1 DETERMINATION OF MECHANICAL BEHAVIOR OF NANOSCALE MATERIALS USING MOLECULAR DYN AMICS SIMULATIONS By SEONGJUN HEO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007
2 2007 SeongJun Heo
3 To my family with love and gratitude
4 ACKNOWLEDGMENTS First of all, I express my sincere gratitude to my advisor, Prof. Susan B. Sinnott, for her support during my graduate study. W ith her kindness, I have join ed the computational group and known the fabulous world of materials simulations Her patience with faith and encouragement helped me overcome the hard times in my research and enabled me to accomplish my research goal. I also thank the members of my supervisor y committee, Prof. Simon R. Phillpot, Prof. Scott S. Perry, Prof. Wolfgang M. Sigm und, and Prof. W. Gregory Sawy er, for their advice and kind support. I am also thankful to all members of Com putational Materials Sc ience and Engineering Focus Group to a great extent, especially Dr. Wen-Dung Hsu, Rakesh Kumar Behera, Taku Watanabe, Donghwa Lee, and Dr. Tao Liang, for helpful academic discussion and a lot of fun during my graduate study. Especially, I want to thank my wife, Hyangran, who has been always with me and helped me persevere throughout my Ph.D study. I also th ank my daughter, Seojin, for bringing me the joy of life. Finally, I wish to show my deepes t appreciation to my be loved parents for their invaluable support and encouragement.
5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ............11 CHAPTER 1 INTRODUCTION................................................................................................................. .13 General Introduction........................................................................................................... ....13 Carbon Nanotubes............................................................................................................... ...13 Structure of Carbon Nanotubes.......................................................................................15 YoungÂ’s Modulus of Nanotubes......................................................................................17 Computational results...............................................................................................17 Experimental results.................................................................................................18 Elastic Properties of Nanotubes.......................................................................................18 Mechanical Properties of Carbon Nanotubes..................................................................19 Carbon Nanotubes Hydrodynamics.................................................................................21 Carbon Nanotube Tribology............................................................................................22 Polyethylene................................................................................................................... ........23 Basic Properties of Polyethylene.....................................................................................23 Polyethylene Tribology...................................................................................................24 2 COMPUTATIONAL METHODS..........................................................................................29 Classical Molecular Dynamics Simulations...........................................................................29 Reactive Empirical Bond Order Potential.......................................................................29 Lennard-Jones Potential..................................................................................................33 Periodic Boundary Conditions........................................................................................34 Predictor-Corrector Algorithm........................................................................................35 Ensemble Details.............................................................................................................37 Temperature Control Methods........................................................................................37 Velocity rescaling.....................................................................................................38 Nos-Hoover............................................................................................................39 Langevin...................................................................................................................40 3 CALIBRATION OF MET HODS FOR NANOMECHANICS..............................................44 Computational Details.......................................................................................................... ..45 Results and Discussion......................................................................................................... ..46 Conclusions.................................................................................................................... .........51
6 4 NANOMECHANICS OF CARBON NANOTUBES............................................................58 Computational Details.......................................................................................................... ..58 Results and Discussion......................................................................................................... ..60 Filling Effect................................................................................................................. ...60 Temperature Effect..........................................................................................................62 Effect of Wall Defects.....................................................................................................63 Conclusions.................................................................................................................... .........65 5 HYDRODYNAMICS OF CARBON NANOTUBES............................................................74 Computational Details.......................................................................................................... ..74 Results and Discussion......................................................................................................... ..75 Conclusions.................................................................................................................... .........77 6 TRIBOLOGY OF CARBON NANOTUBES........................................................................85 Computational Details.......................................................................................................... ..85 Results and Discussion......................................................................................................... ..87 Conclusions.................................................................................................................... .........93 7 TRIBOLOGY OF POLYETHILENE..................................................................................102 Computational Details.......................................................................................................... 102 Results and Discussion......................................................................................................... 103 Conclusions.................................................................................................................... .......104 8 GENERAL CONCLUSIONS...............................................................................................116 LIST OF REFERENCES............................................................................................................. 118 BIOGRAPHICAL SKETCH.......................................................................................................128
7 LIST OF TABLES Table page 1-1 Classification of carbon nanotubes....................................................................................26 2-1 Lennard-Jones parameters for H and C.............................................................................42 2-2 The thermodynamical ensembles and the corresponding independent and dependent variables...................................................................................................................... .......42 5-1 Details of the SWNTs considered......................................................................................78 5-2 The numbers of fluid atom s for the various systems.........................................................78 6-1 Pressures (in GPa) of various two layered CNT bundle systems at different compressive states............................................................................................................. .94
8 LIST OF FIGURES Figure page 1-1 The unrolled honeycomb lattice of a carbon nanotube......................................................27 1-2 Categorization of carbon nanotubes...................................................................................27 1-3 The unrolled honeycomb lattice of a car bon nanotube. The metallic nanotubes are denoted by red solid circles on the map of chiral vectors (n, m).......................................28 1-4 The schematic of the unit cell of polyethylene crystal structure.......................................28 2-1 The energy of Lennard-Jones potential..............................................................................43 2-2 A two-dimensional periodic system...................................................................................43 3-1 Snapshots of a 200 (10,10) hollow singl e-walled CNT that is empty (top) and filled with n-butane molecules (bottom)............................................................................52 3-2 Smoothed force versus strain curves of the empty and n-butane filled nanotubes with different relative percentages of th ermostat atoms (5%, 30%, and 100%)........................52 3-3 Smoothed force versus strain curves of the n-butane filled nanotube. The relative percentage of thermostat atoms varies from 5 % to 100 %...............................................53 3-4 Snapshots of the n-butane filled nanotube using the Langevin thermostat at a strain of 0.1. The relative percentages of thermo stat atoms varies from 5% to 100 %...............54 3-5 The smoothed force versus strain curves of the n-butane filled nanotube at different deformation rates and different relative percentages of thermostat atoms........................55 3-6 Snapshots of the n-butane filled nanotube using the Langevin thermostat at a strain of 0.1 at a deformation rate of 4 m/s..................................................................................55 3-7 The smoothed force versus strain curves of the n-butane filled nanotube using the A) Nos-Hoover, B) velocity rescaling, and C) Langevin thermost ats at different relative percentages of thermostat atoms...........................................................................56 3-8 Snapshots from the compression of the n-butane filled nanotube shown in Figure3-1 at a strain of 0.1 using A) velocity res caling and B) Nos-Hoover thermostats with different relative percentages of th ermostat atoms (5%, 30%, and 100 %).......................56 3-9 System temperature versus strain for the n-butane filled nanotube...................................57 4-1 Snapshots of 100 (10, 10) ho llow CNT for bending and compression..........................66 4-2 Force versus deflection of various hollow and filled CNTs..............................................67
9 4-3 100 long SWNT, C60 filled SWNT, DWNT, and TWNT after buckling......................67 4-4 Force versus strain of hollow CNT, peapod, and DWNT under the compression............68 4-5 Compressed hollow CNT, peapod and DWNT at 0.04, 0.05 0.06 strain...........................68 4-6 The pair distribution functions of A) SWNT and outer CNT part of peapod and DWNT and B) C60 in peapod and inner CNT....................................................................69 4-7 Forces versus deflections of various ho llow and filled CNTs during the bending at various temperatures..........................................................................................................7 0 4-8 Force versus strain of hollow CNT, pea pod, and DWNT at the different temperature.....71 4-9 Buckling force of hollow CNT, peapod, a nd DWNT at different temperatures A) Maximum buckling forces. B) Normalized maximum buckling forces............................72 4-10 Force versus deflection of hollow pris tine CNT and CNTs having various densities of A) vacancy defects and B) functional groups during bending......................................72 4-11 Force versus strain of hollow pristine CNTs and CNTs having A) vacancy defects and B) functional groups of various densities....................................................................73 4-12 Snapshots of various vacancy configuration......................................................................73 5-1 Snapshots a bridged CNT and a cantilevered CNT...........................................................79 5-2 Deflection vs. time of the 20 nm long br idged SWNT A) in vacuum and B) in Ar, Kr, and Xe fluids at 10, 50, and 100 atms, respectively....................................................80 5-3 Analysis of frequency results: A) Deflec tion vs. time and B) frequency after the fastFourier-transformation of the 20 nm l ong bridged SWNT in 100 atm Ar fluid................81 5-4 Frequency vs. fluid pressure of A) bri dged CNTs and B) cantilevered CNTs with the lengths of 10, 20, and 30 nm..............................................................................................82 5-5 Frequency vs. CNT length of A) bridge d CNT and B) cantilevered CNT in vacuum. MD simulational results ( ) and continuum level rod vibration equation ( Â– ) are compared....................................................................................................................... .....83 5-6 Comparison of the frequencies of bri dged 10, 20, and 30 nm long SWNTs in Kr fluid calculated by MD simulations and pred icted by the equation of longitudinal vibrations..................................................................................................................... .......84 6-1 Snapshots of initial structures for CNT tribology..............................................................95 6-2 Compressive forces versus the disp lacement of the topmost diamond surface.................96
10 6-3 Snapshots of cross-sectional and si de view of compressed SWNT, peapod, and DWNT bundle.................................................................................................................... 97 6-4 Force results of various 2-layered CNT bundle systems...................................................98 6-5 Sanpshots, moving distance and rota tion angle of 2-layered CNT bundle........................99 6-6 Moving distance and snapshots of 4-layeres hollow CNT bundle..................................100 6-7 Moving distance and snapshot of hol low CNT bundle in benzene molecules................101 7-1 The schematic diagram of th e PE polymer chain alignment...........................................109 7-2 The lateral forces, the normal forces and the frictional coefficients of the perpendicular and the parallel slid ings of the PE polymer systems.................................110 7-3 Histogram of atomic positions for the pol ymer chains at the sliding interface...............111 7-4 The snapshots of PE system in the perpendicular and th e parallel slidings.....................112 7-5 Shear deflections and shear strain energies of the upper PE substrates...........................113 7-6 Comparisons forces and frictional co efficients between the PE and PTFE.....................114 7-7 Histogram of the number of the bonds formed and broken during sliding......................115
11 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DETERMINATION OF MECHANICAL BEHAVIOR OF NANOSCALE MATERIALS USING MOLECULAR DYN AMICS SIMULATIONS By Seongjun Heo August 2007 Chair: Susan B. Sinnott Major: Materials Science and Engineering It is important to understand the mechanical properties of nanome ter-scale materials for use in such applications as microelectromechan ical systems (MEMS) and nanoelectromechanical systems (NEMS). These properties are difficult to measur e directly using experimental methods due to their small sizes. Computational simulati ons provide important in sights that complement experimental data and lead to improved unde rstanding of the mechanical properties of nanometer-scale systems. Molecular dynamics (MD) simulations, which are used to investigate the properties of materi als at the atomic scale, is used in my research to determine best thermostat managing way for acceptable mechanical behavior of nanoscale systems filling effect on the bending and compressive properties of carbon nanotubes (CNTs) vibrational behavior of bri dged and cantilevered CNT bombarded by external fluid atoms frictional behavior of filled CNT bundles and the effect of external molecules on friction effect of sliding orientations on the tr ibological properties of polyethylene (PE). In all the simulations the reactive empirical bon d-order (REBO) potential combined with the Lennard Jones potential is applied to control inter-atomic interactions. During the MD simulations, thermostats are used to maintain the system temperature at a constant value. Tests indicate that the simulati ons describe the mechani cal behavior of CNTs differently depending on the type of thermostat us ed, and the relative fraction of the system to
12 which the thermostat is applied. The results indicate that Langevin and velocity rescaling thermostats are more reliable for temperatur e control than the Nos-Hoover thermostat. In examining CNT bending and compression, the simulations predict filled CNTs are more resistant to external bending and compressive forces than hollow CNTs. The mechanical properties deteriorate with increases in te mperature and number of CNT wall defects. MD simulations of the vibrational behavior of bridged and cantilevered CNTs are found to match the results of continuum mechanics calcula tions. The principal vibr ation frequency of the CNT is predicted to decrease with increasing nano tube length, gas pressure, and the atomic mass of the external fluid. In studies of CNT tribology, simulations show that two layers of filled CNTs are more resistant to compressive forces and exhibit lowe r friction coefficients during sliding than unfilled CNTs. The friction coefficient increases with the thickness of the CNT layer due to the increase in effective friction interface. The addition of an external, molecular fluid of benzene molecules is predicted to reduce the friction coefficient of CNTs because of the lubr icity of the molecules. Lastly, simulation results illustrate the effect of relative orientation on the tribological properties of polyethylene (PE) sliding surfaces. Th e friction coefficient of perpendicular sliding is much higher than that of parallel sliding based on the polym er chain orientation. The PE exhibits stick-slip motion during sliding regardless of the sliding orientation. In addition, the PE shows no surface morphology change due to the hi gher strength of the PE bonds, which is in contrast to the behavi or of other polymers, such as pol ytetrafluoroethylene (PTFE), which exhibits bond breaking and realignm ent of surface chains along th e sliding direction in the less favorable orientation.
13 CHAPTER 1 INTRODUCTION General Introduction The mechanical properties of materials may vary as the system size changes from the macro-scale to the nano-scale. The miniatur ization of devices demands we understand the mechanical properties of materials at all length scales. Due to their superior mechanical and electrical properties, carbon nanot ubes (CNTs) have been studied extensively in recent years. The responses of CNTs to bending and compressi on, their hydrodynamical behavior in external fluid flows, and their tribological properties are all important for their potential use in some applications; however these mechanical prop erties are still not completely understood. In my research, various mechanical propert ies of nano-sized materials are examined computationally with an emphasis on the following topics: controlling and optimizing thermostats duri ng the mechanical deformation of CNTs understanding how filling CNTs influences th eir responses to bending and compression, examining the hydrodynamic properties of bri dged and cantilevered CNTs while they are bombarded by atoms in an external fluid, determining the frictional coefficients of filled and empty CNT bundles between sliding diamond surfaces and determining the effect of external fluid molecules on these coefficients. In addition, the effect of sliding orientation on th e tribological properties of polyethylene (PE) is examined. Carbon Nanotubes A carbon atom has six electrons to tal. Two of them occupy the 1s orbital and the other four fill the 2s and 2p orbitals; these valence orbitals may hybridize to form the sp3, sp2, or sp orbitals, depending on the bonding structures of the carbon or hydrocarbon polymorphs under consideration. Diamond has four sp3 hybridized orbitals that form four covalent bonds with
14 the neighboring carbon atoms. This results in th e three-dimensional, extended and strongly bound crystal structure responsible for the extremely hard nature of diamond.1 In graphite2, there exist three planar sp2 hybrid orbitals that form three in-plane bonds with an out-of-plane bond. This results in the planar he xagonal structure of the graphene sheet. These graphite sheets bond to one another via van der Waals bonds, wh ich produce a spacing of 0.34 nm between the graphene layers. The bonds in the sp2 orbitals are 0.14 nm long and have a total binding energy of 420 kcal/mol, while those in sp3 orbitals are 0.15 nm with a total binding energy of 360 kcal/mol. Consequently, the covalent carbon-car bon bonds in graphite are stronger than in diamond. Additionally, an out-of-plane orbital distri buted over the graphite planes makes them high electrical and thermal conductors. As a re sult of the weak van der Waals interaction between the graphene layers, graphite is soft in the direction normal to the planes and functions as a good solid lubricant. A CNT may be considered to be Â“a hollow cyli nder formed by roll ing graphene sheetsÂ”.3 The bonding in nanotubes is sp2. However, the curvature makes the three bonds slightly out of plane and the orbital more delocalized outside the tube than is the case in graphite. This gives nanotubes their desirable propert ies, including high strength, high electrical and thermal conductivity, and high chemical and biological activ ity at small diameters. In addition, nanotubes can incorporate topological def ects such as pentagons and heptagons to generate capped, bent, and helical structures. Fullerenes (C60) consist of 20 hexagons and 12 pentagons.4 The bonding in these structures is also sp2, although the bonds show some sp3 character due to the high curvature inherent to the structure. This unique bonding character give s fullerenes some interesting properties, such as high electronegativity.
15 Structure of Carbon Nanotubes A single walled nanotube (SWNT) is illustrated as a graphene sheet rolled into a hollow cylindrical shape with a diameter of about 0.4 Â– 10 nm. If one neglects the two ends of the nanotube, it may be considered to be a one dimensional nano-st ructure because of the large length/diameter aspect ratio, which can be as large as 104-105. The major symmetry classification of a singl e-wall carbon nanotube is as either being achiral (symmorphic) or chiral (non-symmorphic). If the mi rror image of a carbon nanotube parallel to a nanotube axis has an identical shape to the origin al one, it is called an achiral nanotube and there are only two cases where this occurs: armchair and zigzag nanotubes. For chiral nanotubes, the mirror image cannot be the sa me as the original one. These symmetries are specified by the chiral vector Ch ( O A in figure 1-1), which is defined as Ch = n a1 + m a2 ( n m ) (1-1) where a1 and a2 are the real space unit vect ors and n and m are integers that satisfy the condition 0 | m | n It is necessary to consider only 0 | m | n for chiral nanotubes because of the hexagonal symmetry inherent to th e graphene structure. In the unrolled honeycomb lattice of the SWNT, the vectors O A and OB describe the chiral vector Ch and the translational vector T of a carbon nanotube, respectiv ely (Figure 1-1). An armchair nanotube occurs when n = m that is Ch = ( n n ), and a zigzag nanotube occurs when m = 0, or Ch = ( n 0) (Table 1-1). All other ( n m ) chiral vectors correspond to chiral nanotubes. Fi gure 1-2 illustrates th ese three types of carbon nanotubes. The diameter, dt, of a ( n m ) carbon nanotube is described as nm m n a dh t 2 2 (1-2)
16 where a is the lattice constant of the honeycomb lattice, a = 1.44 3 = 2.49 From figure 1-1 the cos and sin can be calculated, where is the chiral angle between Ch and a1 2 2 2 22 2 cos 2 3 sin m nm n m n m nm n m (1-3) Therefore, the chiral angle can be defined with values of in the range 0 | | 30 as: 3 2 3 tan1 m n m (1-4) For example, zigzag and armchair nanotubes correspond to = 0 and = 30, respectively. Another important vector associated with nanotubes is the translational vector T, which is parallel to the nanotube axis a nd normal to the chiral vector Ch. It is expressed in terms of a1 and a2 as: ) (2 1 2 2 1 1t t a t a t T (1-5) where t1 and t2 are integers given by R Rd m n t d m n t 2 22 1 (1-6) where dR is the greatest common divisor of n and m and expressed by 3d. of multiple a is m n if 3 3d of multiple a not is m n if d d dR (1-7) The length of the translational vector is simplified to R hd T 3 (1-8) The unit cell of a 1-dimens ional carbon nanotube correspond s to the rectangle OABÂ’B defined by the chiral vector Ch and the translational vector T. Therefore, the number of hexagons
17 per unit cell N is calculated by divi ding the area of the unit cell, | Ch T |, by the area of the hexagon, | a1 a2 | as: R R hd a L d m nm n N2 2 2 2 2 12 2 a a (1-9) Each hexagon has two carbon atoms. Consequen tly, each unit cell of th e nanotube contains 2 N carbon atoms. YoungÂ’s Modulus of Nanotubes It is not surprising that carbon nanotubes ha ve a high elastic modul us, as do graphene sheets. The elastic moduli of vari ous nanotubes have been predicte d from atomistic calculations and simulations since before the experimental di scovery of nanotubes. Only later were several experimental methods developed that made it possible to measure their elastic moduli. Computational results Using both empirical potentials and first principle methods, Robertson et al .5 examined the strain energies and the elastic properties of nanotubes having radii less than 0.9 nm and predicted an elastic modulus of about 1.06 TPa for SW NTs, which can be dependent on the helical conformation and 1/r2 (where r is the t ubule radius). Yakonson et al .6 assumed the wall thickness to be 0.066 nm because this describes the bucklin g behavior of nanotube walls appropriately and found the YoungÂ’s modulus to be about 5.5 TPa. Other work7 using molecular dynamics assumed a wall thickness of 0. 34 nm, which is same as th e separation distance between neighboring walls in multi-wall nanotubes (MWNTs), obtained values of about 1.0 TPa for the modulus. MWNTs correspond to concentric SW NTs separated by the same van der Waals separation as in graphite. Thus, the difference in the wall thickness influences the cross-sectional area used to determine YoungÂ’s modulus, and th us can significantly influence the calculated values. The universal force field method, includ ing torsional energy terms, was also used to
18 calculate the mechanical properties of nanotubes.8, 9 The YoungÂ’s modulus of a CNT is equivalent to that of in-plane graphite single layer, 1.0 TPa9. For MWNTs, the modulus is dependent on the number of walls.8 BornÂ’s perturbation techni que derived within a lattice dynamic model for nanotubes also gives a valu e of about 1.0 TPa for the YoungÂ’s modulus.10 The elastic moduli of nanotubes are also predicted using fi rst principles techniques, including tight bi nding (TB) methods.11-13 Applying a non-orthogonal TB model, which includes both electronic structure effects and empirical terms, Hernandez et al .11 computed the elastic modulus of CNTs as 1.24 TPa for seve ral achiral and chiral SWNTs. Ozaki et al .12 also used an O( N ) tight binding method to calculate a value of 0.98 TPa for the YoungÂ’s modulus. Using allelectron ab initio calculations with Hatree-Fock 6-31G* level th eory, the predicted YoungÂ’s modulus is higher than 1 TPa for various achiral and chiral SWNTs.13 Experimental results Several experimental methods have been used to measure the mechanical properties of nanotubes. Treacy et al .14 and Krishnan et al .15 estimated a YoungÂ’s modulus of 1.8 TPa for isolated MWNTs and 1.25 TPa for SWNTs by obs erving the amplitude of their freestanding thermal vibrations in a transmission electron mi croscope. Using the bending test within atomic force microscopy, the YoungÂ’s modulus of CN Ts is estimated as about 1.2 Â– 1.3 TPa.16-18 The YoungÂ’s modulus of nanotubes has also been measured by Raman spectroscopy19 and tension testing20, 21 with results ranging fr om 0.27 to 3.6 TPa. Elastic Properties of Nanotubes As discussed earlier, there are three differe nt kinds of forces between carbon atoms characterizing the elastic properties of graphite and carbon nanotubes: strong -bonding and bonding interactions between intralayer C-C bonds and relatively weak interlayer interactions.
19 There are order of magnitude di fferences between these forces, but all three are important for explaining the mechanical behavior of CNTs in different situations. Forming a SWNT by rolling a graphene sheet increases the total energy of the nanotube through the introduction of curvatur e. Therefore, the strain ener gy of the nanotube increases as the nanotube diameter decreases, which means that nanotubes with a small diameter may be less stable than those with a la rge diameter. Due to the sp2 bonding between the carbon atoms, nanotubes have their highest strength along th e direction of the nanotube axis, much like graphene layers 6. Nanotubes, like graphene sheets in graphite, are compliant to forces perpendicular to the hexagonal la ttice in the axial di rection, or tangen tial forces to the -bonding. Therefore, nanotubes may be easily deformed and flattened by applying a force perpendicular to the nanotube axis without breaking any -bonding.22, 23 Although the contribution of -bonding to the elastic energy is much less than that of bonding, the -bonding is essential for the lattice dist ortion, which is known as the Peierls distortion.24 The energy gain for th e interaction between -bonding and the lattice is balanced by the potential energy loss for the lattice. Therefore, minimizing the total energy determines the degree of distortion. The many interesting properties of MWNTs cannot be explained just by the physics of SWNTs. The reason is that there are relatively weak van der Waals interactions between neighboring nanotube layers in MWNTs, which d ecide their relative st acking structure. The structures of two adjacent layers may be incomme nsurate with each other, which can affect the shear properties betw een nanotube shells.25, 26 Mechanical Properties of Carbon Nanotubes The unique mechanical properties of CNTs make them attractive for use as components in nanoelectromechanical systems (NEMS), where th ey would be subjected to various types of
20 forces such as compression, bending, tension, an d torsion. Additionally, CNTs used as tips of atomic force microscopes (AFMs) experi ence both compression and bending during surface contact and scanning. Consequently, there ha s been intense interest in qualitatively understanding and quantifying the m echanical responses of CNTs. For example, Ijima et al.27 found that atomistic simulations of the responses of nanotubes to bending are comparable to images of be nt tubes observed in high resolution electron microscope images. Yakobson et al.6 used similar simulations to examine the mechanical responses of single-walled carbon nanotubes (S WNTs) under axial compression, bending, and torsion. Their results indicated that nanotubes exhibit great fl exibility, and may be severely deformed without breaking any ch emical bonds. Garg and Sinnott28 found that chemical functionalization of nanotube walls leads to lower SWNT buckling forces relative to unfunctionalized SWNTs. The basic mechanical properties of filled SWNTs have been also investigated by Ni et al.29 who found that filling CNTs with fullerenes, CH4 or Ne increases the loads at which bucking occurs and decreases the effect of temperat ure on buckling. In addi tion, Danailov et al.30 predicted that filling SWNTs with Au nanowir es increase the maximum bending force and show the deflection buckling at higher bendi ng force. Similarly, Trotter et al.31 explored the compressibility of CNTs filled with diamond nanowires, smaller nanotubes, C60, CH4, Ne, nC4H10, or n-C4H7 molecules. They observed that na nowire-filled CNTs and MWNTs exhibit similar mechanical responses and that filling CN Ts increase their stiffness during compression. Jeong et al.32, 33 investigated the effect of filling CNTs on the tensional, torsional, combined tensional and torsional, and biaxial tensional an d torsional properties. They observed that these
21 mechanical properties can be improved by filling CNTs, but the failure criteria are different depending on the deformation modes that occur in response to applied forces. Carbon Nanotubes Hydrodynamics Due to their high stiffness and strength, low de nsity, and large aspect ratio, CNTs may be considered to be the ultimate component for NE MS oscillators in ultra-sensitive mass detection34, 35 and radio frequency signal processing36. In signal processing, for ex ample, reducing the size of the resonator increases the frequency and mini mizes energy consumption. For sensors, higher resonant frequency corresponds to higher mass se nsitivity. Using conventional microsized silicon or silicon nitride produced with advanced lithography technology, fundamental frequencies of around 1 GHz can be ach ieved at the tens of nm scale37. The demand for advances in high-frequency NEMS requires ev en higher resonant frequencies. Therefore, some researchers are particularly interested in using carbon nanotubes as resonators or oscillators. For example, an individual carbon nanotube was proposed for use as a nanobalance for nanoscale particles38. The mechanical deflections of cantilevered, MWNTs were measured in a transmission electron microscope It was predicted that the na nobalance can weigh particles in the femtogram to picogram range. There were also attempts to use carbon nanotubes as electomechanical oscillators39, 40. Â‘Guitar-string-like oscillation modesÂ’ of CNTs suspended between two contacts were det ected. It was also shown that the resonance frequency can be tuned. The resonant frequencies of cantilevered and bridged CNTs were also predicted using molecular structural mechanics41-43 and molecular dynamics (MD) simulations44. Recently, it was observed in MD simulations45 that mechanical energy can be transferred between CNTs via resonant effects.
22 Carbon Nanotube Tribology Due to their high strength and low densities, CNTs are being considered for use as additives to lubrication coatings. It is therefore important to unde rstand the responses of CNTs to shear forces. Previ ously, Ni and Sinnott46 predicted that the responses of bundles of SWNTs subjected to compressive and shear forces between two diamond surf aces varied depending on their orientation. Bundles that were oriented hor izontally to the sliding surfaces exhibited sliding and mixed sliding and rolling motions during shea ring, and vertically aligned CNT bundles had higher frictional coefficients than horizont ally aligned bundles. These predictions were subsequently confirmed in experiments by Dickrell et al.47, who observed highly anisotropic tribological reponses that depended on the orientation of MWNT film s. In particular, vertically aligned nanotubes produced very high frictio nal coefficients while horizontally aligned nanotubes give extremely low coefficients. In addition, Karmakar et al.48 investigated the high pressure behavior of CNTs using insitu powder X-ray diffraction (XRD). Their findings indicate that pr essure causes elliptization of SWNT bundles and polygonization of pris tine and Fe-filled MWNT bundles. Falvo et al.49 experimentally observed that indi vidual MWNTs could slide, rotate or roll on mica and graphite surfaces experimentally in response to external forces from proximal probe tips. Simulations by Buldum et al.50 explained this behavior by illustrating how CNTs can have unique equilibrium orientations on graphite surfaces depending on whether the hexagons on the CNT walls are Â‘inregistryÂ’ or not with the hexagons on the graphite surface. When registry is achieved, the CNTs roll and when it is not achieved, they slide in re sponse to applied external forces. This is also consistent with the findings of Miura et al.,51 who showed that SWNT bundles exhibit only sliding behavior on the KCl surf ace because registry cannot be achieved in this system.
23 Polyethylene Basic Properties of Polyethylene Polyethylene52 has the simplest form among hydrocarbon polymers and has the following structure: [-CH2-CH2-]n It is a very widely and comm only used polymer. It was first synthesized by Hans von Pechmann by accident while heating diazomethane in 1898 a nd first industrially produced by Imperial Chemical Industries in 1933 by pol ymerizing the ethylene monomer. Polyethylene can be roughly classified into two main kinds: low density polyethylene (LDPE) that consists of branched molecules and high density polyethylene (HDPE) that has a linear structure. LDPE has a melting point of around 383 Â– 398 K and is 40% crystalline. Its density is about 0.91 Â– 0.92 g/cm3. It is mainly used for packing and wrapping food and textile products. HDPE has a melting point of around 417 Â– 423 K. It is 90 % crystalline and has densities as high as 0.965 g/ cm3. It has higher strength and hardne ss than LDPE and is used for manufacturing toys and househol d articles. Depending on the mol ecular weight, the structure, and the type of branch, PE can also be categor ized into several other categories that include VLDPE (very low density PE) LLDPE (linear low density PE) LDPE (low density PE) MDPE (medium density PE) PEX (cross-linked PE) HDXLPE (high density cross-linked PE) HDPE (high density PE) HMWPE (high molecular weight polyethylene) UHMWPE (ultra high molecular weight PE).
24 Polyethylene Tribology Polyethylene (PE) is one of the most widely used polymers because of its versatility and manufacturability. Its mechanical and tribologi cal properties vary greatly depending on its percent crystallinity, structure, and molecu lar weight. UHMWPE has a molecular weight of millions of grams per mol and a highly crystall ine structure. It exhibits outstanding wear resistance and toughness, and has favorable tribological properties53. Consequently, PE is increasingly used in applications where its tri bological performance is critical, including thin coatings for Si-based MEMS54 and artificial joints55-57. It has been suggested that the frictional properties of polymers are related to their molecular profiles rather than to their chemical compositions. For instance, Pooley and Tabor58 observed that PE and polytetrafl uoroethylene (PTFE) exhibit lo w frictional properties while polymers with bulky side groups such as polypropylene and tetrafluoroethylenehexafluoropropylene(TFE-H FP) have high frictional coefficients They attributed the low friction of PE and PTFE to their smooth molecula r profiles. In addition, Schnherr and Vancso59 found that PE and PTFE surfaces show oriented fibri llar morphologies along the sliding direction using scanning electron microscopy. They also observed th at the orientation of the polymer chain with respect to the scan direction determines the friction force between a scanning force microcopy (SFM) tip and the polymer surface. The influence of polymer chain orientation on fr iction and wear has also been investigated. For example, Vinograd et al .60 examined the tribological beha vior of oriented crystalline polymers, including linear and branched PE, a nd found that oriented polymers exhibit lower friction coefficients and highe r shear modulus than non-orient ed polymers. In addition, Schnherr and Vancso61 employed SFM with chemically modifi ed tips for scanning in directions that are parallel and perpendicu lar to the polymer chains. They observed that friction for a
25 perpendicular scan was about four times larger than that for a para llel scan. Furthermore, Sambasivan et al .62 used nondestructive X-ray absorption sp ectroscopy to determine that a crossshear motion (or figure-eight pattern) generated by a pin-on-disc motion produces more wear on UHMWPE samples than unidirectional sliding becau se many more chains are not aligned with the cross-shear motion direction.
26 Table 1-1. Classifi cation of carbon nanotubes Type Ch Shape of cross section Symmetry Armchair 30 (n, n) Cis -type Dn Ci Zigzag 0 (n, 0) Trans -type Dn Ci Chiral 0 < | | < 30 (n, m) Mixture of cis and trans Cd CN/d
27 Figure 1-1. The unrolled honeycom b lattice of a carbon nanotube. When sites O and A, and B and BÂ’ are connected, a carbon na notube is constructed. Figure 1-2. Categorization of car bon nanotubes: A) (17,0) zigzag, B) (10,10) armchair, and C) (14,5) chiral nanotubes. A B C
28 Figure 1-3. The unrolled honeycomb lattice of a carbon nanotube. The metallic nanotubes are denoted by red solid circles on the map of chiral vectors (n, m). Figure 1-4. The schematic of the unit cell of polyethylene crystal structure
29 CHAPTER 2 COMPUTATIONAL METHODS Classical Molecular Dynamics Simulations In the classical molecular dynamics (MD) simu lations, the motion of a system of particles is predicted by solving Newtoni an motion equations numerically NewtonÂ’s second law states that the force vector Fi applied on an atom i is equal to the product of the mass mi and the acceleration ai as given by i i ima F (2-10) This equation can also be expressed in term s of the gradient of the potential energy. Vi i F. (2-11) Therefore, the force of the atom i can be calculated from the potential energy and, consequently, the acceleration on the atom i can be determined numerically. By integrating the motion equation, the trajectory of particles can be solved, provi ding the positions, velocities, accelerations, and the higher derivatives if needed. From this trajecto ry, the average properties of a system can be determined at any time. Reactive Empirical Bond Order Potential The bond order potential wa s developed by Tersoff63 to model the energetics and dynamics of group IV materials such as carbon and silicon. It was based on the formalism by Abell64, according to which the biding energy of a manybody system can be desc ribed with pair-wise nearest neighbor interactions th at are modified by the local at omic environment. Using this potential, Tersoff calculated the biding energy in Si63, C65, Si-C66, 67, and Si-Ge67 solid-state structures. While the Tersoff pot ential can describe the carboncarbon single, double, and triple bond lengths and energies for hydrocarbons, soli d graphite, and diamond, it cannot describe bonding situations intermediate between single and double bonds, su ch as delocalized bonding in
30 benzene. To correct for this and to correct fo r the non-physical overbinding of radicals, Brenner68 developed an improved form of Tersoff-t ype potential for hydrocarbons. The Morse-type functions for pair interactions us ed in the Tersoff potential, howeve r, go to finite values as the distance between atoms decreases. This means that there is a limit to model processes involving energetic atomic collisions. Th erefore, Brenner and co-workers29 modified the expressions for intra-molecular interactions a nd expanded the fitting database The second generation reactive bond order (REBO) potential gives more accurate bond lengths, energies, and force constants for hydrocarbon molecules. It is also well known that the REBO potenti al can give realistic physical properties of carbon nanotubes.6, 27, 28, 32, 44, 69-81 In the REBO potential based on TersoffÂ’s model, the total binding ener gy is expressed as ii j ij b i i totr E E E, (2-12) where Ei is the energy of atom i and Eb(rij) is the binding energy between atom i and its nearest neighbors, j, given by ii j ij A ij ij R ij br V b r V r E. (2-13) The function VR(rij) corresponds to the pair-wise repulsi ve potential relating to the corecore and electron-electron interactions and the function VA(rij) corresponds to the attractive potential due to the core-elect ron interactions. These functions rely on only the distance, rij, between two atoms, i and j. The analytic forms of th ese functions are given by ijr ij ij c ij Re A r Q r f r V 1 (2-14) 3 1 n r n ij c ij Aij ne B r f r V (2-15)
31 The repulsive term, VR(rij), goes to infinity as the distance, rij, approaches zero and the attractive term, VA(rij), has sufficient flexibility to fit bond properties. A, B, Q, and are twobody parameters that are determined by th e type of interac tion. The function, fc(rij), limits the range of the covalent interactions to insure that the interactions include only the nearest neighbor atoms and is written as max max min min max min min0 2 cos 1 1ij ij ij ij ij ij ij ij ij ij ij ij cD r if D r D if D D D r D r if r f (2-16) where the term, min max ij ijD D defines the distance over which th e value of function varies from one to zero. The bond order function, bij, between atom i and j in equation (2.4) represents the manybody feature of the Tersoff type potential. It includes various chemical effects such as coordination numbers, bond angles, torsion angles, and conjugation effects and it depends on the local atomic environment in which a particular bond is located. Therefore, the REBO potential can describe covalent bond formation and br eakage associated with atomic hybridization alteration by weighing the bond strength. This te rm is most essential for treating chemical reactions, in which the b onding of carbon atoms change, and is expressed by ij ji ij ijb b b b 2 1 (2-17) where ijb and jib are identical forms except exchanging i and j indices. These first two terms are determined by the local coordi nation and bond angles for atoms i and j and are given as
32 2 1 ,, cos 1 H i C i ij j i k R r R r ijk ik c ijN N P e G r f be ik ik e ij ij ijk (2-18) where ijkG cos is a polynomial function that manage s the influence of the nearest neighbors to the bond order according to the bond angle among atoms i j and k and ijk is a fitting parameter used to describe three-body transi tion states around H at oms. The function, Pij is a correction term that explains th e distinct chemistry around atom i, and C iN and H iN are the number of neighboring C and H atoms of atom i and are given by carbon j i k ik c C ir f N, (2-19) The neighboring atoms to atom i, except atom j, can be counted according to the distance between atoms i and k, and the value ranges from 0 to 1. This guarantees that the bond order change is continuous and that Eb is also continuous during bond breaking and reforming. The term ijb is expressed as a sum of two terms: DH ij RC ijijb (2-20) where RC ij is the radial term and DH ij is the dihedral term. The term RC ij relies on whether a bond between the atoms i and j has a radical nature or is a part of a conjugated system. This term is expressed as conj ij t j t i ij RC ijN N N Y, (2-21) where ijY is determined by tricubic spline interpolation and t iN and t jN are the total number of neighboring atoms around atoms i and j, respectively. conj ijN is given by
33 2 2 ,1 carbon j i k jl jl c carbon j i k ik ik c conj ijx F r f x F r f N (2-22) where ik c t k ikr f N x (2-23) and 3 0 3 2 2 2 2 cos 1 2 1ik ik ik ik ikx if x if x x if x F (2-24) The term RC ij depends on the dihedral angle for carbon-carbon bonds, which reflects the torsional effect in the molecule and is expressed by jl c ik c ijkl conj ij t j t i ij DH ijr f r f N N N T2cos 1 (2-25) where ijkl is the torsional angle between atoms i j k and l The function ijT is determined by the tricubic spline interpolation. The term DH ij has some value when the bond between atoms i and j is a double bond and is zero when atoms i and j are not carbon. Lennard-Jones Potential The Lennard-Jones potential82 is a popular choice to descri be van der Waals interactions between atoms or molecules. Ther e are various basic functional form s for this potential, but only the 12-6 form is used for the simulations in this dissertation. This 12-6 form is expressed as 6 124ij ij ij LJr r r V (2-26) where LJV is the cohesive energy, rij is the distance between atoms i and j and and are the Lennard-Jones parameters for particular types of atoms. The 1/r12 term corresponds to the short-
34 ranged repulsive interaction between atoms due to the Pauli principl e, which explains the abrupt energy increase when electron cl ouds overlap each other. The 1/r6 term represents the relatively long-ranged attractive interacti on due to van der Waals dipole-di pole interacti on. Figure 2-1 shows the total energy of the Lenna rd-Jones potential, as well as the repulsive and the attractive terms for the carbon-car bon interaction. For the interaction between different types of atoms, there is the Lorentz-Berelot combining rule to calculate the parameters of the Lennard-Jones potential. If the parameters for the interaction between same t ypes of atom are expressed by AA and AA, the parameters for the interaction between different types of at oms such as A and B are determined by BB AA AB 2 1 (2-27) BB AA AB (2-28) The parameters CC, HH, CC, and HH applied in these studies ar e summarized in table 2-1. Periodic Boundary Conditions Molecular dynamics simulations are genera lly performed on small size systems on the order of nanometers because of the speed of calculations and the available storage. If the property of a very small liquid dr op or a nanocrystal is studied, atoms or molecules can be held together by the cohesive for ce during the simulation. These arrangements, however, are not appropriate for the simulation of bulk materials because the large fraction of atoms or molecules lies on the surface of sample. This surface effect can be overcome by appl ying periodic boundary conditions (PBCs) to the system. As shown in figure 2-2, the primar y cell box, the interested system, is duplicated through space to build an infinite lattice. During th e simulation, as an atom or molecule moves in the primary cell, its periodic image molecule in each neighboring imaginary cell moves in the
35 exact same way. Therefore, if an atom or mol ecule leaves the primary cell, one of its images enter through the opposite face. The number of atom s in the primary cell is always conserved. It is not necessary to track all imaginary molecule s but just the molecules in the primary cell. A given atom or molecule interacts with all other at oms or molecules in this infinite lattice: other atoms or molecules in the primary cell and th e nearest image atoms or molecules in the imaginary cells. For the Lennard-Jon es potential, it is po ssible to perform th e simulation without an atom or molecule interacting with its ow n imaginary duplicate mol ecules in a cubic box of side L 6 Predictor-Corrector Algorithm The finite difference approach is a standa rd method for solving ordinary differential equations such as Newtonian motion equations. Th e predictor-corrector algorithm is one of the high order algorithms and is widely used in MD simulations. The main concept of predictorcorrector is that the positions, velocities, accelerations, and highe r order derivatives of position at time t + t can be estimated by Ta ylor expansion about t in the condition of continuous trajectory. The REBO-MD simulations used in this diss ertation are performed with the third order Nordsieck-Gear predictor-corrector algorithm. The predictor forms are expressed as t b t t b t b t t a t t a t b t t a t t v t t v t b t t a t t v t t r t t rp p p p 2 3 22 1 6 1 2 1 (2-29) where r v a and bare the position, velocity, acceleration, and third derivative of position with respect to time, respectively, of each atom. The superscript p corresponds to the Â‘predictedÂ’ values.
36 From the predicted positions, the interatomic forces at time t + t are calculated by the gradient of potential energy because the predic ted acceleration values are not based on physics. After calculating the correct accelerations, t t ac the comparison of the corrected accelerations with the predicted accelerations can estimate the error size of the prediction step, t t a t t a t t a t t ap c (2-30) Using this error and the predicted results, the posi tions and other derivatives can be corrected and be used to obtain better approximations to the true values using the corrector forms given by t t a c t t b t t b t t a c t t a t t a t t a c t t v t t v t t a c t t r t t rp c p c p c p c 3 2 1 0 (2-31) These corrected values are used to predict the positions and first n derivatives in the next iteration, and then the same procedure is rep eated throughout the entire simulation trajectory. Gear 83, 84 provided the best choices of these coefficients c0, c1, c2, c3 for different predictorcorrector values. The coefficients for a third order predictor-corrector are c0 = 1/6, c1 = 5/6, c2 = 1, and c3 = 1/3. No integration algorithm will provide an exac t solution for a long time because of the truncation of Taylor expansion a nd the round off errors caused by th e finite digit numbers in the computer simulations. Therefore, the appropriate order of truncated equations is chosen to balance the accuracy and the computational speed. In addition, the error for an nth order algorithm is dependent on the time step size. Greater accuracy can be achieved by applying higher order derivatives or using smaller time step, t.
37 Ensemble Details An ensemble is an assembly consisting of a very large number of similarly prepared systems. The collection of thes e systems may be under the same macroscopic or thermodynamic conditions, but with different mi crosopic details. Ensembles are differentiated by three different thermodynamic variables kept constant, as su mmarized in table 2-2. For standard MD simulations, an isolated system is characteri zed by a time-independent translationally and rotationally invariant Hamiltonian. Integration of the classical motion equations for such a system leads to a trajectory mapping a micro canonical (NVE) ensemble of microstates. The condition of the microcanonical ensemble, however does not correspond to the conditions of most experiments. For comparison with experi ment, the following ensembles are more useful. In the canonical ensemble (NVT), the temperature is maintained at a specified average (macroscopic) value through the us e of thermostat algorithms, wh ile the total en ergy (E) of the system can fluctuate. In the isothermal-isobari c ensemble (NPT), the pressure has a specified average value, while the instantaneous volume (V) of the system can change. The grandcanonical ensemble (VT) has a constant volume and temperat ure, but exchanges particles with a surrounding bath. In this case, the chemical potential () of the different species is maintained at a specified value, but the instantaneous particle number (N) can fluctuate. Other ensembles are possible, but of more limite d practical application. Temperature Control Methods A thermostat algorithm is an amendment of MD simulations for generating a thermodynamic ensemble at a constant temperature. A thermostat is used in MD simulations for the following reasons. Matching experimental c onditions (NVT or NPT) Studying thermally dependent processes Evacuating the heat in dissipative non-equilibrium MD simulations
38 Avoiding steady energy drifts by the accumulation of numerical errors The definition of an instantaneous temperature is needed to use a thermostat. Then, this temperature is compared to the reference temperature T0 of the heat bath to which the system is coupled. For the system having N particles, the average internal kinetic energy of the system is related to its macroscopic temperature T through T N k mv Kdf B N i i2 1 2 11 2 (2-32) where m is the mass of particle, v is the velocity of particle, kB is the BoltzmannÂ’s constant, and Ndf is the number of internal degrees of freedom of the system. Similarly, the instantaneous temperature Tins at any time is expressed as K N k Tdf B ins2 (2-33) The average temperature is identical to the macroscopic temperature T. Velocity rescaling Since the macroscopic temperature is a f unction of the average kinetic energy, the velocities of the particles in the system can be ad justed in order to control the system temperature given by ins old newT T v v0 (2-34) where vnew is the rescaled velocity and vold is the velocity prior to rescaling. This method is called the velo city rescaling method, which is the simplest thermostat method. It allows the system to exchange en ergy with the surroundings, but it keeps the kinetic energy of the system constant. Therefore, ther e is no conserved quantity during the simulation. The velocity rescaling method can also lead to an unpredicted problem: a gradual transition of
39 KE from high frequency motions, such as bond st retching and angle bending, into low frequency motions, such as translation and rotation. Nos-Hoover Andersen 85 proposed that the pressure and/or the te mperature can be constant in some MD simulations for better mimicking the experimental situation. In the AndersonÂ’s method, a particle is randomly chosen and its velocity is extracted from the Maxwell distributi on. It was simple but known to have poor efficiency and discontinuous trajectories. AndersonÂ’s extended Lagrangian approach, however, inspired Nos 86, 87 to propose an Â“extended systemÂ” method for the canonical ensemble. He introdu ced a time scale variable s and its conjugate momentum ps, which describe the coupling of system to the heat bath. An additional parameter Q can be regarded as a heat bath mass. Hoover 88 simplified NosÂ’s method by elimin ating the time scaling factor s and introducing a thermodynamic friction coefficient HooverÂ’s modified expression of NosÂ’s method is known as the Nos-Hoover thermostat. For a system containing N atoms, the e quations of motion are expressed by N i B k i i i i i i i i iT k N m p Q p q F p m p q1 21 (2-35) where qi is the position of atom i, pi is the momentum, mi is the atom mass, Fi is the force applied to the atom, and dots denote time derivatives The third equation shows how to control the temperature of system in the Nos-Hoover thermo stat. The difference between the instantaneous kinetic energy and the kinetic energy at the desired temperature determines the sign and magnitude of the frictional coefficient. Dependi ng on the parameter Q, the performance of the
40 Nos-Hoover thermostat can vary. For a small valu e of Q, the system temperature can fluctuate rapidly. With large Q value, sampling of phase space can be inefficient. Langevin A Langevin thermostat considered here is a generalized Langevin equation (GLE) developed from generalized Brownian motion theory 89-91. t t t t mR v F a (2-36) The second term in the right side is the fric tional force corresponding to the frictional dragging between particles. This frictional force, which is proportional to the part icle velocity, has the opposite sign because friction is against mo tion and acts to remove the excess energy. is the frictional coefficient. Based on the Debye continuum model, can be expressed by D 6 1 (2-37) where D is the Debye frequency, which is related to the Debye temperature, TD, by D B DT k (2-38) where kB is the Boltzmann constant and is the Plank constant. The third term in the right side is the random force corresponding to the random co llision between particles. This random force is considered as Gaussian white noise with a zero mean and a width expressed by92 t T k mB 22 (2-39) where m is the mass of particle, T is the desired temperature, and t is the time step. This random force is balanced with the frictional force to maintain the system temperature92. The thermostat atoms follow this Langevin equation of motion in stead of Newtonian motion equations, and their
41 velocities are modified to maintain the instan taneous kinetic temperature to the macroscopic desired system temperature.
42 Table 2-1. Lennard-Jones parameters for H and C93. Atom () / kB (Kelvin) H 2.81 15.0 C 3.35 51.2 Table 2-2. The eight therm odynamical ensembles, and the corresponding independent and dependent variables. Intensive variables ar e the chemical potential for all species (), the pressure (P), and the temperature (T ). Extensive variable s are the number of particles for all species (N), the volume (V), the internal energy (E), the enthalpy (H = E + PV), the Hill energy (L = E Â– iNi), and the Ray enthalpy (R = E +PVÂ– iNi). Independent Dependent Ensemble N V E P T Microcanonical N V T P E Canonical N P H V T Isoenthalpic-isobaric N P T V H Isothermal-isobaric V L N P T Grand-microcanonical V T N P L Grand-canonical P R N V T Grand-isothermal-isobaric P T N V R Generalized
43 2345678 -0.008 -0.004 0.000 0.004 0.008 LJ potential energy (eV)Distance () Total LJ energy Repulsive term Attractive term Figure 2-1. The total energy of Lennard-Jones potential, as well as the repulsive and attractive terms for the carbon-car bon interaction. Figure 2-2. A two-dimensional periodic system Atoms or molecules can enter and leave each box across each of four edges.94, 95 Primary Cell Image Cell Image Cell Image Cell Image Cell Image Cell Image Cell Image Cell Image Cell
44 CHAPTER 3 CALIBRATION OF METHODS FOR NANOMECHANICS Computational simulations are widely used to ev aluate and predict properties of CNTs as it is non-trivial to manipulate CNTs experimentally due to their small size. They have made important contributions to understanding the mechanical6, 8, 27, 69, 71, 74, 76, 96, electronic18, 97-100, chemical77, 101-105, and other properties of nanotubes. Howe ver, there have been some instances when different simulation studies predict different mechanical responses of similar CNT systems. In one set of studies, Trotter et al.106 and Ni et al.29 investigated the compressibility of single-walled CNTs that were either empty or f illed with materials such as fullerenes, neon, and n-butane molecules. The average buckling forces a nd critical strains pred icted for empty (10,10) nanotubes were slightly different in the two studi es, which was attributed to differences in the number of rigid moving atoms and the size of thermostat regions in the two studies. Mylvaganam and Zhang77 found that applying a thermo stat to all the atoms in the CNT except for any rigid atoms is an efficient way to dissipate excessive heat that is generated during the simulated straining of CNTs in tension. However, Trotter et. al.106 observed nonphysical buckling of CNTs that are compressed using MylvaganamÂ’s scheme and Langevin thermostats. Specifically, the CNTs buckled at the interface between the rigid a nd thermostat atoms rather than in the middle, which is predicted to occur when thermostat regions are limited to the CNT ends. One possible reason for these differing findings is variations in the ratio of thermostat atoms, the type of thermostat used, and the degree of equilibration in these various simulations. The differences in the results of these studies indicate the importance of investigating the role of thermostat atom configurations and type s on the results of simulations. Here, we examine the influence of thermostat methods, the number of thermostat atoms, and the rate of deformation on the mechanical behavior of CNTs using classi cal, atomistic, molecular dynamics simulations.
45 Computational Details This study considers hollow or butane fille d (10,10) single-walled CNTs that are 200 long and capped at both ends, as sh own in Figure 3-1, which were investigated by Trotter et al. 106 previously. These CNTs are divided into th ree regions. The outermost atoms on each end of CNTs are held rigid, the next group of atoms as one moves towards the CNT center have a thermostat applied to them to maintain the syst em temperature, and the remaining atoms, as well as all the atoms in the molecules that fill the CNTs, are active atoms that can evolve freely according to NewtonÂ’s equations of motion without any additional constraints and are marked as Â‘normalÂ’ in Figure 3-1. The number of rigid atoms at each end of the CN Ts is held constant in all the simulations considered in this study. However, the relative pe rcentage of thermostat atoms is varied in the simulations as indicated. Additi onally, there different types of thermostats are considered: Langevin107, 108, Nos-Hoover86, 88, and velocity rescaling85, 109. Prior to compressing the CNTs, each system is equilibrated at 300 K until its potential energy is essentially constant with time. Next, the rigid atoms on one end of CNTs are moved at a constant rate towards the opposite end along the axial direction. This results in CNT compression and the instantaneous forces on the rigid atoms at the end that is moved are measured. Although the slower comp ression rate gives more reliable and realistic results, it is reasonable to use the faster compression rate in the MD simulations because they yield produce similar trends and are significantly more computa tionally efficient. Theref ore, all the CNTs in this study are compressed at a rate of 40 m/s except where noted. The time step in all the simulations is 0.2 fs, and a suffi cient number of time steps occur between each displacement step in order to dissipate excess ener gy and maintain the system temp erature at 300 K in all cases.
46 Results and Discussion The relative percentage of thermostat atom s (RT%) in each CNT system is calculated using the following equation: 100 % active thermostat thermostatn n n RT (3-1) where nthermostat is the number of thermostat atoms and nactive is the number of active atoms in each CNT. The value of RT% is varied from 5% to 100% without changing the number of rigid atoms at both ends of each CNT. The effect of the quantity of thermostat atoms on the mechanical properties of empty CNTs is considered and the smoothed force-strain cu rves from the compression of the empty and nbutane-filled CNTs with Langevin thermostats are shown in Figure 3-2. In each case, the shapes of the force curves change in a similar manner with variations in RT%. Regardless of the value of RT%, however, the buckling force of each filled system is always larg er than that of each empty system at the same RT% value. Consequently we can qualitatively evaluate the effect of filling on CNT compressibility i ndependent of the number of thermostat atoms as was done by Trotter et al.106 However, for quantitative comparisons, the number of thermostat atoms should be the same for both the empty and filled CNTs, as is done here. The smoothed force-strain curves from the compression of n-butane filled CNTs where RT% is varied over a wider range of values a nd Langevin thermostats are used are shown in Figure 3-3. In each case the force increases as the CNTs are compressed, and decreases after CNTs buckle in response to the applied forces The maximum buckling force increases as the RT% value increases. In particular, the buckling for ce is about 50 nN at a RT% value of 5%, and about 105 nN at a RT% value of 100%. In contrast the critical strain decreases as the RT% values increase, and range from 0.035 to 0.015. This is due to the fact that the responses of active
47 atoms and Langevin thermostat atom s to the external forces are di fferent. Therefore, CNTs with different RT% can show differe nt mechanical responses in this kind of compression test. The slopes of the force-strain curves shown in Figure 3-3 also decrease prior to buckling as the RT% values increase. Although the YoungÂ’s modulus is not calcula ted directly here, it is possible to say that the YoungÂ’s mo duli of the same CNTs vary w ith changes in the RT% value, as the YoungÂ’s modulus is calculated by dividing the slopes of the for ce-strain curves by the cross-sectional area of the CNT 6-8, 10, 11, 110. This result can also be explained by the different mechanical responses of active and Langevin th ermostat atoms. Thus, taken together, these results indicate that the YoungÂ’s modulus, the maxi mum buckling force, and the critical strain of CNTs determined from MD simulations can be changed by simply altering the fraction of thermostat atoms considered. Snapshots of deformed n-C4H10 filled CNTs at 0.1 strain are shown in Figure 3-4. During compression, the main buckling occurs in the mi ddle of the CNTs, with additional buckling at both tips near the interface between the thermostat and rigid atoms. Interestingly, the degree of deformation decreases as RT% increases. In pa rticular, the position wh ere additional buckling occurs moves towards the middle of the CNT as RT% increases in a manner that corresponds to the shifting of the interf ace between the thermostat and rigid at oms as the number of thermostat atoms increases. Above 30% RT%, nonphysical buck ling occurs at the m oving tip of the CNT. In the case of 100 % RT%, no buckling occurs at the middle and there is just a nonphysical buckling at the CNT end at the interface be tween the thermostat and rigid atoms. When the system temperature is maintained by the Langevin thermostat, the thermostat atoms can have frictional and stochastic force term s applied to them, in addition to forces from the surrounding atoms. As a result, the phenomen a described above are mo st likely due to the
48 fact that the frictional and stochastic force term s applied to the Langevin thermostat atoms act to restrain deformation. It is also likely that the relatively high co mpression rate of 40 m/s exasperates this phenomenon. Therefore the effect of compression rate is also considered by comparing the compression of CNTs at rates of 40 and 4 m/s (see Figure 3-5). At the slower compression rate, the maximum buckling force and critical strain increase simultaneously with increases in the number of thermostat atoms, while, at the faster rate, the cr itical strain decreases and the maximum buckling force increases as th e number of thermostat atoms increase. In addition, the change of the CNTÂ’s modulus is less signifi cant at the slower rate. Figure 3-6 shows that nonphysical buckling at the moving end of the CNTs observed with the Langevin thermostats is avoided at the slower compressi on rate. However, while the results are more reasonable when the deformation rate is low, sl ower rates are more co mputationally demanding. It is therefore important to determine the appr opriate number of thermostat atoms and suitable deformation rates for the non-equilibrium MD simulations to avoid unphysical phenomena. Mylvganam and Zhang77 found that setting all atoms as thermostat atoms (excluding the rigid ones at the CNTs ends) is a more eff ective strategy for dissipa ting excess heat during tensile straining of CNTs. In addition, they evaluated both Evans-Hoover and Berendsen thermostats and concluded that the Berendsen thermo stat is optimum. To inve stigate the effect of thermostat types, Nos-Hoover and velocity-resca ling are considered in addition to the Langevin thermostat discussed above. Velocity rescaling109 is the simplest algorithm that can be used to control the temperature in computational simulations and is an is okinetic type of thermostat. The Berendsen thermostat108 relies on Â“coupling to an external bathÂ”, which is a modified form of velocity rescaling without the stochastic term Importantly, the Langevin thermostat107 also involves
49 coupling of the system to a heat bath at the de sired reference temperature. However, in the case of the Langevin thermostat, this coupling is achieved by introducing friction and stochastic terms, which is equivalent to random noise. In the Evan s-Hoover thermostat, the system temperature is controlled by monitoring the ki netic energy and feeding change s back into the equation of motion by force scaling. The Nos-Hoover thermostat86 also introduces additional dynamical variables by coupling to a heat bath through an applied friction coefficient. However, the equation of motion of the Nos-Hoover does not us e the stochastic term, which it has in common with the velocity rescaling, Bere ndsen, and Evans-Hoover thermostats. The smoothed force-strain curves for the three different thermostats considered by us here are illustrated in Figure 3-7. The figure indi cates how the maximum buckling force and the critical strain vary with the num ber of Langevin thermostat atoms in the CNTs. In the case of the Nos-Hoover and velocity rescaling thermostats, however, the values of the slopes of the forcestrain curves are stable and th e maximum buckling force and the cr itical strain are invariable regardless of the number of thermostat atoms. Additionally, the force-strain curves of the CNTs using Nos-Hoover and velocity rescaling thermostats have si milar trends, and comparable maximum buckling forces and critical strain valu es to CNTs that used Langevin thermostats at RT% of 5%. However, the use of Langevin thermo stat results in significantly smoother forcestrain curves. The snapshots of compressed n-butane filled CN Ts at 0.1 strain with the Nos-Hoover and the velocity rescaling thermostat are shown in Fi gure 3-8. In contrast to deformed CNTs with Langevin thermostats, nonphysical buckling doe s not occur during the compression of CNTs irrespective of the RT% values. Fu rthermore, all CNTs with diffe rent RT% values show similar deformation behavior, which means that the m echanical properties of CNTs to which Nos-
50 Hoover thermostats and velocity re scaling thermostats are applied are stable regardless of the number of thermostat atoms. It is likely that th e stochastic terms in the Langevin thermostat is the cause of the predicted non-buckling behavior during the compression of CNTs using this thermostat. One of reasons for applying thermostat atoms to systems in computational simulations is to imitate the dissipation of heat generated during compression of CNTs. System temperature using different thermostats are examined to see how ef fective each thermostat could be for the heat dissipation. The temperatures of CNTs with 100% thermostat atoms that are shown in Figure 3-7 are plotted in Figure 3-9. Just c onsidering force and stra in, it appears that there is no difference between the Nos-Hoover and veloci ty rescaling thermostats. Howe ver, systems using different thermostats show dissimilar beha vior in the heat dissipation. In particular, the system temperature using Nos-Hoover increases co ntinuously during comp ression of the CNTs, whereas those using the Langevin and velocity re scaling thermostats are relatively constant during the entire deformation pro cess. We also observed that all thermostat schemes show oscillations in the temperature curves when buc kling occurs near a st rain of 0.04, and the amplitude of these oscillations are generally smalle st when the velocity re scaling method is used. Taking into consideration all factors, it is cl ear that the Langevin and velocity rescaling thermostats are effective at managing the sy stem temperature during CNT compression. In addition, the velocity-rescale ther mostat is most efficient contro lling system temperature without inducing nonphysical buckling during the mechani cal deformation of CNTs regardless of the value of RT%. Because the force-strain curve is smoothest using the Langevin thermostat, the Langevin thermostat is an acceptable thermostat to use for the mechanical deformation of CNTs as long as an appropriate value of RT% is used to avoid unphysical deformation modes. Lastly,
51 the Nos-Hoover thermostat is not recommended for use in these types of simulations because of its inability to maintain a relatively consta nt system temperature although the Nos-Hoover thermostat is able to maintain the system temp erature at a constant value during the equilibrium process. Conclusions The mechanical properties of CNTs can vary with the relative amount of Langevin thermostat atoms used in molecular dynamics simulations. Nonphysical buckling occurs during deformation with large RT% values and at high er deformation rates. In contrast, the NosHoover and velocity rescaling thermostats result in physical mechanical deformation modes regardless of the percentage of thermostat atoms. The Langevin and velocity rescaling thermostats are successful at maintaining a c onstant system temperat ure during the compression of CNTs. However, the Nos-Hoover is not able to do so and this is therefore not considered to be suitable for use in simulations of CNT deformation. In conclusion, we have demonstrated the im portance of the choice of thermostat and the fraction of thermostat atoms used in simulations of CNT deformation.
52 Figure 3-1. Snapshots of a 200 (10,10) hollo w single-walled CNT that is empty (top) and filled with n-butane molecules (bottom). Th e thermostat, rigid, moving, and normal regions are designated on each CNT. Th e number of rigid and moving atoms is invariable while the ratio of thermostat to normal atoms is varied as discussed in the text. 0.000.020.040.060.08 0 20 40 60 80 100 120 140 Force (nN)Strain Empty 5 % Empty 30 % Empty 100 % Butane 5 % Butane 30 % Butane 100 % Figure 3-2. Smoothed force versus strain curv es of the empty and n-butane filled nanotubes shown in Figure 3-1 using the Langevin ther mostat with different relative percentages of thermostat atoms (5%, 30%, and 100% ). The dotted and solid lines represent empty and butane filled nanotubes, respect ively. The deformation rate is 40 m/s. Thermostat Thermostat Normal Rigid Moving
53 0.000.020.040.060.080.10 0 20 40 60 80 100 120 140 Force (nN)Strain 05 % 10 % 15 % 20 % 30 % 40 % 100 % Figure 3-3. Smoothed force versus strain curves of the n-butane filled nanotube shown in Figure 3-1 using the Langevin thermostat. The relativ e percentage of thermostat atoms varies from 5 % to 100 % as indicated. Th e deformation rate is 40 m/s.
54 Figure 3-4. Snapshots of the n-butane filled na notube shown in Figure 3-1 using the Langevin thermostat at a strain of 0.1. The relative percentages of thermostat atoms varies from 5% to 100 %. A) 5 %, B) 20 %, C) 30 %, D) 40 %, and E) 100 %. The deformation rate is 40 m/s. A B C D E
55 0.000.020.040.060.08 0 20 40 60 80 100 120 140 Force (nN)Strain 40 m/s 5 % 40 m/s 30 % 40 m/s 100 % 4 m/s 5 % 4 m/s 30 % 4 m/s 100 % Figure 3-5. The smoothed force versus strain curves of the n-butane filled nanotube shown in Figure 3-1 using the Langevin thermostat at different deformation rates (4 and 40 m/s) and relative percentages of th ermostat atoms (5%, 30%, and 100 %). Figure 3-6. Snapshots of the n-butane filled na notube shown in Figure 3-1 using the Langevin thermostat at a strain of 0.1 at a deformati on rate of 4 m/s. The relative percentage of thermostat atoms is A) 30 % and B) 100 %. A B
56 Figure 3-7. The smoothed force versus strain curves of the n-butane filled nanotube shown in Figure 3-1 using the A) No s-Hoover, B) velocity re scaling, and C) Langevin thermostats at different rela tive percentages of thermostat atoms (5% (black), 30% (blue), and 100% (orange)). The deform ation rate is 40 m/s in all cases. Figure 3-8. Snapshots from the compression of th e n-butane filled nanotube shown in Figure3-1 at a strain of 0.1 using A) velocity re scaling and B) Nos-Hoover thermostats with different relative percentages of ther mostat atoms (5%, 30%, and 100 %). 0.000.010.020.030.040.05 0 20 40 60 80 100 120 Force (nN)Strain 0.000.010.020.030.040.05 0 20 40 60 80 100 120 Force (nN)Strain 0.000.010.020.030.040.05 0 20 40 60 80 100 120 Force (nN)Strain A B 5 % 30 % 100 % 5 % 30 % 100 %A B C
57 0.000.020.040.060.080.10 300 400 500 600 700 Temperature (K)Strain Nose-Hoover Velocity rescaling Langevin Figure 3-9. System temperature versus strain fo r the n-butane filled nanotube shown in Figure 31 using the indicated thermostats at a de formation rate of 40 m/s. The relative percentage of thermostat atoms is 100 % for each case.
58 CHAPTER 4 NANOMECHANICS OF CARBON NANOTUBES Here, the mechanical respons es of pristine hollow, C60-filled, n-butane-filled, and multi(dual-, tripleand quadruple-) walled CNTs under bending forces at various temperatures are examined using classical molecular dynamics (MD) simulations. The results are compared to the responses of CNTs with covalently bonded func tional groups and wall vacancies. The results are then compared to those obtained from the application of compressive forces. Computational Details The CNTs considered here are hollow (10, 10) SWNTs, (10,10) SWNTs filled with nbutane, (10,10) SWNTs filled with C60 (peapods), (10,10) SWNTs filled with (5,5) SWCNTs (called a (10,10)@(5,5) doublewalled CNT or DWNT), (15,15)@(10,10)@(5,5) triple-walled carbon nanotubes (TWNTs), (20,20)@(15,15) @(10,10)@(5,5) quadruple-walled carbon nanotubes (QWNTs). All these CNTs are about 10 nm long and capped at both ends except for the outermost (20,20) CNTs in the QWNTs. To examine the effect of vacancies on th e responses of the nanotubes to bending and compression, three different types of vacancies are introduced on the surface walls of the hollow (10,10) SWNTs: a single vacancy, two vacancies th at are next to each other along the nanotube circumference, and two vacancies that are faci ng each other along the nanotube circumference. As determined theoretically in previous studies111-113, each of the three carbon atoms near a vacancy has one dangling bond. In addition, hollow (10,10) SWNTs that have H2C=C groups covalently bound and randomly di stributed along the walls are considered to address their influence on mechanical responses. The density of functional groups considered is 0.05 g/cm3 (or a SWNT/H2C=C weight ratio of 35.08), 0.10 g/cm3 (or a SWNT/H2C=C weight ratio of 17.54), and 0.20 g/cm3 (or a SWNT/H2C=C weight ratio of 8.77).
59 The bending of the CNT is ach ieved by using a three-point be nd test, as illustrated in Figure 4-1A. The reverse bending forces are applie d to the center region and both tube ends of the CNT. The entire force on the center region, F, is equivalent and oppos ite to the summation of the forces applied on both ends of the CNT. Th e bending force is increase d gradually at a rate that increases deflection by 0.1 nm every 100,000 MD time steps. In the bent CNT all the atoms are subjected to classical Newtonian equations of motion along with a velocity rescaling thermostat instead of the Langevin thermostat. Th is is to avoid the restraint effect of the Langevin thermostat80 that can disturb the movement of atoms as a result of the stochastic force term. All systems for CNT compression simulations are equilibrated with a Langevin thermostat.107, 108 In this case the CNT to be compressed is divided into three regions as shown in Figure 4-1B: the outermost 110 atoms on each end of the CNT are held rigid; the next 40 atoms have Langevin thermostats applied to them, and the remaining atoms in the CNT as well as the atoms inside the CNT can evolve freely with no additional constraint s. CNT compression is simulated by shifting the rigid a nd moving atoms at the one end of the CNT towards the other end at a constant rate of 4 m/s, which is an order of magnitude slower than the rate used previously.28, 29, 31 The effect of the compression rate and the relative number of thermostat atoms on the mechanical properties of the CNT was investigated by us previously.80 During compression, the instantaneous forces are meas ured and averaged over the rigid and moving atoms through the last 100 steps of every 1000 time st eps. The strain calculat ed here is the ratio of the instant CNT length to the difference of the initial and instant CNT lengths.
60 Results and Discussion Filling Effect The plot of force versus deflection of vari ous (10, 10) hollow and filled CNTs during the bending test are illustrated in Figure 4-2. Nanotube deflection, x, increases linearly with the bending force in the low force region. Both hollow and filled CNTs undergo buckling with increasing bending force, beyond wh ich they can not support the ex ternal bending force. It is predicted, however, that the bending force of the (5,5)@(10,10) DWNT is larger than that of the (10,10) peapod, which is greater than that of the ( 10,10) SWNT, such that the order of bending deflections is x DWNT > x Peapod > x SWNT. This result indicates that filling can make nanotubes more resistant to external bending forc es and the degree of resistance can be changed depending on the filling materials. Figure 4-2 furt her indicates that the bending forces increase and the maximum bending deflection can change with the number of nanotubes shells. Figure 4-3 shows snapshots from the MD si mulations of the SWNT peapod, DWNT, and TWNT just after buckling occurs. It indicates that the deformed region is spread along a longer region near the center of the SWNT whereas it is more concentrat ed at the center of the peapod, DWNT and TWNT. This is consiste nt with the fact th at the peapod, DWNT and TWNT are more resistant to bending than the SWNT, as illustrated in Figure 4-2. Figure 4-4 illustrates the compressive force versus strain of the SWNT, peapod, and DWNT. In each case, the force increases with compression until buckling occurs, after which the force drops significantly. The filled CNTs exhib it higher buckling forces relative to hollow CNTs irrespective of the filling material. The maximum forces and critical strains of a SWNT, peapod, and DWNT are 58.19 nN at 0.037, 69.60 nN at 0.042, and 111.52 nN at 0.052, respectively. It is clear from Fi gure 4 that the slopes of the fo rce curves prior to buckling are very similar to each other. The elastic modulus of the CNTs can be ca lculated by dividing the
61 slopes of the curves of Figure 4-4 by the cross se ctional area of the CNTs, which means that the elastic modulus of the CNTs are also roughly equal. This finding is in agreement with previous studies that found that the elastic modulus of CNTs are not dependent on filling CNTs29 or the number of inner CNT walls.8 The snapshots of compressed CNTs at strains of 0.04, 0.05 and 0.06 are illustrated in Figure 4-5. At a strain of 0.04, the peapod and DW NT structures maintain their cylindrical shape while the SWNT buckles in the center of the tube. This can also be predicted from the force vs. strain plot in Figure 4-4 because the maximum buckling force of the SWNT is 0.037, which is lower than 0.04. However the maximum buckling forces of the peapod and DWNT are 0.042 and 0.052, respectively, which allows them to support the compressive force without buckling. At a strain of 0.05, the SWNT and peapod structur es have buckled, whereas the DWNT retains its original shape. Finally, at a strain of 0.06, a ll the CNT structures have deformed into similar shapes, except that the SWNT is more flattened than the peapod and the DWNT. That means that van der Waals interaction between the outer na notube wall and fullerene s or inner nanotube makes the structure more resistant to deform ation and results in a higher maximum buckling force and compressive strain. The pair distribution functions (PDFs) in Figure 4-6A further illustrate the effect of van der Waals interaction between the out er nanotube walls and the fulle renes or the inner nanotube walls within the outer nanotubes on the mechanical property of the filled CNTs. Initially, the PDFs of the SWNT and CNT part of the pea pod and DWNT exhibit very similar shapes and carbon-carbon bond distances. However, at the critical strain for each structure, the PDFs shift to shorter distances, which is indi cative that the carboncarbon bond distances in the nanotubes are compressed as a result of external compressive fo rces. In particular, the PDFs of the peapod and
62 DWNT have highly compressed carbon-carbon bond distances. Th is means the peapod and the DWNT structures can withstand more compression and maintain their over all cylindrical shape due to the presence of filling fullerenes and i nner nanotubes. However, the SWNT deforms and has longer carbon-carbon bond distances. This expl ains why the SWNT has the lowest maximum buckling force. Figure 4-6B illustrates the PDFs of the fulle renes and inner (5,5) nanotubes in the DWNT. For the peapod, the carbon-carbon bond distances of the inner fu llerenes are constant during compression, while those of the inner tubes are reduced due to the compressive force. This means that the DWNT is able to support the co mpressive force while maintaining its cylindrical shape by reducing the carbon-carbon bond distances of the inner a nd the outer tubes. However, the fullerenes inside the peapod structure cannot support the exte rnal force and exhibit lower buckling forces than the DWNT. Temperature Effect The compression and bending simulations are repeat ed at different system temperatures to understand the effect on the CNT mechanical res ponses. Plots of forces versus deflection are given in Figure 4-7. The results indicate th at the maximum bending forces of the nanotubes decrease with temperature regardless of whethe r the CNTs are filled or not. The slopes of the curves, however, are roughly consta nt regardless of the temperature, which is consistent with the finding that the YoungÂ’s modulus of the CNTs remains constant during compression as the temperature increases. Jeng et al.114 suggested that the mechani cal properties of CNTs are insensitive to the thermal state of the system at small deformations during tensile testing, but are sensitive under large deformation. This means th at the mechanical properties of CNTs, such as YoungÂ’s modulus, are insensitive to thermal cond itions but the thermal motion of the atoms may be important for obtaining the co rrect maximum buckling force.
63 Figure 4-8 illustrates the for ce versus strain of the SWNT peapod and DWNT at 100, 300, 600, and 2500 K at the same compression rate of 4 m/s. The results indicate that the maximum buckling forces of the compressed CNTs decr eases with increasing system temperature regardless of the filling, in agreement with earlier findings.29 In particular, these systems exhibit low buckling forces at the highe st temperature considered here of 2500 K. In addition, it is observed that the slopes of the curves are similar regardless of the temperature. Figure 4-9A shows the maximum buckling forc e versus temperature, and illustrates how the temperature affects the maximum buckling forces of hollow and filled CNTs. Regardless of the temperature, the maximum buckling forces of the CNTs are in the same order: DWNT > peapod > SWNT. After normalizing the temperatures the forces of the ho llow and filled CNTs are comparable at the lower temperature, as i ndicated in Figure 4-9B. The SWNT, however, is more susceptible to variations in buckling behavi or at higher temperatures than the peapod or the DWNT. Effect of Wall Defects During the synthesis or purification of CNTs de fects, such as vacancies, can be introduced into the nanotube walls. CNTs also can be treat ed to covalently attach chemical groups to the sides115 or ends.116 In these cases the sp2-hybridization of the carbon atoms around the defects and functional groups are altered. To investigat e the effect of wall defects on the mechanical responses of the CNTs, two type s of defects, vacancies and H2C=C functional groups, are introduced on the CNT surface and the effects of these defects on the bending of the CNTs are considered. Figure 4-10 illustrates that the maximum bending forces decreases with the number of wall defects and the density of the functional groups, whic h is very similar to the defect effect on the compressive properties of nanotubes shown in Fi gure 4-11. In the case of bent SWNTs, the
64 vacancy defects are located at the ends of the buc kles. In addition, the buckles always occur in the middle of the SWCNTs during bending because the forces are applied to the middles and the ends of CNTs. Figure 4-11 plots force versus strain of pr istine SWNTs and SWNTs with vacancy defects and covalently bound functional groups of various densities. In the case of wall vacancies, the buckling force decreases with the number of vacancies, while the maximum buckling force of the CNT with a single vacancy def ect is comparable with that of the pristine SWNT and is very similar to the defect effect on the bending prope rties of the nanotubes shown in Figure 4-10. The slopes of the force curves are unchanged as th e number of vacancy defects increases, which means that the elastic modulus of CNTs with one or two vacancies are approximately equal. According to the work of Agrawal et al.117, the YoungÂ’s modulus of nanotubes with vacancy defects decreases with the number of vacancies. Therefore, it is e xpected that the elastic modulus of CNTs would decrease if the density of vacan cies was higher than the numbers considered here. The simulations predict that the vacancies ac t as pinning points of th e buckles that form during compression, as indicated in Figure 4-12. For pristine SWNTs, the buckles form and move and up and down the length of the nanotube before settling near the center. In contrast, for SWNTs with vacancies, the buckl es occur at vacancy defects and do not show any positional movement. In addition, the simulations predict that the initial vacancy defects with three dangling bonds can transform into other configur ations, such as a sing le pentagon with one dangling bond, or a double heptagon with two dangling bonds during compression. According to previous studies, these single vacancy structures are stable up to 4000 K118 for long time scales on the order of 10 hours with an activation energy of around 1 eV.119 However, the structure of a
65 single pentagon with one dangling bond is lowest in energy.120, 121 Therefore, compression can make these single vacancy defects overcome the ac tivation barrier and transform to more stable configurations. It is also observed that c ovalently bound functional groups makes the nanotubes more vulnerable to buckling and the higher the density of functional groups, the more buckles occur at the same strain value. The decrement rates ra nge from 9-12% depending on the densities, which is in a good agreement with previous results.28 In addition, the more functional groups there are on the nanotube walls, the more buckl es occur at the same strain value. The slopes of the force curves are unchanged up to 80 functional groups on the nanotube walls. The simulations thus indicate that the YoungÂ’s modulus of SWNTs is less affected by the density of functional groups than by the number of vacancies. Conclusions Filling SWNTs increases their maximum defl ection and buckling fo rces during bending and compression. MWNTs are predicted to supp ort higher external loads than SWNTs and peapods because the inner CNTs sustain more external forces than the fullerenes. This mechanical behavior of hollow or filled CNTs can deteriorate at high temperature. The chemical modification of the CNT wall through the generation of vacancies or the attachment of functional groups is also predicted to weaken the mechanical responses of CNTs.
66 Figure 4-1. Snapshots of 100 (10, 10) hollo w CNT for (A) bending and B) compression. The rigid, thermostat, and active regions are designated on each CNT. All atoms in CNT are active for the bending test of CNT. Thermostat Thermostat A ctive Rigid Rigid & moving A B F F/2 F/2 x
67 Figure 4-2. Force versus deflection of various 100 (10, 10) hollow and filled CNTs during bending. Figure 4-3. Snapshots of 100 long A) SWNT, B) C60 filled SWNT, C) DWNT, and D) TWNT after the buckling in figure 4-2. QWNT SWNT Pea p od TWNT DWNT 0 10 20 30 40 50 60 010203040 x ()Force (nN) A C B D
68 0.000.020.040.060.080.10 0 20 40 60 80 100 120 DWNT Peapod Force (nN)StrainSWNT Figure 4-4. Force versus strain of 100 (10, 10) hollow CNT, CNT filled with C60 (peapod), and DWNT under the compression rate of 4 m/s. SWNT, peapod, and DWNT are in blue, orange, and green. Figure 4-5. The snapshots of compresse d 100 (10, 10) hollow CNT (upper), C60 filled CNT (middle) and DWNT (lower) at A) 0.04, B) 0.05 and C) 0.06 strain in Figure 4-4. A B C
69 Figure 4-6. The pair distribut ion functions of A) SWNT and outer CNT part of peapod and DWNT and B) C60 in peapod and inner CNT at the initial and the maximum buckling force stages. 1.301.351.401.451.501.551.60 0 5 10 15 20 25 g(r)Distance () initial SWNT initial Peapod initial DWNT compressed SWNT compressed Peapod compressed DWNT 1.301.351.401.451.501.551.60 0 1 2 3 4 5 g(r)Distance () Initial C60 Initial inner CNT Compressed C60 Compressed inner CNT A B
70 Figure 4-7. The plot of forces versus deflec tions of various 100 (10, 10) hollow and filled CNTs during the bending test at the various temperatures (s olid line = 100 K, dashed line = 300 K, and dotted line = 600 K). QWNT TWNT DWNT SWNT 0 10 20 30 40 50 010203040 x ( )Force (nN)Temp. increases Pea p od
71 Figure 4-8. Force versus strain of 100 ( 10, 10) A) hollow CNT, B) CNT filled with C60 (peapod), and C) DWNT at the different temperature (100, 300, 600, and 2500 K) under a rate of compression of 4 m/s. 0.000.020.040.060.080.10 0 20 40 60 80 100 120 Force (nN)Strain SWNT 100K SWNT 300K SWNT 600K SWNT 2500K 0.000.020.040.060.080.10 0 20 40 60 80 100 120 Force (nN)Strain Peapod 100K Peapod 300K Peapod 600K Peapod 2500K A B 0.000.020.040.060.080.10 0 20 40 60 80 100 120 Force (nN)Strain DWNT 100K DWNT 300K DWNT 600K DWNT 2500K C
72 Figure 4-9. Buckling force of 100 ( 10, 10) hollow CNT, CNT filled with C60 (peapod), and DWNT at different temper atures (100, 300, 600, and 2500 K) under a rate of compression of 4 m/s. A) Maximum buck ling forces. B) Normalized maximum buckling forces. Figure 4-10. Force versus deflection of 100 (10, 10) hollow pristine CNT and CNTs having various densities of A) vacancy defects and B) functional groups during bending. 05001000150020002500 0 20 40 60 80 100 120 140 DWNT Peapod Maximum buckling force (nN)Temperature (K) SWNT 05001000150020002500 0.5 0.6 0.7 0.8 0.9 1.0 1.1 DWNT SWNT Normalized buckling forceTemperature (K) Peapod A B 0 2 4 6 8 10 010203040 x ( )Force (nN) Pristine SWNT One vacancy Two close vacancies Two far vacancies 0 2 4 6 8 10 010203040 x ( )Force (nN) Pristine SWNT 10 Functional groups 20 Functional groups 40 Functional groups A B
73 Figure 4-11. Force versus strain of 100 (10, 10) hollow pristine CNTs and CNTs having A) vacancy defects and B) functional groups of various densities under a rate of compression of 4 m/s. Figure 4-12. Snapshots of A) a single vacancy, B) two vacancies next to each other along the nanotube circumference, and C) two vacanci es that are facing each other along the nanotube circumference before compressi on (top) and after buckling (bottom). A B 0.000.020.040.060.080.10 0 20 40 60 80 Force (nN)Strain Pristine SWNT 10 Functional groups 20 Functional groups 40 Functional groups 0.000.020.040.060.080.10 0 20 40 60 80 Force (nN)Strain Pristine SWNT One vacancy Two close vacancies Two far vacancies A B C
74 CHAPTER 5 HYDRODYNAMICS OF CARBON NANOTUBES In this paper, the fundamental vibrational properties of bridged and cantilevered CNTs responding to the external fluids are pr edicted using classical MD simulations. Computational Details The CNTs considered here are 10, 20, a nd 30 nm long (10,10) armchair SWNTs with semi-spherical fullerene end caps. The details of these CNTs are summarized in Table 5-1A. The CNTs are placed in a vacuum or in a noble gas fl uid such as Ar, Kr, and Xe at 10, 50, and 100 atm. The pressures of the noble gases are determin ed according to the following procedure. First, the number of gas particles in each system is cal culated using the equation of state for the ideal gas as summarized in Table 5-2. Then, MD simula tions are used to measure the pressure of each system. To verify the interactions between flui d atoms, these results are compared with the prediction of the modified van der Waals equa tion of state proposed by Redlich and Kwong. 122 This comparison confirms that the fluid at each condition is in the gas re gion of the fluid-phase equilibrium diagram. The entire system is fully equilibrated to 300 K using a Langevin thermostat. In the vibrational simulation of the bri dged CNT, both ends of CNT are fi xed and the other part of the CNT as well as the noble gas atoms move free following NewtonÂ’s equation of motion, as shown in Figure 5-1A. The center region of the CNT is then pulled out by 0.3 nm which is the initial displacement, by applying additiona l force to the atoms in that region. After that, the CNT can vibrate in the vacuum or the noble gas fluid, and the vibrational amplitude and the consequent vibrational frequency can be predicted. For the cantilevered CNT simulations, atoms in one end of CNT are set to be rigid atoms that do not evolve in time according to Newt onÂ’s equation of motion but may be displaced
75 manually. The other part of the CNT and the flui d molecules are changed into active atoms as shown in Figure 5-1B. Then, the CNT is dragged in the fluid by moving th e rigid atoms at one end by 0.0002 nm every time step, which is equiva lent to an overall moving rate of 10 m/s. During the simulation, the deflecti on of CNT is measured which allo ws for the calculation of the vibrational frequency. Results and Discussion Figure 5-2A shows the deflection of 20 nm bridged CNT under vari ous conditions as a function of time. In the case of CNT in vacuum (Figure 5-2A), the period and amplitude of vibration are steady throughout the entire simulation. The simulation of CNT vibration is done in the microcanonical system, where the system en ergy instead of the system temperature is constant. There is no constraint or perturbation from thermostat in the system, resulting in no sign of damping. According to Figure 5-2B, the period and amplitude of th e CNTs vary with the external fluid atoms and system pressure. In th e same noble gas element fluid, the vibrational amplitude of CNT decreases with the increase of fluid pressure. At the same pressure, the amplitude decreases with the mass of noble gas atoms. This can be explained by the collision of particles with the vibrating CNT and the conseq uent energy transfer between CNT and colliding particles. In the higher pressure system, there are more co llisions between CNT and noble gas atoms and the energy can transfer from CNT to fluid particles occurs more quickly. For the heavier noble gas atoms, more energy can transf er during each collision process. Therefore, strong damping is predicted to occur in the hi gh pressure systems, or the systems with the heavier noble gas atom fluid. Through the fast Fourier transformation (FFT) method, the plot of deflection vs. time can be transformed into the vibrational frequency plot as shown in Figure 5-3, from which the primary vibrational frequency is derived. In the case of longer CNT, highe r pressure, or heavier
76 noble gas atoms, the curve of deflection begins to crumple, which results in more major peaks in the frequency plot. In that case, the highest peak is chosen as th e primary vibrational frequency. Figure 5-4 shows the dependency of bridged and cantilevered CNT vibration on the CNT length, the fluid pressure, and the noble gas el ement. As the fluid pressure increases from vacuum to 100 atm, the vibrational frequency of CNT decreases regardless of the type of system, the CNT length, and the noble gas surroundings. The higher pressure, which increases the density of fluid surrounding the CNT, can produce an effect similar to that involved in increasing the mass of the vibrating CNT, thus leading to a frequency decrease. Therefore, the vibrating CNT shows a more significant freque ncy decrease in heavier atom systems than in lighter atom systems. In addition, the vibr ational frequency decr eases with increasing CNT length. For both bridged and cantilevered CNTs, it is observed that the frequency of a 30 nm CNT is about 90% lower than that of 10 nm CNT and a bout 55% lower than that of 20 nm. The frequency of the fundamental transverse vi bration mode of an unstretched rod clamped on both ends, the bridged CNT, and a rod clampe d on one end, the cantilevered CNT, derived from continuum mechanics based considerations123 are given by Y L 2 R 2 4 22 f2 (Bridged CNT) (5-40) Y L 2 R 2 52 3 f2 (Cantilevered CNT) (5-41) where R and L are the radius and length of CNT, respectively, Y is the YoungÂ’s modulus estimated to be 1012 Pa8, 17, and = 1.3 103 kg/m3 8 is the density. The predicted results of CNT vibration in vacuum from this study ma tch well with the predictions of continuum mechanics based equations, as illustrated in Figur e 5-5. While CNTs can be considered to be cylindrical rods due to their stru ctural configurations, the vibrati onal behavior of CNTs in this
77 study is interestingly closer to those of filled solid rod models than of cylindrical rod models. The exact explanation for this is not well understood and thus is not provided here, although this is thought to be a consequence of the st rong carbon-carbon bonding in the CNT walls. It is already observed that the pressure of the external fluid and the mass of fluid particle have an effect on the vibrational frequency of CNTs. Modifying the previous continuum mechanics based equation, here, we propose anal ytical equations to predict the vibrational frequency of bridged and cantileve red CNTs in external fluids: ) log( 2 2 2 4 222P MW Y L R f (Bridged CNT) (5-42) ) log( 2 2 2 52 32P MW Y L R f (Cantilevered CNT) (5-43) where MW is the molecular weight of the fluid and P is the fluid pressure. The value of log (MWP) does not have a unit. For example, the simu lated result of the bridged CNT in Kr fluid matches the prediction from equation (5-3 ), as illustrated in Figure 5-6. Conclusions In summary, using the classical, empirical REBO potential, the vibr ational behavior of bridged and cantilevered CNTs were investigated in external Ar, Kr, and Xe noble gas fluids. The CNT vibrational frequencie s are predicted to depend on th e CNT length as well as the pressure and mass of the fluid atoms. The simu lation result agrees with the predictions of continuum mechanics.
78 Table 5-1. Details of the SWNTs considered. Tube type Length (nm) Diameter (nm) Number of atoms (10,10) 11.06 1.37 1800 (10,10) 22.14 1.37 3600 (10,10) 31.95 1.37 5200 Table 5-2. The numbers of flui d atoms for the various systems CNT type 10 atm 50 atm 100 atm 10 nm 242 820 876 20 nm 486 1446 1491 30 nm 554 1538 1956
79 Figure 5-1. Snapshots from the MD simulations of A) a bri dged 20 nm long (10,10) hollow single-walled CNT in the 100 atm Ar fluid and B) a cantilevered 30 nm long (10,10) hollow single-walled CNT in the 10 atm Kr fl uid. Carbon atoms are in shown in blue, and the Ar and Kr atoms are shown in orange. Thermostat (Velocity scaling) Rigid & moving Other part of CNT and gas atoms are active x Rigid Rigid Other part of CNT and gas atoms are active x A B
80 Figure 5-2. Deflection vs. time of the 20 nm long bridged SWNT A) in vac uum and B) in Ar, Kr, and Xe fluids at 10, 50, and 100 atms, respectively. 050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps) A B Ar Kr Xe 10 atm 50 atm 100 atm
81 Figure 5-3. Analysis of frequenc y results: A) Deflection vs. time and B) frequency after the fastFourier-transformation of the 20 nm l ong bridged SWNT in 100 atm Ar fluid. 050100150200 -4 -3 -2 -1 0 1 2 3 4 x ()Time (ps)0.00.20.40.60.81.0 0.0 0.5 1.0 1.5 2.0 Frequency (THz)Amplitude A B
82 Figure 5-4. Frequency vs. fluid pressure of A) bridged CNTs and B) cantilevered CNTs with the lengths of 10, 20, and 30 nm. 020406080100120 15 20 25 30 35 40 45 50 Frequency (GHz)Pressure (atm) Ar Kr Xe 020406080100120 4 6 8 10 12 Frequency (GHz)Pressure (atm) Ar Kr Xe 020406080100120 0 2 4 6 8 Frequency (GHz)Pressure (atm) Ar Kr Xe 020406080100120 200 220 240 260 280 300 Frequency (GHz)Fluid pressure (atm) Ar Kr Xe 020406080100120 40 50 60 70 80 Frequency (GHz)Fluid pressure (atm) Ar Kr Xe 020406080100120 10 15 20 25 30 35 40 Frequency (GHz)Fluid pressure (atm) Ar Kr Xe 10 nm CNT 20 nm CNT 30 nm CNT A 10 nm CNT 20 nm CNT 30 nm CNT B
83 0.0 1.0x10-82.0x10-83.0x10-84.0x10-8 0.0 2.0x10104.0x10106.0x10108.0x1010 CNT length (nm) CM prediction MD simulationsVibrational frequency (Hz) 0.0 1.0x10-82.0x10-83.0x10-84.0x10-8 0 1x10112x10113x10114x1011 Vibrational frequency (Hz)CNT length (nm) CM prediction MD simulations Figure 5-5. Frequency vs. CNT le ngth of A) bridged CNT and B) cantilevered CNT in vacuum. MD simulational results () and continuum level rod vibration equation (Â–) are compared. A B
84 01020304050 0 100 200 300 400 500 Frequency (GHz)CNT length (nm) vacuum 10 atm 50 atm 100 atm vacuum 10 atm 50 atm 100 atm Figure 5-6. Comparison of the frequencies of bridged 10, 20, and 30 nm long SWNTs in Kr fluid calculated by MD simulations and pr edicted by the equation of longitudinal vibrations
85 CHAPTER 6 TRIBOLOGY OF CARBON NANOTUBES Previous studies30, 31, 79, 81, 124 have also investigated the mechanical properties of filled SWNTs. They indicated that filling makes CNTs more resistant to deformation under external forces, such as compression, tension, torsion, or bending. It is therefor e reasonable to assume that filling CNTs will have an effect on the tribological properties of CNT bundles, but this effect has not been examined systemically. Here, classical MD simulations are used to investigate the effect of filling CNTs and combining them with additive lubricants on th eir mechanical responses. In particular, the responses of CNT bundles subjected to compre ssive and sliding forces between two hydrogenterminated diamond-like carbon (DLC) surfaces is investigated. The molecule used in the simulations as a model lubricant is benzene. Computational Details Three different types of CNT bundles are c onsidered, including those made up of empty SWNTs, nanopeapods, or double-walled nanotubes (DWNTs). The SWNTs are 5.5 nm long, (10,10) armchair type CNTs with end caps that are hemispheres of C60 fullerenes. The nanopeapods and DWNTs are pr epared by introducing C60 or smaller (5,5) SWNTs into the same outer, (10,10) SWNTs, respectively. The bundles c onsist of six or twelve CNTs that are placed horizontally between two hydrogen terminated DLC films on diamond surfaces. In some simulations, benzene (C6H6) molecules are applied as a m odel lubricant fluid. Figure 6-1 illustrates the systems with and without the benzene. Benzene was chosen because it has stacking interactions with th e hexagons in the CNT walls125, 126 and is alike to coat the CNTs without modifying their geometric structure. It is therefore a g ood candidate to investigate the
86 influence of strongly interacting molecular lubricants on the tri bological properties of graphitic materials. The DLC film is generated by heating a diamond film to 8000 K followed by quenching at a rate of 1 1015 K/sec. Each diamond substrate consis ts of six layers of carbon. Periodic boundary conditions are applied parallel to the su rface plane directions to mimic an infinite surface. The DLC/diamond substrat es have dimensions of 5.245 nm6.562 nm in the plane of the surface and contain 12480 atoms each. Before applying any external compressive and sliding forces the system is fully equilibrated using Langevi n thermostats at 300 K. The entire sy stem is then divided into three regions: an active region where atoms evolve according to only Newtonian mechanics, a thermostat region in which atoms are subj ected to Newtonian mechanics and Langevin thermostats, and a rigid region where the atoms ar e not allowed to evolve with time, but may be explicitly moved. Next, the CNT bundles ar e compressed by moving the uppermost rigid hydrogen and carbon atoms toward the lower subs trate. The upper substrate moves by 0.00008 nm every time step, which corresponds to an overall compressive rate of 40 m/s. During compression, the normal forces are measured on the uppermost rigid atom in the upper substrate and averaged over the last 100 of every 625 time steps. Then, the data is smoothed by a 100point adjacent-averaging method. Table I prov ides the pressure on each two-layered CNT bundle system at different compressive states determin ed by the separation distance between the top and bottom DLC films. Following compression, the systems are re-equ ilibrated to eliminate the effect of compression on the system prior to sliding. The systems are then sheared by moving the uppermost rigid hydrogen and carbon atoms parallel to the substrate plane. During sliding both
87 the normal and the lateral forces are recorded. In particular, th e DLC covered diamond substrates move in the negative direction so the measured forces are positive when there are repulsive interactions between the substr ate and the nanotube bundles, and negative when the opposite is true. The friction coefficients of the nanotubes bundles are th en calculated from AmontonsColoumb law127 expressed as N FF F (6-44) where is the frictional coefficient, FF is the frictional or lateral force, and FN is the normal or compressive force. Results and Discussion Figure 6-2 shows the calculated compressive forces versus displacement of the topmost diamond substrate that is moved towards each bundle. In the case of the SWNT bundle, the compressive force increases gradually, drops slig htly, and then increases again. This trend is slightly different from the result of Ni and Sinnott46 since the CNTs here have finite lengths and end caps, while the horizontal t ubes in Ref. 1 where infinitely long by way of periodic boundary conditions. Therefore, the CNTs used here have mo re freedom to rotate a nd slip over each other. During compression the distance between the na notube walls decreases and the cylindrical cross-sectional shape of the na notubes changes from circular to polygonal until the compression displacement has reached 1.0 nm. This is due to the flexibility and low modulus of the empty nanotubes in the direction nor mal to the nanotubes axis.6, 128 At this point, the SWNTs slip over each other in the axial direction, which result s in flattening of the SWNTs. This causes the compressive force to drop slightly, before in creasing again as the shearing progresses. In the case of the nanopeapod bundle, the comp ressive force changes in a trend that is similar to what occurs with the SWNT bundle following initial compression. After 0.5 nm of
88 displacement, the cylindrical cros s-sectional shape of the nanot ubes changes from circular to polygonal and the normal force on the peapod begins to increase more quickly than in the case of the empty SWNT. It is the van der Waals repulsive forces betw een the nanotube walls and the fullerenes that cause the normal fo rce of the peapod to be higher than in the case of the empty SWNTs. It is also predicted that the normal for ce of the DWNT bundle is higher than that of the nanopeapod bundle, and the gap increas es after 1.0 nm of compressive displacement. This can be explained by the different structures of the CNT filling materials, and the fact that the DWNT is, on the whole, more rigid than either the nanopeapods or the empty SWNTs. Figure 6-3 shows snapshots of the comp ressed empty SWNT, nanopeapod, and DWNT bundle structures at a compressive displacement of 1.5 nm. The hollow SWNT is flexible in the direction normal to the nanotube axis22 and there is no filling materi al to provide resistance to external forces. Therefore, all regions but th e end cap areas are flattened and form narrow elliptical shapes in cross-section. This is well matched with the deformation of the horizontally aligned SWNT bundle inves tigated by Ni and Sinnott.46 In the case of the nanopeapod, the C60 fullerenes move in the nanotubes in response to th e external compressive forces. Therefore, the regions with fullerenes form teardrop shapes while maintaining the fullereneÂ’s spherical shapes; empty regions are more severely deformed. Co mpared to the peapod bundle, the inner nanotubes in the DWNTs do not have enough space to move out of the way of the compressing outer tube, which results in DWNTs that are deformed in a more uniform manner than the nanopeapods. This is because the inner nanotubes are flattene d instead of maintaining their cross-sectional spherical shapes as the fulle renes do. The nanopeapods and DWNTs have more carbon-carbon covalent interactions due to the filling materials and van der Waals interactions between the outer nanotubes and the inner fulle renes or smaller nanotubes. Ther efore, the filled CNTs have
89 higher overall external compressi ve forces than the empty SWNTs at the same compressive distance. The sliding of various two-la yered CNT bundle systems is carri ed out at three different compression pressures, low, medium, and high, as summarized in Table 6-1. During sliding, the normal and lateral forces are recorded every 0. 05 nm. The data is th en smoothed using the 100point adjacent-averaging method. Figure 6-4A illustrates th e normal and lateral forces for the smoothed data in the case of the nanopeapod bundle in vacuum as an example. The data indicates that both normal and late ral forces are constant during sliding and increase as the compressive pressure between the sliding surface s increases. The normal force is positive as a result of repulsive interactions caused by compressi on. The lateral force is also positive, meaning there are repulsive interactions between the surf aces and the nanotubes during sliding. The force values in Figure 6-4B are obtaine d by taking the arithmetic averag e of the smoothed force values during the 20 nm sliding. It is obvious that the lateral forces increase as the normal forces increases, irrespective of filling or the addition of benzene as a molecular lubricant. It is also predicted that the filled CNT bundl es exhibit higher normal and la teral forces than the empty SWNT bundles, and that the CNT bundles with molecular benzene lubricants have higher normal and lateral forces than the same systems in the ab sence of benzene at the same compression state. The friction coefficients can be obtained by taking the slopes of the lateral forces as a function of the normal forces. In the lower pr essure region of less than 5 GPa, which is equivalent to a normal force of 200 nN in Figure 6-4B, the lateral force lin early increases as the normal force increases whether the CNT is filled or not. In this region, the friction coefficient of the SWNT bundle is around 0.13 a nd this value is very similar to those of the peapod and
90 DWNT bundles. Thus, at pressures that are most likely to occur in actual applications, there is little difference in CNT frictional behavior regardless of whether they are filled or not. However, when the systems are in the highest state of compression considered in these simulations, the pressure of the filled systems is significantly higher than the pressure of the hollow systems, as indicated in Table 6-1. When these higher pressure data points are taken into account, the friction coefficients of the filled CN Ts increase to up to 0.20 due to the rapid increase of the lateral force, while the friction coefficients of the empty SWNTs remain essentially unchanged at about 0.13 (see Figure 6-4B ). This is because, as shown in Figure 6-3, the filled CNTs maintain spherical cross-sectio nal shapes under compression, which causes them to be stiffer than the empty CNTs. In addition, the contact area between these stiff CNT walls and the sliding surfaces increases as the norma l force goes up. In contrast, the empty SWNTs flatten out at high pressures and offer less resist ance to compression and sliding. Therefore, the friction coefficient of the CNT bundles change more drastically from small values at low pressures to large value at high pressure s for filled CNTs than for empty CNTs. During sliding, the repulsive interaction betw een the sliding substr ate and the nanotubes causes the nanotubes to roll, slide, or rotate. Previous studies49, 50 showed that in the case of a graphite substrate, the nanotubesÂ’ motion was determ ined by their registry with the substrate. In these simulations, however, the nanotubes are in b undles and are placed between DLC films that do not have any long-range order. Therefore, the motion of the CNTs is independent of registry with the sliding surfaces. Figure 6-5A shows snapshots of the comp ressed, hollow SWNT bundle at its initial position and after sliding 5 nm at low compression; different colors are used to distinguish each nanotube in the bundle and indicator atoms are used to verify the type of motion exhibited by the
91 nanotubes. Figure 6-5B illustrates how the dist ances the nanotubes move and their rotational angles change as a function of sliding distance. It clarifies that the nanotubes slide the same amount in the direction of motion of the upper su bstrate. However, not all the nanotubes roll at the same time or to the same degree. The rotation angle is always positive, which means that the rotational direction of every nanot ube is the same as the sliding direction of the substrate. All bundles considered here exhibit the same type of cohesive sliding that is accompanied by independent rolling motions regard less of the filling of the CNTs. While the friction coefficient calculated from lateral and normal forces varies with the degree of comp ression, the motion of the nanotubes for the two-layered system is invari able regardless of the co mpression state. In the case of medium and high compression, the nanotube s move together along the sliding direction and roll independently of each other. A four-layered SWNT bundle system is cons idered to determine the effect of layer thickness on the tribological pr operties of nanotube bundles. At th e lowest compression state of about 0.6 GPa, the friction coefficient is calculat ed to be about 0.04, which is similar to the friction coefficient of the two-layered SWNT bun dle system at low compression. Each nanotube maintains its cylindrical shape th roughout sliding, and a ligns well with the ot her nanotubes in the bundle. The whole carbon nanotube bundle exhibits aligned slidi ng with some independent, nonaligned rotational motion, as il lustrated in Figure 6-6. Theref ore, the whole nanotube bundle moves as one continuous body and the main contribu tion to the friction occu rs at the interfaces between the bundle and the substrates. This behavior is similar to the motion of the two-layered SWNT bundle, indicating that at th is compressive pressure the tr ibological behavior of aligned horizontal nanotube bundles are rela tively independent of thickness.
92 At the higher compression state of 1.0 GPa, the nanotubes glide along the nanotube axis direction and misalign with one another. Therefore the nanotubes are more flattened by the end caps of other nanotubes than they would be othe rwise. Instead of exhi biting aligned sliding motion, each nanotube moves a different slid ing distance depending on its distance from the moving upper substrate, as indicat ed in Figure 6-6. The nanotube s in the topmost layer slide around 15 nm while those in the bottommost layer slid e less than 5 nm. This is in contrast with the aligned sliding motion that occurs at low co mpression, where the sliding distance of all the nanotubes in the bundle is about 7 nm. As a re sult, the main contribution to friction is interactions among nanotubes in the bundle, rather than betw een the bundle and the sliding substrates. In addition, the contact area of th e nanotubes increases with the number of nanotube layers and the degree of compression. These fact ors lead to higher lateral forces at higher compression states for the four-layered na notube bundle system. Ther efore, at higher compression states, the thicker nanotube bundle experi ences higher lateral for ces which result in higher friction coefficients. The effect of molecular fluids on CNT slid ing is also evaluate d at three different compressive pressures, low, medium, and high, which are summarized in Table 6-1. Figure 6-4B indicates that introducing benzene molecules does not change the friction coefficients of the SWNT and peapod bundles. During sliding, each nanot ube rolls independently of the others in a manner that is similar to the rolling of nanotube s in the absence of benzene. Introducing the benzene, however, makes the nanotubes move diffe rently. In particular, na notubes that are closer to the sliding surface readily glide over the lower nanotubes. Consequently, the upper nanotubes slide more rapidly than the lower on es, as indicated in Figure 6-7A.
93 Most of the benzene molecules ar e located between th e nanotube walls and are parallel to the nanotube surfaces, as illustrated in Figure 6-7B These molecules can readily slide over each other during shearing, which leads to lower late ral forces for the CNT/benzene system. As a result of these molecules, however, the upper CN T layer is misaligned re lative to the lower CNT layer and generates higher latera l forces as it slides over the lower layer. Consequently, the presence of the benzene molecules results in similar lateral forces as in the case of the two-layer SWNT bundle system without the molecules. This is the main reason that the friction coefficient does not change much on the introduction of ben zene molecules. This result is different from experimental data that indicate that the intr oduction of benzene molecules on graphite surfaces decreases the friction coefficient of graphite.129, 130 This indicates that the influence of benzene is fundamentally different graphitic sy stems depending on their curvature. Conclusions The simulations considered here predic t that filled CNT bundles can support higher compressive forces than hollow CNT bundles. At low pressures, the fill ed CNT bundles exhibit similar friction coefficients to hollow CNT bun dles. However, the fille d CNT bundles exhibit higher friction coefficients at high pressures th an do empty CNTs because they are stiffer and retain their spherical shape to a greater extent under compression. The thickness of the nanotube layer does not influence the predic ted friction coefficients at low compressive pressures, but does affect them at higher compressive pressures. In addition, the simulations illustrate how the introduction of additive molecules affects the friction behavior of CNT bundles.
94 Table 6-1. Pressures (in GPa) of various two layered CNT bundle systems at different compressive states. The displacements of the upper surface rela tive to the lower surface needed to achieve these pressures are also given. Compression state (Displacement of upper surface relative to lower surface) Low (0.5 nm) Medium (1.0 nm) High (1.5 nm) SWNT 0.89 1.22 2.81 Nanopeapod 1.53 3.32 8.58 DWNT 1.92 4.40 10.26 SWNT/benzene 1.36 1.98 5.03 Nanopeapod/benzene 1.50 3.86 10.29
95 Figure 6-1. Snapshots of initial structures for CN T tribology. A) four layered CNTs and B) two layered CNTs with benzene molecules between diamond-like-carbon-diamond mixed substrates. Outermost carbon and hydrogen at oms at the both substrate are fixed and the next carbon atoms are the th ermostat atoms. The other at oms in substrates as well as all atoms in CNTs and benzene fluid are active atoms A B
96 0.00.51.01.5 0 100 200 300 400 500 600 DWNT Peapod Force (nN)Displacements (nm) SWNT Figure 6-2. Compressive forces versus the di splacement of the topmost diamond surface moving towards various nanotubes bundles.
97 Figure 6-3. Snapshots of cro ss-sectional view of compressed CNT bundles (left) and the side view (right) of A) SWNT, B) peapod, and C) DWNT in the bundle structures at high compression. The DLC substrates are omitted for the clarification. A B C
98 0100200300400 0 20 40 60 80 Lateral force (nN)Normal force (nN) SWNT Peapod DWNT SWNT in Benzene Peapod in Benzene Figure 6-4. Force results of various 2-layered CNT bundle sy stems: A) The normal and the lateral forces vs. the shearing distances of the peapod bundle system at low (left), medium (middle) and high (right) compre ssive states. B) The normal forces and lateral forces averaged from the plots in A). 05101520 0 100 200 300 400 Force (nN)Distance (nm) 05101520 0 100 200 300 400 Distance (nm) 05101520 0 100 200 300 400 Distance (nm) Lateral force Normal force 100 point smoothing curve of lateral force 100 point smoothing curve of normal force A B
99 Figure 6-5. Shearing of 2-la yered CNT bundle: A) Snapshots of low compressed SWNT bundle between DLC substrates at th e initial shearing position (lef t) and after shearing 5 nm (right). Some carbon atoms are shown in ball model for the indication of sliding and rolling. B) Plots of the moving distance (t op) and the rotation angle (bottom) of each SWNT vs. the shearing distances at th e low compressed SWNT system. CNT 1 (), 2 () and 3 () are the upper CNTs and 4 (), 5 () and 6 () are the lower CNTs. 05101520 0 5 10 15 20 Moving distance (nm) CNT 1 CNT 2 CNT 3 CNT 4 CNT 5 CNT 6 SubstrateShearing distance (nm)05101520 0 100 200 300 400 500 Rotation angle ()Shearing distance (nm) CNT 1 CNT 2 CNT 3 CNT 4 CNT 5 CNT 6 A B
100 Figure 6-6. Shearing of 4-laye res hollow CNT bundle: A) Plots of the moving distance vs. the shearing distance of the low compressed (l eft) and the medium compressed (right) four layered CNT bundle between DLC substr ates. Each mark represents each CNT and they are in different colors and shap es: red circles (top la yer), green triangles (2nd), blue squares (3rd), and orange diamonds (botto m). B) The snapshots of low compressed 4 layered SWNT bundle between DL C substrates at the initial shearing position (left) and after shear ing 20 nm (right). Some carbon atoms are shown in ball model for the indication of sliding and rolling. 05101520 0 5 10 15 20 Moving distance (nm)Shearing distance (nm) 1st layer 2nd layer 3rd layer 4th layer Substrate05101520 0 5 10 15 20 Moving distance (nm)Shearing distance (nm) 1st layer 2nd layer 3rd layer 4th layer Substrate A B
101 05101520 0 5 10 15 20 CNT 1 CNT 2 CNT 3 CNT 4 CNT 5 CNT 6 Substrate Shearing distance (nm)Moving distance (nm) Figure 6-7. Shearing of hollow CNT bundle in benzene molecules: A) Plots of the moving distance of each empty SWNT vs. sliding distances under medium compression. CNT 1 (), 2 () and 3 () are upper CNTs and 4 (), 5 () and 6 () are lower CNTs in the bundle. B)Snapshot of two-laye red nanotube bundle system surrounded by benzene. The figure illustrates the way in which the benzene molecules surround the SWNTs. A B
102 CHAPTER 7 TRIBOLOGY OF POLYETHILENE Here, we report on the results of classical molecular dynamics (MD) simulations of the sliding of two crystalline PE surfaces against one another in two different orientations. The goal is to gain insight into the mechanisms behind PE polymer friction and the way in which it is affected by the relative sliding or ientations of the surfaces. We al so compare the results to those of comparable PTFE surfaces to determine the influence of polymer type on the results. Computational Details The unit cell of the crystalline PE system cons idered in the simulations is shown in Figure 1-4. There are 17 ethene (C2H4) monomers in each polymer chain and 12 chains in each surface layer. Both the upper and lower sliding substrates consist of seven layers of PE chains. Figure 71 illustrates the PE system employed in the simulations. Periodic boundary conditions94, 95 are applied within the planes of the surfaces to simu late infinite PE surfaces. The dimension of each substrate is 4.4 4.4 nm and the thickness is about 3.4 nm Each chain is connected to its four nearest chains with two cross-link units such that the final cross-link density is 3.58/nm3 1 depending on the way in which it is calculated. Th ere are 20,880 atoms in the system as a whole. As a result of its density and crystal structure, this PE system can be expected to have properties that are similar to those of ultrah igh molecular weight PE (UHMWPE) which is known to be crystalline, have lo w friction coefficients, and be re sistant to wear. The outmost 0.5 nm thick regions of both the upper and lower surfaces are held rigid and not allowed to evolve in time over the course of the MD simulations. Mov ing towards the interface between the upper and lower surfaces, the next 0.5 nm thick regions co ntain atoms whose temper ature is regulated by a 1 Number of cross-links per unit volume
103 Langevin thermostat,40 which can maintain the system temperature at a constant value throughout the simulations and mimic the heat dissipation behavior of the much larger experimental systems. The rest of the atoms in the system are allowed to evolve according to NewtonÂ’s equations of motion withou t any additional constraints. The surfaces of the two layers are initially 1 nm apart. The entire system is fully equilibrated to 300 K and then compressed by moving the rigid atoms of the upper surface toward the lower surface by 2.010-6 nm every time step of 0.2 fs, which is equivalent to an overall compressive rate of 10 m/s. After 825,000 time steps of compression, corresponding to a displacement of 1.65 nm, a pre ssure of around 250 MPa is achiev ed. Following compression, the friction study is performed by sliding the upper su rface against the lower su rface at a sliding rate of 10 m/s while the lower surf ace remains in place. Two differ ent sliding directions are considered here; perpendicular sliding for which the sliding direction is perpendicular to the aligned direction of the polymer chains (direction a in Figure 7-1) and parallel sliding for which the sliding direction is parallel to the aligned direction of the polymer chains (direction b in Figure 7-1). The normal and lateral forces during s liding are calculated at the rigid atoms at the top of the upper surface. The upper surface always m oves in a negative late ral direction, so the forces are positive when there are repulsive interactions between the surfaces. The friction coefficient of the PE system is calculated using the classical Am ontons friction equation127 N FF F (7-45) where is the friction coefficient, FL is the lateral force, and FN is the normal or compressive force. FL and FN are obtained directly from the MD simulations.
104 Results and Discussion Figure 7-2 presents the lateral forces, norma l forces, and friction coefficients of the perpendicular and lateral configurations as func tions of the sliding di stance of the upper PE surface. The lateral and normal forces change rapidly during the in itial stages of both perpendicular and parallel slidi ng because the polymer chains go through an initial relaxation process. For the case of the perpendicular orient ation this involves the chains initially bunching together and elastically straining in response to the applied shearing fo rces, but not actually sliding. For the case of the parallel orientation the initial rela xation involves the chains moving from their initial positions to more stable, relaxed positions. After about 2.5-5.0 nm of sliding, these forces attain steady state and are relatively constant as the top and bottom surfaces sliding against each other in both configurations. All of the tribological prope rties of polymers are analyzed after this initial relaxation. The simulations of the two different orientations start from the same in itial structure with a normal force of 5 nN. However, the perpendicular and the parallel cases exhibit opposite trends in initial normal force evolution, as indicated in Figure 7-2. In partic ular, after the initial relaxation period, the normal for ce during perpendicular sliding has a median value of about 7.5 nN, which is about 50% higher than the initial normal force. In contrast, during parallel sliding the normal force following the initial relaxation period has a median value of about 2.5 nN, which is about half the initial normal force. This difference can be explained by the change in the interfacial structure that occurs during relaxation. In the case of perpendicular slid ing, the polymer chains at the interfaces slide over each other. As a result, the substrates ar e compressed when the chains of the upper surface reach the peaks of those of the chains of the lower surface, causing the normal forces to increase. When the chains of the upper su rface move into the depressions between the chains of the lower
105 surface, they are not able to fully optimize their position because of the sliding. Thus, overall the normal forces increase slightly dur ing perpendicular slidi ng. In contrast, in the case of parallel sliding the interfacial chains of the upper surface move into and then remain in the depressions between the chains. Therefore the nor mal forces of the parallel slid ing remain at a constant value that is lower than the initial normal forces following relaxation. This behavior can be verified by examining the changes in the vertical, y, positions of the surface chains at the interfaces, which are summarized in Figure 7-3. During compression, the surface chains in the upper substrate are aligne d to and overlapped with those in the lower substrate, which results in the initial overl ap as shown in Figure 7-3A. There are fewer overlapping atoms after the initial relaxation peri od in the perpendicular configuration in Figure 7-3B, while the opposite is true fo r the parallel conf iguration in Figure 7-3C In other words, the surfaces are more compressed during perpendicular sliding following relaxation, but are slightly expanded during parallel slid ing following relaxation. Following relaxation, the lateral force in th e case of perpendicula r sliding exhibits a repeating, oscillating be havior that is consistent with stic k-slip motion that occurs when the sliding surfaces initially remain pinned to each other until some critical force is reached. At this point the surfaces begin to slide past each othe r and the pattern of pinning and sliding repeats continuously.131 During sliding in the parallel direction, th e lateral force also exhibits stick-slip behavior, although the magnitude of the pinning force is much lower than in the case of perpendicular sliding. This can be explained by the different structural configuration along the sliding direction. During perpendicular sliding, e ach surface chain in the lo wer substrate acts as a barrier for the movement of surface chains in the upper substrate due to the corrugation
106 configuration. In contrast, in the case of parallel sliding there is st ructural constraint not from the chain alignment but from the atomic configurati on over which the surface chains must slide. Figure 7-2 summarizes the fric tion coefficients for both or ientations. In the case of perpendicular sliding, the fric tion coefficient has an average value of about 0.694 0.165 and exhibits stick-slip motion. Duri ng parallel sliding, stick-slip motion is also observed but the average value is about 0.061 0.124, or an orde r of magnitude lower th an in the case of perpendicular sliding. Snapshots from the MD simulations during slidin g are shown in Figure 7-4. They represent the initial structures (A is the schematic view along chain direction for the perpendicular sliding and AÂ’ is that normal to chain direction for the pa rallel sliding), the structure at the maxima of the lateral forces (B and BÂ’), and the minima of the lateral forces (C and CÂ’), respectively. The upper and lower surfaces are presented in different colors so that the interface can be more clearly distinguished. Figure 7-4 i llustrates how pinning dist orts the chains to a significant extent in the case of perpendicular sliding, and to a much smaller extent in the case of parallel sliding. This degree of pinning is related to the friction coefficients discusse d above. It is also noting that at the maximum force, the polymer substrate is more deformed than at the minimum of force. This is a result of the elastic strain built up during the pinning (stick), and is released during the sliding (slip). Both the perpendicular and parallel configur ations are also evaluated with regard to shearing deflection, (the distan ce by which the polymer substrate is deformed during shearing), and the shearing strain energy, as illustrated in Figure 7-5. The PE chains experience more shear deflection due to pinning during perpendicular sliding than during para llel sliding, which corresponds to higher shear strain energy. This creat es a greater lateral for ce in the perpendicular
107 sliding than in the parallel sliding, resulti ng in the higher friction coefficient for the perpendicular sliding. It is worthwhile to assess how the molecula r structure of the polymer influences these responses. We therefore now compare these PE results with results for PTFE reported by us previously132. It should be noted that the structures of the PE and PTFE are very similar to each other and have identical surface ar eas, numbers of layers, and cross-link densities. However, due to different unit cell sizes, the number of mers per chain and the number of chains per layer are slightly different (17 monomers in each PE ch ain versus 18 monomers in each PTFE chain, and 12 chains per layer in each PE surface versus 10 ch ains per layer in each PTFE surface). In both cases, the same pressure of about 250 MPa is applied. The trends in the normal and lateral forces w ith sliding distance are similar for PTFE and PE (see Figure 7-6). The major difference between these two polym ers is that the PTFE system does not show any stick-slip motion during either perpendicular or parallel sliding.132 According to the Tomlinson model,131 stick-slip motion can be observed only if the system is not too firm or the interaction between the sliding surfaces is st rong enough. Therefore, the different motions of PE and PTFE during sliding can be explained by differences in th e bulk structures of PE and PTFE or in the interactions between the PE-PE and PTFE-PTFE interfaces. Sliding in the PE and the PTFE systems produce different amounts of bond formation and breaking, as summarized in Figure 7-7. In the case of PE, a small number of bonds are formed and broken during both the perpe ndicular and the parallel slidi ng events. For PTFE, however, tens of bonds are broken and formed during slid ing; the PTFE perpendi cular sliding event in particular leads to around 70 bonds being broken. Th is may be explained in terms of the electron affinity of fluorine, which at 328 kJ/mol is mu ch higher that that of hydrogen, which is 72.8
108 kJ/mol. This leads to weaker carbon-carbon bonds in each PTFE chain than that in each PE chains. This is consistent with our MD simulati ons using the REBO potential, which predict that the PE chain has a tensile strength 90% higher th an that of the PTFE chain. Therefore, the different strengths of the PE and the PTFE chains, particularly those of both cross-links, result in different overall strengths for these systems. As a result of these differences the PE syst em is stiff enough to exhibit stick-slip motion during sliding and exhibits litt le bond breaking or deformation regardless of the orientation across the sliding interface. In contrast, the PTFE system is too pliable for stick-slip motion; instead, the chains break and/or realign in the direction of sliding to relieve the high forces associated with sliding.132 In the case of perpendicular slidi ng the average friction coefficient of the PE is 0.73, which is comparable to but slightly lower than that of PTFE (0.60). In the case of parallel sliding, the friction co efficient of the PE (0.063) is significantly lower than the PTFE value of 0.35. The significant difference in the latte r case is directly related to the differences in stiffness discussed above. Conclusions The effect of the sliding orientation on the tribological properties of the PE system was analyzed using classical molecular dynamics simulations. The friction coefficient of the perpendicular sliding is higher that that of the parallel sliding due to the higher lateral forces experienced during perpendicular sliding. The PE system exhibits stick-slip motion at the interface during both perpendicular sliding and parallel sliding. This is due to the inherent strength of the PE chains and the cross-links within the PE systems. Comparison of these results to comparable results for PTFE illustrates the way in which the chemical characteristics of the polymer influence the results.
109 Figure 7-1. The schematic diagram of the PE polymer chain alignment. a b
110 Figure 7-2. The lateral forces, the normal fo rces, and the frictional coefficients of the perpendicular and the parallel slid ings of the PE polymer systems Perpendicular Parallel 05101520 0 5 10 15 Forces Normal LateralNormal & lateral forces (nN)Sliding distance (nm) 05101520 0 5 10 15 Forces Normal LateralNormal & lateral forces (nN)Sliding distance (nm)05101520 0.0 0.5 1.0 1.5 Frictional coefficientSliding distance (nm)05101520 0.0 0.5 1.0 1.5 Frictional coefficientSliding distance (nm)
111 Figure 7-3. Histogram of atomic positions for the polymer chains at the sliding interface A) at the initial state, B) after th e perpendicular sliding, and C) after the parallel sliding -14-12-10-8-6-4-2 0 20 40 60 80 100 120 # of atomsY position Upper Lower -14-12-10-8-6-4-2 0 20 40 60 80 100 120 # of atomsY position Upper Lower -14-12-10-8-6-4-2 0 20 40 60 80 100 120 # of atomsY position Upper Lower A C B
112 Figure 7-4. The snapshots of PE system in A) the perpendicular and B) the parallel slidings as shown in Figure 7-2 To better s ee the change of the PE subs trate, three stripe regions are designated not by specific chains but by distance and are set as opaque while the other regions are set as transparent. A B C AÂ’ BÂ’CÂ’ A B
113 Figure 7-5. Comparison of A) the shear deflection of the uppe r PE substrates and B) the resulting shear strain energies during both the perpendicular and the parallel slidings 0510152025 10-1100101102103 Shear strain energy (eV)Shear distance (nm) Perpendicular Parallel 0510152025 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Shear deflection (nm)Shear distance (nm) Perpendicular Parallel A AÂ’ CÂ’ C BÂ’ B CÂ’ C BÂ’ B A AÂ’ A B
114 Figure 7-6. Comparisons of the lateral and the normal forces, and the frictional coefficients during the perpendicular and parallel slidin gs between the PE and PTFE systems. 05101520 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Frictional CoefficientDistance (nm) PE perpendicular PTFE perpendicular 05101520 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Frictional CoefficientDistance (nm) PE parallel PTFE parallel 05101520 -10 -5 0 5 10 15 20 Normal force (nN)Distance (nm) PE PTFE 05101520 -10 -5 0 5 10 15 20 Normal force (nN)Distance (nm) PE PTFE 05101520 -5 0 5 10 15 Lateral force (nN)Distance (nm) PE PTFE 05101520 -2 -1 0 1 2 3 4 Lateral force (nN)Distance (nm) PE PTFE Perpendicular Parallel
115 Figure 7-7. Histogram of the number of the bonds formed and broken during perpendicular and the parallel sliding of the A) PE and B) PTFE systems PerpendicularParallel -10 0 10 20 30 40 50 60 70 80 Shearing mode# of bonds Forming Breaking Breaking Forming PerpendicularParallel -10 0 10 20 30 40 50 60 70 80 # of bondsShearing mode A B
116 CHAPTER 8 GENERAL CONCLUSIONS In the work reported here, the fundamental m echanical properties of nanoscale materials are investigated using classical MD simulations. New insights are obtained, including of the role on thermostat atoms on the predicted mechanical properties of CNTs, of the compressive and bending properties of CNTs, of CN T vibrational behavior in respons e to external fluids, of the frictional properties of CNT bundle when the nanotubes are filled or when the bundles are surrounded by external molecules, and of the effect of sliding orientation on PE friction. In the MD simulations, the mech anical properties of the CNTs can be predicted differently depending on the relative amounts of Langevin th ermostat atoms. Nonphysical buckling occurs during deformation with a large amount of Langevi n thermostats and at higher deformation rates. In contrast, the Nos-Hoover and velocity rescal ing thermostats result in physical mechanical deformation modes regardless of the percentage of thermostat atoms. The Langevin and velocity rescaling thermostats are successful at maintain ing a constant system temperature during the compression of CNTs. However, the Nos-Hoover is not able to do so and this is therefore not considered to be suitable for use in simulations of CNT deformation. In short, the importance of the choice of thermostat and the fraction of th ermostat atoms used in simulations of CNT deformation is demonstrated. Filling SWNTs increases their maximum defl ection and buckling fo rces during bending and compression. For example, MWNTs are predic ted to support higher external loads than SWNTs and peapods because the in ner CNTs sustain more external forces than the fullerenes. This mechanical behavior of hollow or filled CNTs can deteriorate at high temperature. The chemical modification of the CNT wall through the generation of vacancies or the attachment of functional groups is also pred icted to weaken the mechan ical responses of CNTs.
117 The vibrational behavior of bri dged and cantilevered CNTs is investigated in the presence of external Ar, Kr, and Xe fluids The frequency depends on the length of CNT as well as the pressure and particle mass of fluid. The simulati on results in vacuum agree with the predictions of continuum mechanics. The CNT tribology simulations predict th at filled CNT bundles can support higher compressive forces than hollow CNT bundles. At the same pressures, the filled CNT bundles exhibit lower frictional coefficients than ho llow CNT bundles. The th ickness of the nanotube layer does not influence the predic ted friction coefficients at low compressive pressures, but does affect them at higher compressive pressures. In addition, the simulations illustrate how the introduction of additive lubricants can redu ce the frictional coefficient of CNT bundles. The effect of the sliding orientation on the tr ibological properties of the polyethylene (PE) system is analyzed. The frictional coefficient for sliding chains aligned in a perpendicular manner is higher than that of s liding chains aligned parallel to one another due to the higher lateral forces experienced during perpendicular slid ing. The PE system exhibits stick-slip motion at the interface during both perpendicular and paralle l sliding. This is due to the inherent strength of the polymer chains and the cros s-links within the PE systems. Also, the sliding results for the PE systems show no change in surface morphology, whereas comparable pol ytetrafluoroethylene (PTFE) systems show bond breaking and rea ligning of polymer chains along the sliding direction. In short, MD simulations can establish succe ssfully fundamental mechanical behavior of nanoscale materials, especially bending and compressing behavi or and vibrational motion of CNTs as well as tribological properties of CNT bundle and PE system. These findings can be used to assist the application of nanoscale materials and devices.
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128 BIOGRAPHICAL SKETCH Seongjun Heo was born in Seoul, Korea, in 1973. He received his B.S. and M.D. in Department of Materials Engineering fr om Hanyang University, Korea in 1997 and 1999 respectively. As a research engineer, he had been working at the se miconductor division of Samsung Electronics from 1999 to 2003. He started his study for Ph.D degree in the Department of Materials Science and Engineer ing at the University of Flor ida in 2003. Since then he has been working in the Computa tional Materials Science and E ngineering Focus Group (CMSFG) with Dr. Susan B. Sinnott. He has been intere sted in the computational simulation for the mechanical and electronic propert ies of carbon nanotubes as well as the tribological properties of materials using molecular dynamics (MD) simula tions and density functional theory (DFT) calculations. Since 2006, he has been a member of the American Vacuum Society and he received an Honorable Mention in Materials Science in the 2006 Joint Symposium of the Florida Chapter of the American Vacuum Societ y and the Florida Society for Microscopy.