nanofluidics can be applied to critical areas of engineering, medicine, and
chemistry that will no doubt create new research, design, and manufacturing jobs.
2.5 Recent Research on Fluid Flow through Carbon Nanotubes
Fluid flow in and around nanotubes and nanostructures is a growing area of
research. Work has been done which looks at liquid argon flow along the outside of a
carbon nanotube to investigate drag forces and drag coefficients [11]. Molecular
dynamics (MD) simulations were performed and the results were compared to both
empirical equations based on experiments conducted on macroscale cylinders, as well as
finite element analyses based on Navier-Stokes equations for flow past cylinders. This
work was different from ours in that their MD setup used argon atoms initially evenly
spaced and given the identical initial speed. In addition, their work focused on only rigid
nanotube structures. Their report concluded that classical continuum mechanics cannot
be used to calculate drag coefficients on a nanotubes surface. First, it was shown that as
the flow velocity decreased, the difference increased between the drag coefficients of the
MD simulations and the two other cases. Second, when the flow velocity in the MD
simulation was high, the drag coefficient was lower than the other two cases due to argon
slippage on the nanotube surface. Third, single- and double-walled nanotubes having the
same outer diameter were compared and found to have nearly the same drag forces. This
suggests that the inside tube in the double-walled case does not interact with the argon
fluid molecules.
13
Other work has been done investigating the effect of size on the flow rate of water
through a carbon nanotube using MD simulations [12]. Contrary to what would be
expected to happen, it has been shown that as the diameter of a nanotube decreases and
the fluid viscosity increases, the liquid flow rates can be orders of magnitude faster than
flow at the macroscale. In this simulation water molecules placed inside various sized
rigid single-walled carbon nanotubes were given an initial acceleration to create a
uniform axial flow inside the tube. As the flow slows down the shearing stress between
the tube wall and the water molecules was calculated. This work differs from ours by
using only rigid nanotubes, by varying the fluid flow rate, and by applying an initial
acceleration to each atom and then removing the force. The results showed that the shear
stress is the key factor in concluding that the viscosity of the confined liquid water is a
function of both tube size and flow rate.
Researchers have also looked at different ways to push fluid through nanotubes
and nanostructures. One method uses MD simulation to investigate driving a fluid down a
nanoscale channel using a temperature gradient at the fluid-solid interface [13]. This
phenomenon is known as thermal transpiration and generates a thermophoretic force. In
this work four different periodically-repeating nanochannel arrangements were
considered in order to optimize the device configuration. This work differs from ours by
using platinum wall surfaces instead of carbon nanotubes, and by giving the argon atoms
random initial velocities. Their results showed one of the systems considered produced a
significant pumping effect. It used a thermally-insulating slip wall which conducts little
heat and has a weak interaction with the fluid molecules. This allowed it to develop a
large temperature gradient yet guide the flow easily with minimal disturbance.
14
Investigations have also looked at the effect of gravity force on argon liquid
movement through nanochannels [14]. A molecular dynamics simulation modeled the
flow as confined between two molecular walls. A uniform acceleration was applied
parallel to the walls, inducing a flow as if the liquid argon were falling due to gravity.
This simulation differed from ours in that nanochannels of various sizes were used
instead of nanotubes, randomly distributed initial velocities were assumed for the argon
atoms, and two different amounts of argon atoms were used. They discovered that the
density profiles have a peak value near the wall surface, indicating that the liquid
particles are packed much more closely near the wall surface than that far away from the
wall. They also found that at small gravity forces the axial velocity profile in the channel
was small due to stronger interactions between the fluid and wall. However, at higher
gravity forces, the differences of velocity profiles among the strong and weak fluid/wall
interactions became smaller.
Another analysis has looked at the flow of helium and argon fluid through carbon
nanotubes [15]. The nanotubes investigated had varying diameters and lengths, and were
either held rigid (frozen) or allowed to relax (dynamic). This simulation differed from
ours in that the helium and argon atoms were given five different initial velocities, and
there were four different amounts of fluid atoms present. They found that the relaxed
nanotubes slowed down the fluid flow faster than a rigid tube. They attribute this to the
nanotube movement disturbing the motion of the nearby fluid atoms, which more quickly
randomizes the fluid motion and leads to collisions with the tube. They also saw that
helium flows through the nanotube easier than argon. This is due to argon being 10 times
15
more massive than helium, and can therefore create larger amplitude vibrations in the
nanotube.
16
CHAPTER III
THE METHODS AND THE MODEL
3.1 Molecular Dynamics
Molecular dynamics (MD) is a computer simulation method used by scientists to
theoretically study the movement of atoms and molecules over a period of time. It acts to
bridge the gap between theory and experiment. MD has become widely used within the
last few decades as high-speed computers became available. MD uses a mathematical
model (theory) to run a simulation (experiment) from which results can then be analyzed.
The technique works by integrating the equations of motion of a set of atoms or
molecules which are interacting with each other. It uses Newton’s Second Law of Motion
r
r
d 2 ri
Fi = mi 2
dt
3.1
r
where Fi is the force on each atom due to it’s interactions with neighboring atoms, mi is
r
the atom’s mass, and ri is the position vector of the atom i. The acceleration can be
integrated numerically over time to find the atom’s velocities and positions. As positions,
velocities, and accelerations change with time, all trajectories of each atom are
developed.
17
This microscopic information, such as pressure, temperature, and energy is
converted to macroscopic information by statistical mechanics. Statistical mechanics can
be applied to a set of trajectories to come up with an average value of the atoms’
properties. This is done using a statistical ensemble, a very large set of similar systems
which are considered all at once. Ensembles have different microscopic states but have
identical macroscopic or thermodynamic states. Common statistical ensembles used in
simulations are the canonical or constant-NVT (N = number of particles, V = volume, T
= temperature), micro-canonical or constant-NVE (E = energy), and isothermal-isobaric
or constant-NPT (P = pressure). An ensemble average is average taken over a large
number of replicas of the system considered simultaneously. It is assumed that the time
average equals the ensemble average. The basic idea is that if one allows the system to
evolve in time indefinitely, that system will eventually pass through all possible states.
One goal, therefore, of a molecular dynamics simulation is to generate enough
representative arrangements of the parts of the structure such that this equality is
satisfied. If this is the case, experimentally relevant information concerning structural,
dynamic and thermodynamic properties may then be calculated using a feasible amount
of computer resources [16].
The first step in this project was to learn how to use the computer software tools
available. The machine used was a Silicon Graphics, Inc. (SGI) computer workstation
running the UNIX operating system. Loaded on this machine was Cerius2, a popular
software package used for molecular modeling [17,18]. The project would focus on using
this one piece of software which has a graphical user interface and scripting
(programming) capabilities.
18
3.2 Energy Considerations
The first decision to make in the design of this experiment was to pick an
appropriate fluid to flow through the nanotube. The element argon (Ar) was chosen due
to its full outer shell of electrons that makes it form a simple liquid with no complicated
interactions. Argon is a relatively abundant and inexpensive product that is used in many
applications. By using argon as the fluid, a simple, usable, and practical model was
created.
Once the choice of fluid was determined the next step was to investigate the
behavior of the argon atoms as they interacted with a carbon nanotube. A simple way to
do this was to unravel a carbon nanotube to form a graphite sheet. Graphite is a name
given to a form of carbon that is arranged in such a way that it forms flat sheets that can
stack up on top of one another. A square graphite (or graphene) sheet was created in
Cerius2 50 Å (1 Å = 1/10 nm) wide on each side by inserting a carbon atom into a blank
workspace and linking it to neighboring carbon atoms. Next a single argon atom was
placed at a location 15 Å above the middle of the graphite sheet. The graphite was
constrained in place and then the argon atom was moved toward the sheet in steps of one
Å. See Figure 3.1.
19
Figure 3.1 - Minimum potential energy distance for graphite sheet
The potential energy of the system was calculated at each step. In addition, the
variation of total potential energy as a function of argon atom – graphene sheet separation
was obtained. The results are shown in Figure 3.2.
Single Argon Atom - Graphite Sheet
Energy vs. Distance Relationship
880
Hollow Site
Bridge Site
Corner Site
Energy (kcal/mol)
870
860
850
840
830
820
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Distance from surface (Angstrom)
Figure 3.2 – Total potential energy of the argon atom / graphite sheet system
Figure 3.2 shows that as the argon atom approaches the graphite sheet the
potential energy of the system begins to decrease and reaches a minimum at a distance of
20
3.4 Å from the sheet. As the atom moves closer, the energy of the system grows rapidly.
This information meant that an argon atom tends to move to a state of minimum potential
energy located at a distance of 3.4 Å from the graphite sheet.
Figure 3.3 - Argon atom locations above graphite sheet
This calculation was performed three times, each time placing the argon atom
directly above the hollow site, bridge site, and corner site with similar results. See Figure
3.3. The results were similar and it could be seen that moving the argon atom in the
horizontal directions would be easier as different site energies are close. The hollow site
has the lowest potential energy and therefore argon atoms will prefer that location.
Now that it was known how argon behaved near a sheet of carbon, the next step
was to investigate how an argon atom reacted with an actual carbon nanotube. To do this
a pre-assembled single-walled carbon nanotube of radius 6.78 Å and length 50 Å was
placed into the cell. The nanotube was created to the desired dimensions by a computer
program provided by the research advisor. The nanotube was fixed in place and an argon
atom was inserted into the model at a distance 15 Å from the outside surface of the tube.
21
The argon atom was moved toward the tube in one-Å increments and the total potential
energy of the system was calculated. See Figure 3.4.
Figure 3.4 - Minimum potential energy distance for carbon nanotube
The results are shown in Figure 3.5.
Single Argon Atom - Carbon Nanotube
Energy vs. Distance Relationship
3795
Middle - Outside
End - Outside
Middle - Inside
End - Inside
Energy (kcal/mol)
3790
3785
3780
3775
3770
3765
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Distance from tube surface (Angstrom)
Figure 3.5 – Total potential energy of the carbon nanotube system
22
Figure 3.5 shows that as the argon atom approaches the carbon nanotube the
potential energy of the system begins to decrease and reaches a minimum around 3.4 Å
from the tube. As the atom moves closer to the tube, the energy of the system grows
rapidly. Similar to the graphite sheet, this information said that an argon atom tends to
move to a state of minimum potential energy at a distance of 3.4 Å from the nanotube.
This calculation was performed four times, each time placing the argon atom either at the
ends of the tube or at middle of the tube for locations both inside and outside the tube.
Similar results were seen for each location. It could be concluded that moving the atom to
different locations around the nanotube would not affect the distance at which minimum
potential energy occurs.
Once a location had been established for where the argon atoms would like to
settle near the nanotube, it was time to determine whether an argon atom could somehow
pass through the wall of a nanotube. If argon could move across the nanotube wall
through the gaps between the carbon atoms, then the tube would behave like a leaky
garden hose. To investigate this possibility, the previous calculations were extended in
which an argon atom was moved closer to the surface of both a graphite sheet and a
carbon nanotube until it passed through to the other side. The total potential energy at
each step is shown in Figure 3.6.
23
Pass-Through Test: Single Argon Atom - Graphite Sheet
Energy vs. Distance Relationship
160000
140000
Energy (kcal/mol)
120000
100000
80000
60000
40000
20000
0
-3
-2
-1
0
1
2
3
Distance from surface (Angstrom)
Figure 3.6 – Pass-through potential energy of graphite sheet system
The results showed that the amount of energy it would require to force an argon
atom through the wall of a carbon nanotube, equivalent to about 566 Mega Joules or 157
kilowatt hours, was so great that one could safely assume that it would not happen.
The next step in this project was to determine how argon atoms placed near a
carbon nanotube would behave when the energy of the system was minimized. A
minimum energy state is the desired state a system tends to find when the energy applied
to keep it in a certain configuration is removed. This part of the project was performed by
creating a carbon nanotube and placing argon atoms along the outside and inside axes of
the tube, as seen in Figures 3.7 and 3.8.
24
Figure 3.7 - Argon initial placement, end view
Figure 3.8 - Argon initial placement, side view
The energy of the system was then minimized until the argon atoms came to rest.
The argon atoms moved to form concentric rings a distance of 3.4 Å from the nanotube
all along the length of the tube. See Figure 3.9.
25
Figure 3.9 - Minimized energy state, end view
The results show that for locations near the nanotube, argon tends to settle at a
distance of about 3.4 Å away from the tube surface. This agreed with the results from the
previous steps discussed above. By this point in the project it had been determined how a
system of argon atoms and carbon nanotube would behave. The next step was to
construct a model to move fluid through the tube.
3.3 Building a Model
The first step needed to create a fluid-flow model was to construct argon fluid.
This was accomplished by first forming a solid argon crystal. Argon is a solid at
temperatures less than 83.8 K (-308.8 °F) and a gas at temperatures higher than 87.3 K (302.5 °F). This meant that in order to use argon as a fluid in the model, the temperature
had to be in a narrow 4 K band between 83 and 87K.
The simulation cell was heated at constant pressure (NPT) from its initial
temperature of 0 K (-460 °F) up to 85 K by running a molecular dynamics simulation.
Molecular dynamics (MD) can recreate on a computer a system of atoms which can
26
interact with each other and move around as they would in nature for a specified amount
of time under given constraints of temperature, pressure, and volume. We can turn solid
argon into a liquid using MD by increasing the temperature of the unit cell in increments
of 5 K and 500 MD steps per increment. As the argon cell was slowly heated one could
observe how the crystal structure changed from argon atoms rigidly constrained in fixed
positions to where the argon atoms began to move about and become less orderly. By the
time the temperature had reached 85 K the argon atoms had moved all around and
become disordered, resembling a fluid.
At this point the simulation cell of argon fluid needed to be enlarged to be able to
hold a completely immersed carbon nanotube. The dimensions of this fluid box were
critical because this model was using periodic boundary conditions. Periodic boundary
conditions meant that if a box of argon fluid with finite dimensions of width x, height y,
and length z was placed into an empty model, it essentially fills the entire space by
repeating itself in every direction with individual cells of fluid. While only one cell of
fluid is simulated, there were invisible identical cells stacked all around it. If one atom
exited the boundary on one end of the fluid cell, it would immediately re-enter on the
opposite end of the same cell. To create the correct size of fluid cell, the information
found from the first experiments investigating locations of minimum potential energy
was used. The first nanotube had a radius of 6.78 Å and argon atoms tended to have no
reaction to the nanotube at distances of 15 Å and greater. To ensure the argon atoms in
the fluid cell would not interact with the nanotubes in the neighboring fluid cells the
dimensions were set to 54 Å wide by 54 Å high by 126 Å long. These dimensions were
multiples of the small simulation cell of the argon fluid.
27
Once the argon fluid was created and given the necessary size and shape it was
time to insert the carbon nanotube into it. A model of a nanotube with radius 6.78 Å and
50 Å in length was placed into the fluid model. The nanotube was then moved into
position midway in the width and height directions and flush with one end of the cell, as
seen in Figure 3.10.
Figure 3.10 - Nanotube placed into argon fluid
When the nanotube was moved into position it potentially overlapped many argon
atoms. To move the argon atoms away from the walls of the tube the energy of the new
system was minimized. Care had to be taken to ensure that the energy minimization was
used sparingly. If used too much the argon fluid would begin to loose its fluid properties
and turn back into a solid, a state of lower energy. Therefore the energy minimization
was run for only five steps at a time. After each minimization it could be seen that the
system’s potential energy dropped by several orders of magnitude. As the energy of the
system began to level out after 10 minimizations as the argon atoms moved away from
28
the walls of the nanotube the minimization process was terminated. Finally, a 500-step
NVT MD simulation was run to equilibrate the system and ensure the argon was still in a
fluid state.
The next step was to find a way to get the argon fluid to flow through the
nanotube. The approach used was to push the argon into the tube like a piston in a
cylinder. To do this a graphite sheet was constructed with the same width and height of
the fluid box and inserted vertically into the model at the end opposite the nanotube. See
Figure 3.11.
Figure 3.11 - Graphite wall placed into model
Before the wall was inserted, a small vertical band of argon atoms was deleted at
the location where the wall was to be placed to ensure there would be no overlapping of
atoms.
In order for the graphite wall to be able to push the argon atoms through the
nanotube it needed to be kept rigid. This was done by selecting the entire wall and fixing
its atoms in place. In the first simulation the entire nanotube was also constrained. In this
29
way the nanotube would not be pushed out of the box of argon fluid when the wall began
pushing the argon atoms toward it.
Now the model was complete and it was time to figure out a way to push the
argon fluid through the nanotube. One method would be to select the graphite wall and
move it in small increments in the direction of the nanotube. Once the wall had moved
one increment, MD could be run for a few hundred steps. This process could be repeated
many times until the wall moved across the length of the fluid cell to reach the entrance
of the nanotube. Doing this would be a laborious and time-consuming process. A short
computer program was created to automate this task.
The color of the argon atoms was varied in bands like a rainbow so that later
when the atoms became spread apart and mixed up due to the MD, one could easily know
from where an argon atom started. See Figure 3.12.
Figure 3.12 - Identifying argon atoms with different colors
The graphite wall was moved, followed by MD steps, and the process was
repeated over and over until the wall had moved across the fluid cell. The wall was set to
move a step-size distance of 0.1 Å in the axial direction and run 500 MD simulations per
30
step. The temperature of the system was held at 90 K and the volume of the system was
held constant.
Simulations of argon fluid flow were run for four different carbon nanotube
configurations. The first configuration was a single-walled nanotube that was rigidly
constrained along its entire surface. The second configuration was a single-walled
nanotube that was rigidly constrained at its ends only, while the rest of its surface was in
a relaxed state, free to move about. The third and fourth configurations were the same as
the first two except a double-walled nanotube was used instead.
The double-walled nanotube used in this simulation was created by inserting into
the argon fluid two concentric single-walled nanotubes. Once the tubes were placed into
the model a few MD steps were run to move any overlapping argon atoms away from the
surfaces of the nanotubes.
When the MD simulation on each model was complete, the resulting model
showed argon atoms pushed out of the fluid box in the direction of wall movement. To
correct this picture, a program was created that would take any argon atoms that had
passed out of the boundary of the fluid box and place them back in their correct location
inside the box. This could be done due to the periodic boundary condition imposed on the
model. The program was written to find the position of every argon atom in the model
and compare its x, y and z coordinates to the width, height, and length dimension of the
fluid box, respectively. If the argon atom had a location outside the box it was moved
back inside by a distance of the difference between its location and the box’s dimensions.
This produced the correct picture of the model.
31
A sample simulation can be seen in Figures 3.13, 3.14 and 3.15. Figure 3.13
shows the model at the beginning of the simulation before the wall has moved toward the
nanotube. Figure 3.14 shows a side-view of the simulation when it is 30% complete.
Figure 3.15 shows how the model appeared after the simulation had completed.
Figure 3.13 - Beginning of MD simulation
Figure 3.14 - MD simulation 30% complete
32
Figure 3.15 - MD simulation complete
33
CHAPTER IV
ARGON FLUID FLOW THROUGH SINGLE-WALLED CARBON NANOTUBES
4.1 Collecting Data for Analysis
In this chapter we begin by explaining how the results of the simulation were
collected. Reasons are given for why the results are presented in a particular way. Next,
the results are shown for the single-walled carbon nanotubes. Different combinations of
nanotube diameter for both rigid and relaxed configurations are analyzed. Screenshots of
tube deformation for both end and side views are shown. Argon fluid position and argon
fluid pressure are calculated and plotted.
As mentioned in Chapter II, each argon atom’s position was incrementally saved
as an (x, y, z) coordinate data point as the simulation was performed. The trick was to
find a way to take this data and transfer it to a PC for analysis.
Each argon atom was assigned a unique ID number by the Cerius2 software.
However the ID numbers were not in any particular order. For instance, if there were
3000 argon atoms in the system, they were not numbered 1, 2, 3…2998, 2999, 3000.
They were numbered randomly such as 5, 6, 13 …345, 347, 351…4563, 5345, 5487.
This made it difficult to write a simple program to differentiate carbon atoms from argon
atoms by ID number.
34
Several steps were taken to solve this problem. First, the model’s computer file
was opened in Cerius2 showing the entire argon fluid, carbon nanotube, and graphite wall
system at its initial state before any simulation was run. Next, each rainbow colored band
of argon atoms between the moving graphite wall and the entrance of the nanotube was
individually selected. The coordinates of each colored band was already known from the
previous computer code which had given the argon atoms their respective colors in an
earlier step. Using this information, to start, all atoms colored “red” were selected and
kept in place, while all the remaining atoms in the system were temporarily deleted. Now
with only the “red” atoms on the screen the computer was told to list all atom ID
numbers. This list was output to a text file and saved for future use. Next, the system file
was reloaded and the next colored argon band in the sequence, “green”, was selected and
the ID numbers saved. This was continued until all the colored argon atoms’ ID numbers
were saved separately by groups of color.
Next, the positions of each colored band of argon atoms could be tracked at
different increments during the simulation from start to finish. Snapshots of the
simulation had been saved every 50 steps from the start at step 0 to the end at step 700. It
was decided to focus on just a few increments of the simulation: the initial step 0, step
250, step 500, and the end, step 700. This would keep the analysis simple and still
sufficiently show how the argon fluid flowed over time.
A new program was written to open the snapshot files for steps 0, 250, 500, and
700. Next the list of colored argon atoms’ ID numbers was pasted into the program and
instructed to list the (x, y, z) coordinates for each color at each step and save the results in
a text file. For example, the program would open the snapshot file at step 500, use the ID
35
numbers to find all the “yellow” colored atoms, and output as a list each one’s position at
that point in the simulation. This process was repeated until each argon color band was
recorded for each step during the simulation for each configuration of single/doublewalled and rigid/relaxed nanotubes. Now the argon fluid flow was completely known
from start to finish.
Next the argon position data files were transfer to a PC where they could be
analyzed. MATLAB was chosen as the software to plot the argon positions because of its
ease of programming and its powerful plotting capability. A MATLAB program was
written to take all the colored argon (x, y, z) coordinate positions and recreate the
snapshot of the simulation model at each increment. From this the data could be used to
generate any type of plot to show how the argon atoms were moving through and around
the nanotube. It was decided to simplify the output data to show only a narrow band of
argon atoms, as wide as the diameter of the nanotube, which would show a cross-section
view of the model. This cross-section would show argon flow both through the inside and
around the outside of the nanotube, which allowed for easy comparing and contrasting.
At the same time it would exclude flow that was far from and only around the outside of
the tube.
An 11th order polynomial curve fit was applied to show an average position for
the color bands as they moved during the simulation. A rectangle was drawn on the plot
to show the correct size and location of the carbon nanotube cross-section. The argon was
pushed from right to left by the moving wall (not shown in plots).
36
4.2 Rigid Single-Walled Carbon Nanotubes
This section will discuss the results of argon fluid flow through carbon nanotubes
in the rigid, single-walled configuration.
4.2.1 (10,10) Carbon Nanotube
Figures 4.1 and 4.2 show screenshots of the (10,10) rigid SWNT taken at the end
of the simulation. Figure 4.1 looks down the axis of the nanotube from the end. It shows
the nanotube as a purple ring rigidly constrained in place, surrounded by argon atoms of
different colors both inside and outside the tube. The argon atoms inside the tube have
settled along the axis of the tube and formed one concentric circular ring shape around
the nanotube axis. The argon atoms outside the tube have also arranged themselves in
concentric rings around the nanotube. Notice the argon rings inside and outside are
equidistant from the surface of the nanotube, and also equidistant from the row of argon
atoms running down the nanotube axis. Argon liquid becomes ordered like a solid inside
this atomic scale confinement.
37
Figure 4.1 – End view, rigid (10,10) SWNT
Figure 4.2 is a side view of the (10,10) rigid SWNT. A cross-section of the
nanotube has been taken which removed all argon atoms around the outside of the tube to
clearly show the argon atoms inside the tube. The nanotube shows its armchair
configuration and holds its rigid shape by the constraints placed on it by the model. Here
the first two colored bands of argon atoms from outside the end of the nanotube, white
and orange, can be seen to have been pushed into the tube from the right by the moving
wall. The white color has progressed through to the middle of the nanotube, while the
orange reached only about one-third of the way inside. Three to four argon atoms placed
side-by-side is all that a (10,10)-sized nanotube can accommodate.
38
Figure 4.2 - Side view, cross-section, rigid (10,10) SWNT
Figure 4.3 plots how the argon atoms moved in and around the (10,10) rigid
SWNT. It shows the progress of the simulation at four points in time: the beginning (step
0 of 700), one-third of the way through (step 250 of 700), two-thirds through (step 500 of
700), and at the end (step 700 of 700). At step 0 the moving wall is at a location on the
horizontal z-axis of 198 Å, at step 250, 123 Å, at step 500, 78 Å, and at step 700, 53 Å.
The mouth of the nanotube is located at 49.2 Å. Step 0 shows the initial argon atom
configuration of colored bands. White is nearest to the mouth of the nanotube, followed
from left to right by black (orange), light blue, yellow, dark blue, green, and red. The
moving wall (not shown) is immediately to the right of the red band of argon. The axes
mark the distance in units of Å. The moving wall therefore travels from z = 120 Å to z =
50 Å on the horizontal axis, a distance of 70 Å, or 0.1 Å per step for a total of 700 steps.
The rectangle in Figure 4.3 represents the size, shape, and location of the nanotube.
39
The argon atoms shown in Figure 4.3 are taken from a cross-section of the model
that was as wide as the diameter of the smallest nanotube investigated. In this case the
smallest-diameter tube was a (10,10) with a diameter of 13.56 Å. Therefore a slice 13 Å
wide, centered at the axis of the tube, was used on each nanotube discussed in this paper.
Figure 4.3 – Argon movement through rigid (10,10) SWNT
Figure 4.3 shows a number of interesting results. First, the colored bands begin to
move toward the nanotube uniformly and evenly spaced as seen in step 250. But as the
simulation continues, the bands become jagged and mixed together. By step 700 the
bands have become so mixed up that one can not distinguish the original configuration.
Second, the argon flows farther around the outside of the nanotube than through the
inside. The argon atoms on the outside are not constrained by either the walls of the
40
nanotube or the small tube diameter which is only a few argon atoms in width. Third, the
three bands closest to the mouth of the nanotube, white, black (orange), and light blue,
flow the farthest into the nanotube. Forth, the white, black (orange), and light blue bands
move fairly quickly into the tube, but slow down and never make it completely through
the tube. This could be due to the lack of additional argon atoms in front of the moving
wall to provide pushing power by the time the simulation is nearing completion and the
wall is near the end of its stroke.
The local hydrostatic pressure of the argon fluid was calculated using the equation
P=−
1
∂E
=−
3V
∂V
∑ r φ ′ (r )
ij ij
4.1
ij
i< j
where rij = distance between argon atoms i and j, φij′ = the derivative of potential energy
with respect to position, E = total potential energy, and V = volume of a region under
investigation. In this calculation only argon atoms were considered. The equation takes
into account the pressure on the argon fluid “plug” or region from both neighboring argon
and carbon atoms. Each dot on the graph represents the pressure value for that region.
The cell was divided along the horizontal z-axis into 32 regions with a length of
approximately four Å per region.
Figure 4.4 shows the hydrostatic pressure of the argon fluid for the rigid (10,10)
case. It plots the pressure inside the nanotube along its length. A positive pressure value
means that the fluid is being compressed and wants to expand, and vice versa. The same
position data that was used to create Figure 4.3 was used to develop the pressure values
for the same four points during the simulation: initial wall position at step 0 located at
123 Å along z-axis (blue line), step 250 at 98 Å (black line), step 500 at 73 Å (red line),
41
and step 700 at 53 Å (green line). A rectangle is drawn in the bottom left corner of the
figure to show the position and length of the nanotube during the simulation. The
horizontal axis shows the distance in Å, with 0 Å < z < 50 Å as the location of the
nanotube, and 50 Å < z < 80 Å as the area just before the entrance to the nanotube. The
vertical axis shows the hydrostatic gauge pressure of the argon fluid in units of GPa.
First, Figure 4.4 shows the pressure of the argon fluid as it nears the entrance of
the nanotube is around zero GPa. This indicates that the argon is not facing much force
and wants to neither expand nor contract. Second, at step 0 the pressure inside the
nanotube is negative. This negative pressure indicates the fluid wants to be pulled into the
nanotube. Third, as the simulation progresses and fluid enters the nanotube the pressure
increases significantly at steps 250, 500, and 700. This indicates the fluid is being
squeezed by the moving wall as it tries to push the argon into the tight fit of the small
diameter tube. This explains why the fluid has difficulty flowing completely through the
nanotube as seen in Figure 4.3, step 700. Additional pressure contour plots are provided
in Appendix A.
42
Figure 4.4 – Argon fluid hydrostatic pressure inside rigid (10,10) SWNT
Figure 4.5 plots the hydrostatic pressure of the argon fluid outside the nanotube
along its length during the simulation. Here it can be seen that the pressure on the argon
fluid is nearly zero outside the nanotube at each step of the simulation. This means the
fluid is free to move easily around the outside of the tube. This agrees with Figure 4.3,
where the argon around the outside of the tube moved farther than those inside the tube.
43
Figure 4.5 – Argon fluid hydrostatic pressure outside rigid (10,10) SWNT
4.2.2 (20, 20) Carbon Nanotube
Figures 4.6 and 4.7 show screenshots of the (20,20) rigid SWNT taken at the end
of the simulation. Figure 4.6, like Figure 4.1, looks down the axis of the nanotube from
the end and shows the nanotube as a purple ring rigidly constrained in place surrounded
by argon atoms. Again the argon atoms inside the tube have settled along the axis of the
tube. But in this case the argon has formed three concentric circular ring shapes around
the nanotube axis rather than just one. This is due to the larger diameter of the (20,20)
tube which is twice the size of a (10,10) tube, allowing space for more rings to form yet
still stay the minimum distance away from the nanotube walls. The argon atoms outside
the tube have once again arranged themselves in concentric rings around the nanotube.
44
Just like the (10,10) case, the argon rings inside and outside are equidistant from the
surface of the nanotube, and also equidistant from the row of argon atoms running down
the nanotube axis.
Figure 4.6 - End view, rigid (20, 20) SWNT
Figure 4.7 is a side view of the (20,20) rigid SWNT. Here each of the seven
colored bands of argon atoms from outside the end of the nanotube, white, orange, light
blue, yellow, dark blue, green and red, can be seen to have been pushed into the tube
from the right by the moving wall. The white and orange colors have moved the farthest
through the nanotube, with the white nearly to the end of the nanotube. Eight to nine
argon atoms placed side-by-side can fit into a (20,20)-sized nanotube.
45
Figure 4.7 - Side view, cross-section, rigid (20,20) SWNT
Figure 4.8 plots how the argon atoms moved in and around the (20,20) rigid
SWNT. First, the colored bands again begin to move toward the nanotube uniformly and
evenly spaced as shown in step 250. But this time the bands do not become so mixed
together by the end of the simulation as in the rigid (10,10) case. This is likely due to the
larger diameter tube which allows the argon to flow easier through tube. Second, the
argon still flows farther around the outside of the nanotube than through the inside. Even
though the nanotube has a larger diameter, it still offers more resistance to flow through
the inside than around the outside. Third, the three bands closest to the mouth of the
nanotube, white, black (orange), and light blue, also flow the farthest into the nanotube.
Forth, the white, black (orange), and light blue bands move fairly quickly into the tube,
but again slow down and never make it completely through the tube. They come to a stop
at nearly the same location as in the rigid (10,10) case.
46
Figure 4.8 - Argon movement through rigid (20,20) SWNT
Figure 4.9 shows the hydrostatic pressure of the argon fluid for the rigid (20,20)
case. At steps 250 and 500 the pressure both before the entrance to the nanotube and
inside the tube is nearly zero. This indicates the fluid is not facing much resistance to
flow and the argon can enter and flow through the tube, which agrees with Figure 4.8.
However, by step 700 the pressure begins to increase inside the tube. This means the
argon is experiencing resistance, which agrees with Figure 4.8, step 700 where the atoms
have slowed to a stop. At every step the pressure is lower for the rigid (20,20) case than
the rigid (10,10) case, meaning the argon fluid flows better through a larger diameter
tube.
47
Figure 4.9 - Argon fluid hydrostatic pressure inside rigid (20,20) SWNT
4.3 Relaxed Single-Walled Carbon Nanotubes
This section will discuss the results of argon fluid flow through carbon nanotubes
in the relaxed, single-walled configuration.
4.3.1 (10,10) Carbon Nanotube
Figures 4.10 and 4.11 show screenshots of the (10,10) relaxed SWNT taken at the
end of the simulation. The relaxed nanotube appears differently than the rigid kind
previously discussed. Figure 4.10 shows the relaxed nanotube with a purple ring on the
ends of the tube and gray in the middle. The purple bands rigidly constrain the tube in
48
place so that it is not pushed out of place by the moving wall. The grey middle is not
constrained and is allowed to deform due to the interaction with its neighboring carbon
atoms and the surrounding argon atoms. By the end of the simulation it no longer is a
perfectly circular shape, but instead has some areas that have moved toward the center of
the tube and others that have moved away from the center.
In Figure 4.10 the argon atoms both inside and outside the tube appear nearly
random and barely resemble the concentric rings of the rigid (10,10) case. The reason for
this unequal spacing is due to the deformation of the nanotube wall. The argon inside the
tube is found to be disordered. We do not have a conclusion that it is like a liquid or a
solid.
Figure 4.10 - End view, relaxed (10,10) SWNT
49
Figure 4.11 is a side view of the (10,10) relaxed SWNT. The cross-section shows
how the sides of the nanotube have been deformed. It also shows the argon atoms moving
up and down to match the tube movement, compared to Figure 4.2 where the argon was
neatly arranged. Here green and red argon atoms have been pushed into the nanotube, in
addition to the white and orange argon atoms in the rigid (10,10) case. This may be due
to way in which the nanotube deformed at different times during the simulation. The red
and green atoms may have happened to be near the mouth of the nanotube when the
deformation allowed them to move inside. The white color has progressed farthest
through the inside of the nanotube, but not nearly as far as the in the rigid (10,10) case.
Again the tube deformation likely impeded the flow of argon. Here barely three atoms
can fit side-by-side, creating a tighter fit than the rigid (10,10) case.
Figure 4.11 - Side view, cross-section, relaxed (10,10) SWNT
50
Figure 4.12 plots how the argon atoms moved in and around the (10,10) relaxed
SWNT. Figure 4.12 shows a number of results similar to the rigid (10,10) case. Again
the colored bands begin to move toward the nanotube uniformly and evenly spaced as
seen in step 250. And again as the simulation continues, the bands become jagged and
mixed together by step 700. The argon flows farther around the outside of the nanotube
than through the inside due to the small tube diameter, as clearly seen in step 500. And
again the three bands closest to the mouth of the nanotube, white, black (orange), and
light blue, flow the farthest into the nanotube, but not as far as seen in step 500. Here the
white and black (orange) bands move fairly quickly into the tube, yet slow down and
never make it completely through the tube. In contrast to the rigid (10,10) case, the light
blue color does not progress as far as into the tube, as shown in step 500. In step 700 the
position data becomes erratic. This may be due to the nanotube deformation that is
pushing and pulling the argon in different directions and constraining the space inside the
nanotube.
51
Figure 4.12 - Argon movement through relaxed (10,10) SWNT
Figure 4.13 shows the hydrostatic pressure of the argon fluid for the relaxed
(10,10) case. The plot shows a sharp peak at the entrance of the nanotube at steps 250 and
700. In the middle of the nanotube there is negative pressure at every step. These features
are likely due to the deformation of the nanotube walls, which pinch together in the case
of high pressure or expand apart in the case of low pressure. When compared to the rigid
(10,10) case, the relaxed (10,10) case has a similar looking pressure outside the mouth of
the nanotube. But the pressure inside the tube appears much different, having mostly
negative values for the relaxed case whereas the rigid case had all positive pressure
values. This must mean that the ability of the nanotube to deform can significantly
change the way fluid flows though it. The peaks near the mouth of the nanotube represent
52
a crowding effect due to the moving wall. As argon atoms become stuck just inside the
mouth of the tube the pressure builds. Meanwhile, further inside the tube there are fewer
argon atoms and therefore they have more room to move about, causing the pressure to
drop.
Figure 4.13 - Argon fluid hydrostatic pressure inside relaxed (10,10) SWNT
4.3.2 (20,20) Carbon Nanotube
Figures 4.14 and 4.15 show screenshots of the (20,20) relaxed SWNT taken at the
end of the simulation. In Figure 4.14 the argon atoms both inside and outside the tube
again appear to be random, but under closer review one can see the three inner concentric
rings and one outer ring similar to the rigid (20,20) case. This unequal spacing can be
attributed to the deformation of the relaxed nanotube wall. Sections of the nanotube wall
53
have moved toward the center axis of the tube and others have moved away from the
center.
Figure 4.14 - End view, relaxed (20,20) SWNT
Figure 4.15 is a side view of the (20,20) relaxed SWNT. The cross-section clearly
shows the deformed sides of the nanotube. Here argon atoms from each color band have
been pushed into the nanotube just like the rigid (20,20) case. But the argon has not
moved through the tube as far as in the rigid (20,20) case, likely due to the tube
deformation constraining the fluid flow. Again the white color has progressed farthest
through the inside of the nanotube to nearly the half-way point, but not as far as the in the
rigid (20,20) case where it reached the end of the tube.
54
Figure 4.15 - Side view, cross-section, relaxed (20,20) SWNT
Figure 4.16 plots how the argon atoms moved in and around the (20,20) relaxed
SWNT. First, the colored bands again begin to move toward the nanotube uniformly and
evenly spaced as shown in step 250, just like every other SWNT case previously
discussed. By step 500 the bands are still uniformly spaced and the argon flow around the
outside has moved farther down the tube than inside. The flow has not moved as far as in
the rigid (20,20) case, due to the tube deformation. However, the flow has moved farther
than in the relaxed (10,10) case due to a larger tube diameter. In step 700 the flow does
not become as erratic-looking as the relaxed (10,10) case, likely due to the larger
diameter tube allowing the argon more room to move uniformly and orderly.
55
Figure 4.16 – Argon movement through relaxed (20,20) SWNT – 0th Pumping
Figure 4.17 plots the hydrostatic pressure of the argon fluid inside the nanotube
along its length during the simulation. Here it can be seen that again the pressure on the
argon fluid is nearly zero before the mouth of the nanotube at each step of the simulation.
But the pressure inside the tube is higher than the rigid (20,20) case. This agrees with the
position data, which showed the flow was better through the rigid (20,20) tube which saw
no tube deformation.
56
Figure 4.17 – Argon fluid hydrostatic pressure inside relaxed (20,20) SWNT
4.3.3 Repeated Pumping through (20,20) Carbon Nanotube
An additional simulation was run to investigate if repeated pumping action could
finally push the argon fluid completely through the length of the nanotube and out the
end. The relaxed (20,20) case was selected for this test for two reasons. One, the larger
diameter compared to the (10,10) case would allow the argon to flow better through the
tube. Two, the relaxed configuration rather than rigid would more accurately represent
the characteristics of an actual nanotube. The pumping action was achieved by running
the original relaxed (20,20) simulation to completion. Next, the argon atoms that had
been pushed out of the “box” were selected and moved back to their correct position
according to the periodic boundary conditions placed on the model. Then, the moving
57
wall of graphite was selected and moved back to its original position so that it could
begin its pumping stroke again. After two addition pumping simulation runs (Pump “1”
and “2”), the argon fluid was finally pushed through the nanotube.
Figures 4.18 and 4.19 show the pumping actions taken on the relaxed (20,20)
SWNT. In Figure 4.18 the argon fluid starts out at step 0 in nearly the same position as
the original simulation left off in Figure 4.16 step 700. By the time Pump1, step 700 is
reached the white, orange (black) and light blue argon atoms are still inside the nanotube.
One addition pumping action was then run, as shown in Figure 4.19.
Figure 4.18 - Argon movement through relaxed (20, 20) SWNT – 1st Pumping
By the time Pump 2, step 700 was reached the white, orange (black) and light
blue color bands have been completely pushed out the end of the nanotube. From this it
58
can be said that a nanotube is not always plugged and fluid flow through nanotubes is
possible.
Figure 4.19 - Argon movement through relaxed (20,20) SWNT -2nd Pumping
Figure 4.20 shows the average hydrostatic argon fluid pressure at step 500 that
was calculated and plotted to compare to the results of the pumping position data. The
three lines indicate the three pumping cycles: the original simulation, called Pump 0
(black line, P0), the first pumping action, Pump 1 (red line, P1), and the second pumping
action, Pump 2 (green line, P2). Figure 4.20 shows the pressure inside the nanotube along
its length. The pressure does not deviate much from zero, and along most of the length is
a negative value. This indicates the fluid is not being compressed, but instead is allowed
to flow relatively freely down the length of the tube. Each of the three pumping cases has
59
nearly the same pressure values, indicating that the flow is experiencing the same
environment in each simulation. The deep trough at the right of Figure 4.20 shows the
location of the moving wall piston during each of the simulations. These pressure values
were calculated at step 500, at which time the moving wall would be about 20 Å to the
right of the mouth of the nanotube. The extremely low pressure at this location is due to
the argon atoms just behind the moving wall rushing to fill in the empty space created by
the wall as it travels from right to left.
Figure 4.20 – Argon fluid average hydrostatic pressure inside relaxed (20, 20) SWNT
during three successive pumping cycles, taken at same step 500 of simulation
60
In conclusion, several main points can be made about argon fluid flow through
SWNT. First, argon flows better around the outside of the nanotube than through the
inside. This occurs because the fluid inside the nanotube is constrained by the tube walls
and the argon atoms must stay a minimum distance away from the carbon atoms. Second,
argon flows better through rigid tubes than relaxed because the relaxed tubes have
deformed walls which impede the fluid flow. Third, argon flows better through (20,20)
tubes than (10,10) tubes due to a larger diameter which allows more argon atoms to gain
access to the inside of the nanotube. Forth, repeated pumping action is required to push
argon atoms completely through the length of the nanotube. Pumping places a new set of
argon atoms between the mouth of the nanotube and the moving wall piston. This
provides the pushing power to move the argon further down the nanotube interior.
61
CHAPTER V
ARGON FLUID FLOW THROUGH DOUBLE-WALLED CARBON NANOTUBES
5.1 Rigid Double-Walled Carbon Nanotubes
This section will discuss the results of argon fluid flow through carbon nanotubes
in the rigid, double-walled configuration.
5.1.1 (10,10)@(15,15) Carbon Nanotube
Figures 5.1 and 5.2 show screenshots of the (10,10)@(15,15) rigid DWNT taken
at the end of the simulation. Figure 5.1 looks down the axis of the nanotube from the end.
It shows the nanotubes as two purple rings rigidly constrained in place, surrounded by
argon atoms of different colors both inside and outside the tube. The argon atoms inside
the inner (10,10) nanotube have settled along the axis of the tube and formed one
concentric circular ring shape around the nanotube axis, just like the rigid (10,10) SWNT
case. The argon atoms outside the tube have also arranged themselves in concentric rings
around the nanotube, as in the previous rigid SWNT cases. Again the argon rings inside
and outside are equidistant from the surface of the nanotube, and also equidistant from
the row of argon atoms running down the nanotube axis. The difference in the DWNT
cases is the presence of a ring of argon atoms between the (10,10) and the (15,15)
nanotubes. These atoms came to be in this location when the DWNT was placed into the
62
argon fluid. During the subsequent energy minimizations and molecular dynamics these
argon atoms maintained their position between the nanotube walls.
Figure 5.1 - End view, rigid (10,10)@(15,15) DWNT
Figure 5.2 is a side view of the (10,10)@(15,15) rigid DWNT. A cross-section of
the nanotube has been taken which removed all argon atoms around the outside of the
tube to clearly show the argon atoms inside the tube. Here only the white colored argon
atoms from outside the end of the nanotube have been pushed into the tube from the right
by the moving wall. One green and one dark blue atom are near the mouth of the
nanotube, but their entrance likely occurred at the end of the simulation and does not
appear to be part of the fluid flow through the tube. The white color has progressed
through to the half-way point of the nanotube just as in the rigid (10,10) SWNT case.
Between the nanotube walls are a few light blue argon atoms. These atoms were initially
63
in that position and did not move in or out of the tube during the simulation. Nor did new
colored argon atoms enter from the right. This indicates the space between the nanotubes
is too narrow to conduct argon fluid flow.
Figure 5.2 - Side view, cross-section, rigid (10,10)@(15,15) DWNT
Figure 5.3 plots how the argon atoms moved in and around the (10,10)@(15,15)
rigid DWNT. The rectangles in Figure 5.3 represent the size, shape, and location of the
DWNT. There are similarities to the rigid (10,10) SWNT case. One, the colored bands
begin to move toward the nanotube uniformly and evenly spaced as seen in step 250, but
mix together by step 700. Two, the argon flows farther around the outside of the
nanotube than through the inside. This is clearly seen in the white, black (orange), and
light blue lines in step 500. There are contrasts to the rigid (10,10) SWNT case as well.
At step 500 the white argon atoms are the only colored band that has entered the inner
64
(10,10) tube, whereas in the SWNT case both white and black (orange) had entered by
that time. This agrees with the cross-section view shown in Figures 5.2. This reduction in
flow in the rigid (10,10)@(15,15) DWNT compared to the rigid (10,10) SWNT must be
explained by the presence of the outer (15,15) nanotube. A DWNT has more carbon
atoms than a SWNT, which at the mouth of the nanotube may act to repel the argon
atoms from entering the tube due to the minimal energy distance.
Figure 5.3 – Argon movement through rigid (10,10)@(15,15) DWNT
Figure 5.4 shows the hydrostatic pressure of the argon fluid for the rigid
(10,10)@(15,15) case. Here the pressure outside the mouth of the nanotube is nearly zero
during each step. But the fluid inside the nanotube has positive values, and by step 700
65
reaches quite a high pressure. This spike in pressure at the end of the simulation may be
caused by the moving wall piston pushing the argon fluid up against the DWNT nanotube
having a larger cross-sectional area than that of a SWNT. In general these results
resemble those of the rigid (10,10) SWNT case. This is expected because the fluid is
flowing through essentially the same inner rigid (10,10) tube.
Figure 5.4 - Argon fluid hydrostatic pressure inside rigid (10,10)@(15,15) DWNT
5.1.2 (15,15)@(20,20) Carbon Nanotube
Figures 5.5 and 5.6 show screenshots of the (15,15)@(20,20) rigid DWNT taken
at the end of the simulation. Again the argon atoms inside the tube have settled along the
axis of the tube. But in this case the argon has formed just two concentric circular ring
66
shapes around the nanotube axis rather than just one. This is due to the diameter of the
(15,15) tube, which is larger than a (10,10) tube but smaller than a (20,20) tube. From
this it can be said that for rigid nanotubes, there is room for one inner ring of argon atoms
in a (10,10), two rings in a (15,15), and three rings in a (20,20). The argon atoms outside
the tube have once again arranged themselves in concentric rings around the nanotube.
And again the argon rings inside and outside are equidistant from the surface of the
nanotube, and also equidistant from the row of argon atoms running down the nanotube
axis. As in the (10,10)@(15,15) rigid DWNT case there is a ring of argon atoms between
the two nanotubes that was there since the tubes were placed into the fluid at the
beginning.
Figure 5.5 - End view, rigid (15,15)@(20,20) DWNT
Figure 5.6 is a side view of the (15,15)@(20,20) rigid DWNT. Here some of each
of the seven colored bands of argon atoms from outside the end of the nanotube has been
67
pushed into the tube. This also occurred in the rigid (20,20) SWNT case. But in this case
fewer of each color entered the tube. Those that did enter did not travel as far through the
tube. This is likely due to the smaller inner diameter of the (15,15) nanotube restricting
the flow compared to a (20,20). Only seven to eight argon atoms placed side-by-side can
fit into a (15,15)-sized nanotube The white and orange colors have moved the farthest
through the nanotube. No colored argon atoms flowed into the space between the
nanotube walls, just as in the rigid (10,10)@(15,15) DWNT case.
Figure 5.6 - Side view, cross-section, rigid (15,15)@(20,20) DWNT
Figure 5.7 plots how the argon atoms moved through and around the
(15,15)@(20,20) rigid DWNT. The flow is more uniform and evenly spaced than the
rigid (10,10)@(15,15) DWNT as seen in steps 500 and 700. As in every previous case,
the argon flows farther around the outside of the nanotube than through the inside, seen
clearly in step 700. In contrast to the rigid (10,10)@(15,15) DWNT, the inner diameter of
68
the (15,15) inner nanotube must be large enough such that the presence of the additional
outer (20,20) nanotube does not interfere with the fluid flow through the tube.
Figure 5.7 - Argon movement through rigid (15,15)@(20,20) DWNT
Figure 5.8 shows the hydrostatic pressure of the argon fluid for the rigid
(15,15)@(20,20) case. Here the pressure inside the tube at steps 250 and 500 is near zero
or slightly negative, indicating the fluid is flowing through the tube without obstruction.
This agrees with the movement of the argon atoms in Figure 5.7. Overall the pressure
inside the tube is lower at every step than in the rigid (10,10)@(15,15) case. This is due
to the larger diameter (15,15) inner tube allowing less-restricted flow than a (10,10) tube.
69
Figure 5.8 - Argon fluid hydrostatic pressure inside rigid (15,15)@(20,20) DWNT
5.2 Relaxed Double-Walled Carbon Nanotubes
This section will discuss the results of argon fluid flow through carbon nanotubes
in the relaxed, double-walled configuration.
5.2.1 (10,10)@(15,15) Carbon Nanotube
Figures 5.9 and 5.10 show screenshots of the (10,10)@(15,15) relaxed DWNT
taken at the end of the simulation. Figure 5.9 shows the nanotubes as two purple rings
rigidly constrained in place at their ends with the relaxed section shown in grey
surrounded by argon atoms of different colors both inside and outside the tube. The argon
atoms inside the inner (10,10) nanotube have formed one nearly concentric circular ring
70
around the nanotube axis. This ring is more circular than the relaxed (10,10) SWNT case,
due to two related reasons. First, the outer (15,15) nanotube keeps the inner (10,10) tube
from deforming too much. Second, this lack of inner wall deformation keeps the argon
fluid in the (10,10) tube from moving out of the circular shape to match the deformed
wall. The argon atoms outside the tube have also arranged themselves in concentric rings
around the nanotube, as in the previous relaxed SWNT cases. Again the argon rings
inside and outside are equidistant from the surface of the nanotube walls. The major
difference between this relaxed DWNT case and the rigid DWNT cases is the absence of
argon atoms between the inner and outer nanotubes. These argon atoms were selected and
deleted on purpose before the simulation began. The reason for this was to ensure the
model was created to most accurately represent what a real-world relaxed DWNT would
look like. It was decided that the space between the two nanotubes was too narrow to
accept argon atoms during the simulation, so they were left out to begin with.
Figure 5.9 - End view, relaxed (10,10)@(15,15) DWNT
71
Figure 5.10 is a side view of the (10,10)@(15,15) relaxed DWNT. The crosssection of the nanotube shows how the argon atoms were removed between the inner and
outer nanotubes before the simulation began. By the end of the simulation only one light
blue argon atom has entered this space at the left side at the end of the nanotube. This is
likely do to random argon movement at the end of the simulation. By that time most of
the argon fluid would be to the left of the nanotube and must have had enough reactive
force to push the argon backwards into the tube. In this cross-section the lack of nanotube
wall deformation in the vertical direction can be seen. The deformation is not as
pronounced as in the previous relaxed SWNT cases. A few white and two green argon
atoms have been pushed into the inner nanotube by the moving wall. The white argon
atoms are both fewer in number and do not travel as far into the tube as in the rigid
(10,10)@(15,15) SWNT case. This is due to the deformation of the relaxed nanotube
walls pinching the tube and restricting the flow. In the rigid (10,10)@(15,15) DWNT
three to four argon atoms can fit side-by-side inside the inner tube, but in the relaxed case
there is room for only two or three. The relaxed (10,10) SWNT had more colored argon
bands present inside the nanotube compared with this case. However, in both that case
and this one the argon moved only about one-forth of the way into the nanotube from the
right side.
72
Figure 5.10 - Side view, cross-section, relaxed (10,10)@(15,15) DWNT
Figure 5.11 plots how the argon atoms moved in and around the (10,10)@(15,15)
relaxed DWNT. After starting off as a uniform flow in step 250, by step 500 only the
white argon atoms have entered the inner tube while the other colors pass around the
outside. This agrees with what is shown above in Figure 5.10.
73
Figure 5.11 - Argon movement through relaxed (10,10)@(15,15) DWNT
Figure 5.12 shows the hydrostatic pressure of the argon fluid for the relaxed
(10,10)@(15,15) case. Here the dominating feature is a spike of high pressure near the
exit of the nanotube which occurs at each step in the simulation. The pressure there is
nearly three times larger than the values recorded in any other previous case. This may be
caused by a few argon atoms entering the space between the nanotubes from the left-hand
side, as seen in Figure 5.10. The location of these argon atoms matches the location of the
pressure spike. These argon atoms interacting with the relaxed nanotube walls could
cause deformation which would pinch the tube and create the pressure buildup.
74
Figure 5.12 - Argon fluid hydrostatic pressure inside relaxed (10,10)@(15,15) DWNT
5.2.2 (15,15)@(20,20) Carbon Nanotube
Figures 5.13 and 5.14 show screenshots of the (15,15)@(20,20) relaxed DWNT
taken at the end of the simulation. Inside the inner (15,15) tube the argon atoms appear
random, but looking closely one can see two slightly concentric rings like in the rigid
(15,15)@(20,20) DWNT case. This again suggests the argon atoms are reacting to the
deformation of the relaxed nanotube walls. The outer (20,20) relaxed nanotube does not
deform as much as the inner (15,15) tube. This is similar to the relaxed (10,10)@(15,15)
DWNT case, where the outer (15,15) tube did not change shape as much as the inner
(10,10) tube. As a result, the argon atoms around the circumference of the outer nanotube
form a more concentric ring shape similar to the rigid nanotube cases. Lastly, a few argon
75
atoms appear to be in the area between the inner and outside nanotubes. These atoms are
only near the entrance or exit of the nanotube a do not represent a real flow though the
tubes in that space.
Figure 5.13 - End view, relaxed (15,15)@(20,20) DWNT
Figure 5.14 is a side view of the (15,15)@(20,20) relaxed DWNT. The crosssection shows argon atoms from each colored band have entered the inner nanotube from
the right side. This flow has both more argon atoms and flow which progresses further
down the tube than the relaxed (10,10)@(15,15) DWNT. But this flow has both fewer
argon atoms and the flow does not go as far into the tube as the rigid (15,15)@(20,20)
DWNT. In this case the argon atoms have noticeable spaces between them. There is
room for five to six argon atoms side-by-side in the inner (15,15) tube. Compared this to
the rigid (15,15)@(20,20) DWNT, which had more tightly arranged argon atoms with
very little space between them, as well as room for six to seven argon atoms in the inner
76
tube. The two light blue argon atoms between the two nanotube walls at the end of the
tube were once again pushed in by the interacting argon that had gone around the outside
and settled around the left side of the DWNT.
Figure 5.14 - Side view, cross-section, relaxed (15,15)@(20,20) DWNT
Figure 5.15 plots how the argon atoms moved in and around the (15,15)@(20,20)
relaxed DWNT. In step 500 one can see that the flow has progressed further into the tube
than the relaxed (10,10)@(15,15) DWNT, but not as far as in the rigid (15,15)@(20,20)
DWNT case.
77
Figure 5.15 – Argon movement through relaxed (15,15)@(20,20) DWNT
Figure 5.16 shows the hydrostatic pressure of the argon fluid for the relaxed
(15,15)@(20,20) case. The dominant feature is the pressure spike near the tube exit at
each simulation step, similar to the relaxed (10,10)@(15,15) case. Here the pressure rises
to 500 GPa, not as high as the (10,10)@(15,15) case but still much higher than the
highest pressure values in any other case. Again this may be explained by the presence of
argon atoms that have entered the space between the nanotube walls at the tube exit. A
few of these argon atoms are shown in the cross-section view in Figure 5.14 and match
the location of the pressure spike.
78
Figure 5.16 - Argon fluid hydrostatic pressure inside relaxed (15,15)@(20,20) DWNT
Next the deformed shape of the double-walled relaxed nanotubes was compared
to the single-walled relaxed tubes. This was done to see if the pressure spikes seen in the
relaxed (10,10)@(15,15) and (15,15)@(20,20) cases shown above in Figures 5.12 and
5.16 were indeed due to the pinching and narrowing of the inner shell nanotube. Figure
5.17 shows the cross-section of a relaxed (10,10)@(15,15) DWNT beside a relaxed
(10,10) SWNT taken at the end of the simulation at step 700. We can see from this figure
that the inner (10,10) tube in the DWNT on the left has more deformation than the
(10,10) SWNT on the right.
79
Figure 5.17 – Ends of relaxed (10,10)@(15,15) DWNT vs. relaxed (10,10) SWNT at
finish of simulation
Figure 5.18 shows the cross-section of a relaxed (15,15)@(20,20) DWNT beside
a relaxed (15,15) SWNT taken at the end of the simulation at step 700. Here can clearly
that the inner (15,15) tube in the DWNT on the left has more deformation than the
(15,15) SWNT on the right.
Figure 5.18 – Ends of relaxed (15,15)@(20,20) DWNT vs. relaxed (15,15) SWNT at
finish of simulation
80
In contrast, Figure 5.19 shows the cross-section taken in the middle of the
nanotube rather than at the end for a relaxed (15,15)@(20,20) DWNT. Here the shape of
the tube has not deformed as much as at the end. This explains why the pressure graph in
Figure 5.16 does not have a spike of high pressure at the midpoint of the nanotube.
Figure 5.19 – Midpoint of relaxed (15,15)@(20,20) DWNT at finish of simulation
81
CHAPTER VI
CONCLUSIONS
The purpose of this work was to study the flow of fluid through carbon nanotubes.
To do this, molecular dynamics simulations were preformed, atomic structure of
nanotubes and argon fluid were analyzed, and pressures at different spatial regions were
calculated. Nanotubes of different sizes and configurations were placed in a cell of argon
fluid. A moving wall of graphite continuously pushed the argon atoms towards the mouth
of the nanotube. We compared and contrasted nanotubes of different diameters, singlewalled and double-walled nanotubes, and rigid and relaxed tube walls. Repeated pumping
action was tried to see if argon could be pushed through the length of the nanotube.
This work differs from previous investigations in several ways. Earlier studies
assigned initial velocities to the fluid atoms; used only rigid nanotubes; varied the fluid
flow rate; applied an initial acceleration to each atom and then removed the force; looked
at platinum wall surfaces instead of carbon nanotubes; used nanochannels of various sizes
instead of nanotubes; assigned randomly distributed initial velocities for the argon atoms;
used two different amounts of argon atoms; used helium and argon atoms together;
assigned five different initial velocities; used four different amounts of fluid atoms.
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This work has answered several questions. First, is it possible for fluid to be
pushed through a nanotube? In each case in can be seen that, yes, the argon fluid entered
the mouth of the nanotube and moved a distance down the length of the tube. The argon
flows further around the outside of the nanotube than through the inside in every case.
Second, how does the flow appear different in nanotubes of different diameters? It
was shown that as the diameter of the tube increased the volume of argon atoms inside
the tube increased. Also, the argon atoms moved further down the inside length of the
tube in larger diameter nanotubes.
Third, how does flow compare between a nanotube with relaxed walls and one
with rigid walls? The results showed that the flow moved further into the nanotubes
having rigid walls than those with relaxed walls. A relaxed nanotube can not
accommodate as many argon atoms because the deformation of the tube’s walls act to
pinch off the flow of fluid.
Forth, can fluid be continuously pushed down the length of the nanotube in a
pumping action? It was shown that by moving the graphite wall back to its initial position
and running the simulation again, it was possible to push the argon atoms through the
length of the tube in the case of the rigid (20,20) single-walled nanotube.
Finally, does the flow differ in a single-walled nanotube from that of a doublewalled nanotube? The results show that for nanotubes of the same diameter and wall
constraint, fluid flows through a single-walled nanotube better than a double-walled tube.
Future work can take many directions. It would be interesting to compare the
simulations discussed above to those with longer nanotubes, infinite-length nanotubes,
larger diameter nanotubes, and fluids other than argon.
83
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APPENDIX
PRESSURE CONTOUR PLOTS
Figure A.1 - Argon fluid hydrostatic pressure inside rigid (10,10) SWNT, Step 250
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Figure A.2 - Argon fluid hydrostatic pressure inside rigid (10,10) SWNT, Step 500
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Figure A.3 - Argon fluid hydrostatic pressure inside rigid (10,10) SWNT, Step 700
88