Index

Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively.

A

Antibonding molecular orbitals, 56, 66
Apex angle, 371–372, 377
on mechanical behaviors of carbon nanocones (CNCs), 436–438
Arc-discharge evaporation method, 1, 2, 2
Armchair CNTs, 4, 122, 235
critical diameters versus tube lengths for, 240f
fundamental frequencies of, 240f
Armchair nanorings, 412–413, 414f, 417f
Armchair single-walled carbon nanotubes, 4f, 49, 56–57, 60, 71, 74, 79t, 216
bifurcation strain for, 216f
critical diameters versus tube lengths for, 240f
Atomic finite element method, 249–250
coupling with atomistic-continuum method, 251–254
bridging domain method, 252
bridging scale method, 252–254
quasicontinuum method, 251–252
Atomic structure of CNTs, 2, 3–5
Atomistic simulation, 12–13, 141, 206, 206, 228, 235–239, 411
Atomistic-continuum method, 15, 15–16, 228, 411
coupling of atomic finite element method with, 251–254
bridging domain method, 252
bridging scale method, 252–254
quasicontinuum method, 251–252
Atomistic-continuum theory, 141, 146–150
Bernoulli–Euler beam model, 216–221
buckling and postbuckling behaviors, 186–207
axial buckling and postbuckling behaviors of SWCNTs, 206–207
axial buckling of SWCNTs, 190–193
buckling behavior of SWCNTs upon bending, 198–206
hydrostatic pressure-induced structural transitions of SWCNTs, 186–190
torsional buckling of SWCNTs, 193–198
Cauchy–Born rule, 145–146
fracture nucleation, 208–216
bifurcation strain and fracture strength, 214–216
onset of bifurcation, 213–214
prebifurcation, 213
in SWCNTs, 212–213
Young’s modulus of SWCNT, 214
mesh-free computational framework, 142–143, 159–186, 172–178
advantages and disadvantages of, 143–144
algorithm, stability of, 181
bending test, 185–186
discrete equation, integration scheme for, 178–179
domain of influence (DOI) of nodes, 168–169
equilibrium solution, procedures for, 181–182
essential boundary conditions, enforcement of, 179–181
mesh-free shape function, 168
MK interpolation, 169–172
uniform tension, 182–184
validation studies, 182
weight functions, 163–168
single-walled carbon nanotubes (SWCNTs), 150–159
elastic properties, 155–158
global buckling of, 222–225
pressure–radial strain curve, 158–159
structural parameters, 153–154
transformation of, 150–153
vibration characteristics, 226–240
analysis of free vibration characteristic of carbon nanostructures, 227
atomistic simulation, 228
chiral effect and critical diameter, 232–240
free vibration frequency, edge effect on, 230–232
free vibration of SWCNTs, 228–229
quasicontinuum model, 227–228
Axial buckling of SWCNTs, 190–193, 206–207
Axial compression, 13–14, 49
CNT bundles under, 104–108
global buckling of SWCNTS under, 222–225
twisted CNT bundles under, 110–112
Axial tension, 97
CNT bundles under, 102–104
twisted CNT bundles under, 112–118

B

B3LYP method, 358–359
Beam models, 216–218
Euler–Bernoulli beam model, 216–221, 218, 302, 302–303, 303, 308, 308
nonlocal elastic beam models for flexural wave propagation in DWCNTs, 306–309
Timoshenko beam model, 226–227
Bending buckling, 198, 198–202
Bending test, 185–186, 255–256
Berendsen thermostat, 122–126, 126
Bernoulli–Euler beam model, See Euler–Bernoulli beam model
Bernoulli–Euler beam-bending theory, 261
Bifurcation, onset of, 213–214
Bifurcation strain and fracture strength, 214–216
Biotechnology, 39–40
BN nanotubes (BNNTs), 91, 95, 398–399
BN-doped SWCNTs, 54, 57–59
Boltzmann equation, 396
Boltzmann factor, 357, 364, 364
Born–Karman model, 303–304
Boron nitride nanotubes (BNNTs), 398–399
Bravais multilattice, 146, 210
Brenner potential, 50, 50–51, 51t, 53, 153, 254–255
Bridging domain method, 252
Bridging scale method, 252–254
Buckling and postbuckling behaviors, 186–207
See also Buckling of CNTs
axial buckling and postbuckling behaviors of SWCNTs, 206–207
bending buckling, 198–202
critical buckling curvature, 202–206
effect of loading methods, 202
axial buckling of SWCNTs, 190–193
buckling behavior of SWCNTs upon bending, 198–206
hydrostatic pressure-induced structural transitions of SWCNTs, 186–190
torsional buckling of SWCNTs, 193–198
Buckling of CNTs, 266–281
DWCNT, 275–278
explicit solution for, 267–268
with vdW interaction, 269–272
particular case of DWCNTs without vdW interaction, 268–269
SWCNTs, 198–206
vdW interaction after buckling, 273–275
vdW interaction before buckling, 272–273
Buckling of CNTs bundles, 97–118
axial compression, CNT bundles under, 104–108
axial tension, CNT bundles under, 102–104
twisting effects, 109–118
Buckling shapes, 421–426, 425f

C

Cantilevered SWCNC, 369, 371, 371–372, 372–373, 372f, 372f, 373–374, 373f, 374f
Carbon nanocoils, 427–434
critical rising angles, 433–434
equilibrium structures, 430–433
geometric structures, 427–428
maximum rising angles, 428–429
stable characteristics of carbon nanosprings, 429–430, 431f, 432f, 433f
Carbon nanocones (CNCs), 368–369, 435–441
effect of apex angle on mechanical behaviors of, 436–438
effect of cutting tip’s length on buckling of, 439–441
Carbon nanorings, 411–426, 411, 412
armchair nanorings, 412–413, 414f, 417f
buckling shapes, 421–426
critical tension displacements, 419–421
molecular dynamics (MD) simulation, 412
zigzag nanorings, 413–419, 416f, 417f
Carbon nanosprings, stable characteristics of, 429–430, 431f, 432f, 433f
Carbon nanostructures, analysis of free vibration characteristic of, 227
Cauchy–Born rule, 15, 141, 145–146, 208, 210, 249
Chemical vapor deposition (CVD), 2, 25, 27, 388
Chiral single-walled carbon nanotubes, bifurcation strain for, 217f, 217f
Clamped–clamped SWCNT, 305f, 313, 316–317
Classical molecular dynamics (MD) simulations, See Molecular dynamics (MD) simulations
Clean energy devices, 402–403
Close-capped single-walled CNTs, 128–129
Coatings and films, 37–38
Coaxial CNTs inside boron–nitride nanotube, structural stability of, 87–97
binding energy, 92–95
electronic structure and bonding model, 95–97
modeling method, 87–89
structural parameters, 89–92
Commercial suppliers of CNTs, 6
Companies developing and/or selling CNT products, 9t
COMPASS, 340, 412
Complex Bravais crystals, 141
Composite materials, 36–37
Computational model, 49–53
Continuum models, 13–15, 261, 301
buckling of CNTs, 266–281
buckling of DWCNT, 275–278
buckling of MWCNTs, 278–281
DWCNTs with vdW interaction, 269–272
explicit solution for DWCNTs, 267–268
general MWCNTs, 266–267
particular case of DWCNTs without VDW interaction, 268–269
vdW interaction after buckling, 273–275
vdW interaction before buckling, 272–273
continuum shell model, 264–266
nonlocal beam and rod models for vibration of SWCNTs, 302–306
nonlocal elastic beam models for flexural wave propagation in DWCNTs, 306–309
nonlocal elastic shell model, 309–311
van der Waals interaction, explicit formulas for, 263–264
vibration characteristics of CNTs, 281–298
Donnell shell model for the vibration of MWCNT, 281–283
radial vibration analysis of MWCNT, 283–298
vibration characteristics of CNTs, 311–317
wave propagation of CNTs, 317–331
nonlocal elastic beam models for flexural wave propagation, 317–324
nonlocal shell model for elastic wave propagation, 324–331
Continuum theory, 208
atomistic-continuum theory, 146–150
Contribution ratio of the higher-order terms, 224, 225f, 225f
Conveying fluid, 336–355
driving water molecules along a diameter-gradient SWCNT, 348–355
driving water molecules along a SWCNT, 340–348
Coriolis mass flowmeters, 339, 339–340
Covalently carbon hexagonal network, 1–2
Critical rising angle (CRA), 431–432, 433–434
Critical tension displacements, 416–418, 419–421
Cutting tip’s length, effect of
on buckling of carbon nanocones (CNCs), 439–441
Cyclohexylpyrrolidone (CHP), 31–32
Cylindrical shell model, 261–262, 262, 269f, 280–281

D

Density functional theory (DFT), 54, 61, 61–62
calculation, 356
Development and current situation of CNTs, 6
Diffuse element method (DEM), 142
Dimethylformamide (DMF), 31–32
Direct imposition method (DIM), 179–180
Discrete equation, integration scheme for, 178–179
DMol3 software, 68, 88–89
Donnell shell model for vibration of MWCNT, 263, 281–283
Double-walled CNTs (DWCNTs), 13, 226–227, 262–263
buckling of, 275–278
explicit solution for, 267–268, 283–285
nonlocal elastic beam models for flexural wave propagation in, 306–309
particular case of DWCNTs without VDW interaction, 268–269
with vdW interaction, 269–272
Driving water molecules
along a diameter-gradient SWCNT, 348–355
along a SWCNT, 340–348

E

Elastic properties of CNTs, 54–76
effects of vacancy defect reconstruction on elastic properties, 60–68
effects on Young’s modulus of single-walled CNTs, 67–68
model and methods, 61–62
in single-walled CNTs, 62–67
Young’s moduli of single-walled CNTs with grafts, 68–76
Young’s modulus of single-walled CNTs with impurities, 54–60
analysis of the results, 55–60
effect of impurities on single-walled CNTs, 54
model of SWCNTs with impurities, 54
Elastic wave propagation, nonlocal shell model for, 324–331
Electron paramagnetic resonance spectroscopy (EPR), 32
Element-free Galerkin (EFG) method, 142
Energy storage and environment, 38–39
Essential boundary conditions, enforcement of, 179–181
Euler–Bernoulli beam model, 216–221, 218, 302, 302–303, 303, 308, 308
Experimental tools in preparing and testing CNTs, 23
application prospect and researching significance, 35–40
biotechnology, 39–40
coatings and films, 37–38
composite materials, 36–37
energy storage and environment, 38–39
microelectronics, 38
mechanical properties of CNTs, 33–35
preparation methods, 24–27
arc discharge and laser ablation, 24–25
chemical vapor deposition, 25
CNT quality, 26–27
growth mechanism of CNTs, 26
testing technologies, 27–33
other characterization techniques, 32–33
photoluminescence (PL) spectroscopy, 32
Raman spectroscopy, 28–30
UV-Vis-nIR absorption spectroscopy, 30–32

F

Field effect transistors, 399–400
5–1DB defect, 61, 62, 62–64, 66–67
Flexural wave propagation, nonlocal elastic beam models for, 317–324
Fracture nucleation, 208–216
bifurcation strain and fracture strength, 214–216
onset of bifurcation, 213–214
prebifurcation, 213
in SWCNTs, 212–213
equilibrium equation for CNT, 213
Young’s modulus of SWCNT, 214
Fracture of CNTs, 119–122
Free longitudinal vibration, 302, 305, 311, 317
Free torsional vibration, 302, 305, 311, 317
Free transverse vibration, 302, 305, 311, 317
Free vibration frequency, edge effect on, 230–232
Free vibration of SWCNTs, 228–229
Frequency shift, analysis of, 370–380

G

G band, 30, 392–393
G' band, 392–393
Gas adsorption properties of CNTs, 355–356
Gauss integral method, 220
Gaussian quadrature rule, 179
Ghost force, 252, 252
Gibbs free energy, 357, 365–366
Global buckling of SWCNTS under axial compression, 222–225
Gram-Schmidt orthogonalization method, 179–180
Graphene nanoribbons (GNRs), 396
Graphene-based SET, 400f
Graphene–polymer nanocomposites, 403
Green–Kubo approach, 396
Growth mechanism of CNTs, 26

H

Hall effect, 398
Hamilton’s principle, 227, 370
Hexagonal boron nitride (h-BN) sheets, 398–399
Higher-order Cauchy–Born rule, 145
High-quality CNTs, 2
Hook’s law, 309
Hydrocarbon gases, 2
Hydrogen storage, 355–368
computational methodology and physical models, 357–359
enthalpies and free energies of the reaction, 363–368
reaction pathway of atomic hydrogen interaction with CNT, 359–363

I

Indium tin oxide (ITO), 37–38
Intertube van der Waals force, 110, 111f

K

Kriging, 169
Kronecker delta property, 179

L

Lagrange multiplier method, 143, 179–180, 254
Laser ablation technique, 2, 24–25
Lennard–Jones potential, 53, 263
Lomer dislocation, 251

M

Mass detection, 368–380
analysis of resonant frequency and frequency shift, 370–380
initial equilibrium SWCNC, 370
Mass production, 2, 6
MATERIALS STUDIO molecular modeling software packages, 340
Matrices grad and σ, 149–150
Maximum rising angle (MRA), 428, 428
Maxwell speed distribution, 67–68
Maxwell–Boltzmann distribution, 412
Maxwell–Boltzmann law, 126
Mechanical behaviors of CNTs, theories for, 12–16
atomistic simulations, 12–13
continuum models, 13–15
hybrid approaches, 15–16
Mechanical properties of CNTs, 33–35
Mesh-free computational framework, 142–143, 159–186, 172–178, 204
advantages and disadvantages of, 143–144
algorithm, stability of, 181
bending test, 185–186
discrete equation, integration scheme for, 178–179
domain of influence (DOI) of nodes, 168–169
equilibrium solution, procedures for, 181–182
essential boundary conditions, enforcement of, 179–181
mesh-free shape function, 168
moving least-squares approximation, 161–163
MK interpolation, 169–172
uniform tension, 182–184
validation studies, 182
weight functions, 163–168
Microelectronics, 38
MK interpolation, 169–172
Modified variational principle, 143
Molecular dynamics (MD) simulations, 49, 301, 396, 412
buckling of CNTs bundles, 97–118
CNT bundles under axial compression, 104–108
CNT bundles under axial tension, 102–104
twisting effects, 109–118
close-capped single-walled CNTs, 128–129
computational model, 49–53
elastic properties of CNTs, 54–76
effects of vacancy defect reconstruction on elastic properties, 60–68
Young’s moduli of single-walled CNTs with grafts, 68–76
Young’s modulus of single-walled CNTs with impurities, 54–60
fracture of CNTs, 119–122
open-ended multiwalled CNTs, 133–135
open-ended single-walled CNTs, 130–132
structural stability and buckling of CNTs, 76–97
structural stability of coaxial CNTs inside boron–nitride nanotube, 87–97
SWCNTS and MWCNTs, buckling of, 76–87
thermal stability of CNTs, 122–135
Molecular mechanics (MM) simulations, 416
Morse-type potential, 50
Moving least square (MLS), 142, 143, 161–163, 179
Multiwalled CNTs (MWCNTs), 6–12, 34, 34, 38–39, 81–87, 261, 261, 262–263, 266–267, 301
buckling of, 76–87, 278–281
development and current situation of, 6
Donnell shell model for vibration of, 281–283
open-ended, 133–135
radial vibration analysis of, 283–298
explicit solution for DWCNT, 283–285
explicit solution for TWCNT, 285–298
stress–strain curves for individual MWCNTs, 35f

N

Nanoelectromechanical systems (NEMS), 335
Newton’s method, 175, 175–176
Newton’s second law, 49–50
Newton–Raphson method, 181
N-methylpyrrolidone (NMP), 31–32
Nodal interpolation method (NIM), 179–180
Nonlocal atoms, 251
Nonlocal elastic beam models
for flexural wave propagation, 317–324
for flexural wave propagation in DWCNTs, 306–309
for vibration of SWCNTs, 302–306
Nonlocal elastic shell model, 309–311
Nonlocal shell model for elastic wave propagation, 324–331
Nonlocal Timoshenko beam theory, 13–14, 226–227
Nosé method, 412

O

ONIOM method, 356–357, 358–359, 359
Open-ended CNTs, 68
Open-ended multiwalled CNTs, 133–135
Open-ended single-walled CNTs, 130–132, 131f
Organic light-emitting diode (OLED) displays, 38

P

Penalty function method, 143, 179–180
Photoluminescence (PL) spectroscopy, 32
π bond, 60
Poisson’s ratio, 155–158
Postbuckling behaviors of SWCNTs, 206–207
Postbuckling of DWCNT, 261–262
Prebifurcation, 213
Prebuckling pressure, 262, 271t, 273, 273f
Preparation methods, 24–27
arc discharge and laser ablation, 24–25
chemical vapor deposition, 25
CNT quality, 26–27
growth mechanism of CNTs, 26
Pressure–radial strain curves, of SWCNTs, 158–159
Producers of CNT powders and dispersions, 7t

Q

Quality, of CNT, 26–27
Quantum generalized tight-binding MD (GTBMD), 76–77
Quantum mechanics (QM) treatment, 356–357
Quasicontinuum method, 227–228, 251–252

R

Radial breathing mode (RBM), 28, 29f
Radial vibration analysis of MWCNT, 283–298
explicit solution for DWCNT, 283–285
explicit solution for TWCNT, 285–298
Raman spectroscopy, 26, 28–30, 392–393, 393f
Reactive empirical bond order (REBO) potential, 51–52, 52, 53, 54, 301–302
Resonant frequency, analysis of, 370–380, 373f
Ru’s vdW interaction model, 261–262

S

Scanning probe microscopy (SPM), 391–392
Scanning tunneling microscopy (STM), 6–12
Selected-area electron diffraction (SAED), 32–33
Sensors, 400–401
σ bond, 60
Single vacancy defect ratio
effect of, on Young’s modulus of SWCNTs, 67–68
Single-atom vacancy defect, tensile failure of SWCNTs with, 256–257
Single-layer graphene, 388f, 392–393
Single-wall CNCs (SWCNCs), 368–369, 369, 369
initial equilibrium, 370
Single-walled carbon nanotubes (SWCNTs), 1, 1, 3f, 26, 34, 76–81, 117, 141, 256, 256f, 261, 261, 262–263, 301, 305–306, 335, 336–337, 411
axial buckling of, 190–193, 206–207
buckling behavior of SWCNTs upon bending, 198–206
bending buckling, 198–202
critical buckling curvature, 202–206
effect of loading methods, 202
buckling of, 76–87
calculation model of, 55f
chirality and diameter of, 3–4
close-capped, 128–129
diameter-gradient
driving water molecules along, 348–355
doped with BN, 55f
driving water molecules along, 340–348
effect of impurities on, 54
effect of single vacancy defect ratio on Young’s modulus of, 67–68
fracture nucleation in, 212–213
equilibrium equation for CNT, 213
free vibration of SWCNTs, 228–229
global buckling of, under axial compression, 222–225
with grafts, Young’s moduli of, 68–76
hydrostatic pressure-induced structural transitions of, 186–190
with impurities, Young’s modulus of, 54–60
nonlocal beam and rod models for vibration of, 302–306
open-ended, 130–132
postbuckling behaviors of, 206–207
structural and elastic properties of, 150–159
elastic properties, 155–158
pressure–radial strain curve, 158–159
structural parameters, 153–154
transformation of SWCNTs, 150–153
SWCNT–water system, 343–345
tensile failure of, with single-atom vacancy defect, 256–257
torsional buckling of, 193–198
Young’s modulus of SWCNT, 214
Smooth particle hydrodynamics (SPH) method, 142, 142
Sodium dodecylbenzenesulfonate (SDBS), 31–32
Solid-state NMR, 32
Strain energy versus the bending angle, 185f
Stress–strain curves for individual MWCNTs, 35f
Structural stability and buckling of CNTs, 76–97
structural stability of coaxial CNTs inside boron–nitride nanotube, 87–97
SWCNTS and MWCNTs, buckling of, 76–87
Substrate-based growth techniques, 388

T

Taylor expansion, 250
Technologically relevant applications, 335
conveying fluid, 336–355
driving water molecules along a diameter-gradient SWCNT, 348–355
driving water molecules along a SWCNT, 340–348
hydrogen storage, 355–368
computational methodology and physical models, 357–359
enthalpies and free energies of the reaction, 363–368
reaction pathway of atomic hydrogen interaction with CNT, 359–363
mass detection, 368–380
analysis of resonant frequency and frequency shift, 370–380
initial equilibrium SWCNC, 370
Tensile failure, 254–257
bending test, 255–256
tensile failure of SWCNTs with single-atom vacancy defect, 256–257
Tension test, 182
Tersoff potential, 50
Theoretical analysis, 12
Thermal conductivity of CNTs, 23
Thermal stability of CNTs, 122–135
Thermogravimetric analysis (TGA), 26
Thin film spectroscopy, 31–32
Timoshenko beam model, 13–14, 226–227, 303, 313, 319–322
Torsional buckling of SWCNTs, 193–198
Transmission electron microscopy (TEM), 26
Triple-walled CNTs (TWCNTs), 13–14, 226–227, 262
explicit solution for, 285–298
Twisted CNT bundles under axial compression, 110–112
twisted CNT bundles under axial tension, 112–118
Twisting effects of CNTs bundles, 109–118
2-D graphene and white graphene, 387
application prospects, 399–403
clean energy devices, 402–403
field effect transistors, 399–400
graphene–polymer nanocomposites, 403
sensors, 400–401
fundamental properties and general behaviors, 393–398
electronic properties, 393–394
mechanical properties, 394
optical properties, 395
thermal properties, 395–398
preparation methods and testing technologies, 389–393
characterizing graphene flakes, 390–391
preparation methods, 389–390
Raman spectroscopy, 392–393, 393f
scanning probe microscopy (SPM), 391–392
recent research advance in, 398–399

U

Universal force field (UFF), 88–89
UV-Vis-nIR absorption spectroscopy, 30–32

V

Vacancy defect reconstruction
effects on elastic properties, 60–68
Van der Waals (VdW) interaction, 105–107
after buckling, 273–275
before buckling, 272–273
DWCNTs with, 269–272
explicit formulas for, 263–264
particular case of DWCNTs without vdW interaction, 268–269
Velocity Verlet integration, 412
Vibration characteristics of CNTs, 281–298, 311–317
Donnell shell model for vibration of MWCNT, 281–283
radial vibration analysis of MWCNT, 283–298
explicit solution for DWCNT, 283–285
explicit solution for TWCNT, 285–298
Voltage-contrast SEM (VC-SEM) images electronic conduction pathways, 32–33

W

Water molecules, driving
along a diameter-gradient SWCNT, 348–355
along a SWCNT, 340–348
Wave propagation of CNTs, 317–331
nonlocal elastic beam models for flexural wave propagation, 317–324
nonlocal shell model for elastic wave propagation, 324–331
Weight functions, 163–168
Wiedemann–Franz law, 395–396

X

X-ray photoelectron spectroscopy (XPS), 32

Y

Young’s modulus, 34, 119, 155–158, 156f, 156f
of MWCNT, 261
of single-walled CNTs, 54–60, 68–76, 214

Z

Zigzag CNTs, 13, 235
Zigzag nanorings, 413–419, 416f, 417f
Zigzag nanotube, 4, 4f
Zigzag single-walled carbon nanotubes, 153–154
bifurcation strain for, 215f
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