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Carbon Nanotubes Carbon Nanotubes A Theoretical study of Young's modulus Kolnanorör En teoretisk studie av Youngs modul Tore Fredriksson Health, Science and Technology Physics 30 Thijs Holleboom Lars Johansson 2014-06 Faculty of Health, Science and Technology Department of Engineering and Physics Carbon Nanotubes A Theoretical study of Young's modulus Supervisor: Author: Thijs Holleboom Tore Fredriksson Examiner: Lars Johansson January 29, 2014 Abstract Carbon nanotubes have extraordinary mechanical, electrical, thermal and optical properties. They are harder than diamond yet flexible, have better electrical conductor than copper, but can also be a semiconductor or even an insulator. These ranges of properties of course make carbon nanotubes highly interesting for many applications. Carbon nanotubes are already used in products as hockey sticks and tennis rackets for improving strength and flexibility. Soon there are mobile phones with flexible screens made from carbon nanotubes. Also, car- and airplane bodies will probably be made much lighter and stronger, if carbon nanotubes are included in the construc- tion. However, the real game changers are; nanoelectromechanical systems (NEMS) and computer processors based on graphene and carbon nanotubes. In this work, we study Young's modulus in the axial direction of carbon nanotubes. This has been done by performing density functional theory calculations. The unit cell has been chosen as to accommodate for tubes of different radii. This allows for modelling the effect of bending of the bonds between the carbon atoms in the carbon nanotubes of different radii. The results show that Young's modulus decreases as the radius decreases. In effect, the Young's modulus declines from 1 to 0.8 TPa. This effect can be understood because the bending diminishes the pure sp2 character of the bonds. These results are important and useful in construction, not only when using carbon nanotubes but also when using graphene. Our results point towards a Young's modulus that is a material constant and, above a certain crit- ical value, only weakly dependent on the radius of the carbon nanotube. Graphene can be seen as a carbon nanotube with infinite radius. I Acknowledge First I want to thank my supervisor Thijs Holleboom for the many hours of discussions on how to make this the best thesis it can be. Also, Krister Svensson for his many insights into carbon nanotubes that he has shared with me. My fiancee deserves one huge thank you for her support during this whole ordeal. Finally to the rest of my family and my friends for making the hard times durable, and the good times even better. II Contents 1 Introduction 5 2 Carbon 7 2.1 Diamond . .8 2.1.1 Structure . .8 2.1.2 Properties . .8 2.1.3 Lonsdaleite . .8 2.2 Graphite . .9 2.2.1 Structure . .9 2.2.2 Properties . .9 2.3 Fullerenes . 10 2.3.1 Buckminsterfullerenes . 10 2.3.2 Carbon nanotubes . 10 2.3.3 Nanobuds . 11 2.3.4 Nanofoam . 11 2.3.5 Carbyne . 11 2.4 Graphene . 12 2.4.1 Electronic Properties . 12 2.4.2 Mechanical Properties . 13 2.4.3 Optical Properties . 14 3 Carbon nanotube 15 3.1 Discovery . 15 3.2 Categories of carbon nanotubes . 17 3.2.1 Chirality . 17 3.2.2 Single walled . 18 3.2.3 Multi walled . 19 3.3 Properties . 20 1 CONTENTS 2 3.3.1 Strength . 20 3.3.2 Electrical properties . 21 3.3.3 Thermal properties . 22 3.4 Extreme carbon nanotubes . 22 3.5 Future uses for graphene and carbon nanotubes . 24 3.5.1 Material additives . 24 3.5.2 Nanomechanics . 24 3.5.3 Nanoelectromechanical systems . 25 3.5.4 Solar cells . 26 3.5.5 Energy storage . 26 3.5.6 Biomedicine . 27 3.6 Toxicity and biocompatibility graphene and carbon nanotubes 29 4 Orbital Hybridization 30 4.1 spn Hybridization . 30 4.1.1 sp Hybridization (linear) . 31 4.1.2 sp2 Hybridization (trigonal) . 32 4.1.3 sp3 Hybridization (tetrahedral) . 32 4.2 π bond............................... 33 5 Mechanical properties 34 5.1 Tensile strength . 34 5.2 Young's modulus . 34 6 Density Functional Theory 38 6.1 Born Oppenheimer Approximation . 39 6.2 The theorems of Hohenberg Kohn . 39 6.2.1 The first Hohenberg Kohn theorem . 40 6.2.2 The second Hohenberg Kohn theorem . 40 6.3 Kohn Sham equations . 41 6.4 Local Density Approximation . 42 7 Augmented Plane Wave Method 44 7.1 Linearized APW . 45 7.2 Full Potential Linearized Augmented Plane Wave . 46 8 Results 48 8.1 ELK . 48 CONTENTS 3 8.1.1 Periodic Structures and unit cells . 49 8.2 Graphene . 51 8.3 Carbon nanotubes . 51 8.3.1 Zigzag . 51 8.3.2 Armchair . 53 8.4 Y(r) . 54 8.5 In situ Si . 54 9 Discussion and Conclusions 56 A Tables 58 List of Tables 3.1 Mechanical properties . 21 8.1 E() = a + bx + cx2 ........................ 53 8.2 Y (r)................................ 54 A.1 E() - Graphene . 58 A.2 E() - Zigzag r = 3:21301 . 59 A.3 E() - Armchair r = 2:74452 . 60 A.4 E() - Armchair r = 3:41653 . 60 A.5 E() - Armchair r = 4:09065 . 61 A.6 E() - Armchair r = 4:76598 . 61 A.7 E() - Armchair r = 5:44211 . 62 A.8 E() - Armchair r = 6:79564 . 62 A.9 E() - Armchair r = 6:79564 . 63 4 Chapter 1 Introduction \Graphene - a single layer of carbon atoms - may be the most amazing and versatile substance available to mankind" - Graphene Flagship When Iijima 1991 [16] made carbon nanotubes known to the broad scientific community, he set a revolution in motion. Albeit slow at first but from 2004, when Geim and Novoselov [28] successfully uncovered one free layer of graphene, there has been a boom seldom seen. This was accentuated by graphene being the first of EU's flagships1, granting e1 B over a ten year period, the biggest research initiative ever. Graphene is one monolayer of carbon atoms packed into a honeycomb lattice. It was the first two dimensional material to be discovered and is the mother of all graphitic allotropes. It can be wrapped up into zero-dimensional struc- tures known as fullerenes. Stacked layer upon layer to three-dimensional graphite. Rolled up into one dimensional carbon nanotubes, the structure we have chosen to investigate further. In mid 1930s Peierls and Landau [29, 21] taught us that two dimensional materials can not exist as they are thermodynamically unstable. However these strange two dimensional materials where still theoretically interesting 1http://graphene-flagship.eu/ 5 CHAPTER 1. INTRODUCTION 6 thus they were discussed and researched for purely academic reasons. In 1991 Iijima published a high impact article [16]. Here he claimed to have seen carbon structures shaped like needles, this was not the first evidence that carbon nanotubes existed. Despite not being first, Iijimas carbon needles, triggered a first boom in the interest around low dimensional materials. This boom was enough for some to disregard Peierls and Landau's results. In 2004 Konstantin Novoselov and Andre Geim [28], successfully extracted a single layer of graphene from the graphite in a pencil, using only scotch tape. The ensuing second boom this created was immense, making carbon and its allotropes one of the biggest fields of research. The discovery yielded Novoselov and Geim the Nobel prize in physics 2010. Despite many tries, no one is yet successful in unambiguously determining the Young's modulus of carbon nanotubes. Reported values are ranging between 0.6-5.5 TPa [38]. However, they seem to be converging to 1 TPa [26]. This is our try to cast some light on the subject, and hopefully some new insights can emerge from our work. Chapter 2 Carbon Carbon only exists as single atoms in extreme temperature environments. In ambient environment carbon reacts with other atoms to stabilize, to form multi atomic compounds. The most important of these compounds must be attributed to organic chemistry and carbons r^olein makeup the nucleic acids in DNA. Also, the proteins which are the building blocks for all life, thus making carbon the basis of life. Now, a short description of how carbon bonds to carbon forming what is known as allotropes1, especially graphene and carbon nanotubes. The most fascinating about the carbon allotropes is the multitude of dif- ferent properties they display, most notably electrical, range from strongly insulating to nearly perfect conductors. Thermal, most thermally conduct- ing, and mechanical, diamond is the hardest naturally occurring material and graphite one of the softest, carbyne has the highest Young's modulus ever measured. A short description of the most common and interesting carbon allotropes, together with some of their extreme properties follows below. 1Carbon allotropes include, but is not limited to; amorphous carbon, graphite, dia- mond, fullerenes (buckyballs, carbon nanotubes, carbon nanobuds and nanofibers), lons- daleite, glassy carbon, carbon nanofoam, graphene and linear acetylenic carbon (carbyne) 7 CHAPTER 2. CARBON 8 2.1 Diamond In an environment with extreme pressure (4.5-6.0 GPa) and temperatures (900-1300 ◦C), carbon forms the compact allotrope diamond. 2.1.1 Structure The Diamond lattice is a face centered cubic crystal structure, where each atom is bonded tetrahedrally to four other carbon atoms in a sp3 bonding, see chapter 4, thus making a three dimensional network of puckered six mem- bered rings of atoms; it is the same structure as silicon and germanium, but due to the strength of the carbon-carbon bonds, it is the hardest naturally occurring material in terms of resistance to scratching. 2.1.2 Properties Hardness 10 000 kg mm−2, scratch resistance 160 GPa and a Young's modulus of 1.22 TPa.
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