FLUID FLOW THROUGH CARBON NANOTUBES: A NEW MODELING AND SIMULATION APPROACH. A Thesis. Presented to. The Graduate Faculty of The University of Akron

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 Rigid Single-Walled Carbon Nanotubes...37 4.2.1 (10,10) Carbon Nanotube...37 4.2.2 (20,20) Carbon Nanotube...44 4.3 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 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 5.2 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 0 th Pumping...56 4.17 Argon fluid hydrostatic pressure inside relaxed (20,20) SWNT...57 4.18 Argon movement through relaxed (20,20) SWNT 1 st Pumping...58 4.19 Argon movement through relaxed (20,20) SWNT -2 nd 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 C h r r = n 2.1 1 a1 + n2 a2 where a 1 and a 2 are basis vectors for graphite. C h is an example of a chiral vector. Each nanotube is specified by the two integers, n 1 and n 2. In Figure 2, (n 1,n 2 ) = (5,1). Figure 2.2 - Schematic of nanotube structure Zigzag tubes are denoted as (n,0), armchair as (n,n), and chiral as (n 1,n 2 ). 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 C h. This perpendicular vector is 6

r r r {( n1 + 2n2 ) a1 ( 2n1 + n2 ) a2} T = 2.2 q where N q = 3N ( n ) ( ) 1 n2 mod 3 if ( n n ) mod( 3) 1 2 2.3 and N is the greatest common divisor of n 1 and n 2. The larger the value of T, the longer the nanotube will be. The nanotube radius is given by Ch R = = 2π 3d 2π n 2 1 + n 2 2 + n n 1 2 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 vρd Re = 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 128µ LQ P = 2.6 4 πd 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 2 d F r r = i i m 3.1 i 2 dt where F r i is the force on each atom due to it s interactions with neighboring atoms, the atom s mass, and r i is the position vector of the atom i. The acceleration can be m i is 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 Cerius 2, 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 Cerius 2 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. 880 Single Argon Atom - Graphite Sheet Energy vs. Distance Relationship Energy (kcal/mol) 870 860 850 840 Hollow Site Bridge Site Corner Site 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. Energy (kcal/mol) 3795 3790 3785 3780 3775 Single Argon Atom - Carbon Nanotube Energy vs. Distance Relationship Middle - Outside End - Outside Middle - Inside End - Inside 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

160000 Pass-Through Test: Single Argon Atom - Graphite Sheet Energy vs. Distance Relationship 140000 120000 Energy (kcal/mol) 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 Cerius 2 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 Cerius 2 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 11 th 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 i< j ( ) E 1 P = = r ij φ ij r V 3V ij 4.1 where r ij = distance between argon atoms i and j, φ ij = 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 0 th 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 1 st 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 -2 nd 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. 82

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 86

Figure A.2 - Argon fluid hydrostatic pressure inside rigid (10,10) SWNT, Step 500 87

Figure A.3 - Argon fluid hydrostatic pressure inside rigid (10,10) SWNT, Step 700 88