Computational Physics of Carbon Nanotubes

Hashem Rafii-Tabar9780521853002, 0521853001

Carbon nanotubes are the fabric of nanotechnology. Investigation into their properties has become one of the most active fields of modern research. This book presents the key computational modelling and numerical simulation tools to investigate carbon nanotube characteristics. In particular, methods applied to geometry and bonding, mechanical, thermal, transport and storage properties are addressed. The first half describes classic statistical and quantum mechanical simulation techniques, (including molecular dynamics, Monte Carlo simulations and ab initio molecular dynamics), atomistic theory and continuum based methods. The second half discusses the application of these numerical simulation tools to emerging fields such as nanofluidics and nanomechanics. With selected experimental results to help clarify theoretical concepts, this is a self-contained book that will be of interest to researchers in a broad range of disciplines, including nanotechnology, engineering, materials science and physics.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Contents……Page 9
Preface……Page 11
1 Introduction……Page 15
Part I……Page 27
2 Formation of carbon allotropes……Page 29
2.1 Diamond……Page 32
2.2 Graphite……Page 33
2.3 Fullerenes……Page 36
2.4 Carbon nanotubes and nanohorns……Page 37
2.4.1 Geometry of a graphene sheet……Page 38
2.4.2 Geometry of an SWCNT……Page 41
2.4.3 Construction of an SWCNT……Page 46
2.4.4 Connection between curvature and size of an SWCNT……Page 49
2.4.5 Structure of an MWCNT……Page 51
2.4.6 Structure of an SWCNH……Page 53
2.4.7 Remarks on measured and computed propertiesof SWCNTs, MWCNTs and SWCNHs……Page 55
3 Nanoscale numerical simulation techniques……Page 57
3.1.1 Microstate and phase space……Page 58
3.1.2 Equations of motion of a Raw-point……Page 59
3.1.3 Gibbs ensemble and ensemble average……Page 61
3.1.5 Grand-canonical ensemble……Page 63
3.1.7 Isothermal–isobaric ensemble……Page 64
3.1.9 Liouville’s theorem……Page 65
3.1.10 Correspondence between time and ensemble averages……Page 66
3.1.11 Fluctuations in statistical-mechanical ensembles……Page 67
3.1.12 Fluctuations in a canonical (NVT) ensemble……Page 68
3.1.13 Fluctuations in an isothermal–isobaric (NPT) ensemble……Page 70
3.1.14 Fluctuations in a grand-canonical (VT) ensemble……Page 71
3.1.15 Equipartition of energy and virial theorems……Page 72
3.2 Key concepts underlying the classical molecular dynamics (MD) simulation method……Page 75
3.2.1 Structure of an MD simulation code……Page 76
3.2.2 Molecular dynamics simulation in a canonical ensemble……Page 80
3.3.1 The Monte Carlo method in a canonical ensemble……Page 85
3.3.2 The Metropolis method……Page 87
3.3.3 Conducting an MC simulation in a canonical ensemble……Page 89
3.3.5 Conducting an MC simulation in an isothermal–isobaric ensemble……Page 90
3.3.7 Conducting an MC simulation in a grand-canonical ensemble……Page 92
3.4 Ab initio molecular dynamics simulation methods……Page 94
3.4.2 The Car–Parrinello molecular dynamics (CPMD) method……Page 102
4.1 Interatomic potential energy function (PEF)……Page 105
4.2 Force-field (molecular mechanics) method……Page 108
4.3.1 The Tersoff analytic bond-order many-body PEF……Page 109
4.3.2 The Brenner first-generation bond-order many-body PEF……Page 111
4.3.3 The Brenner second-generation bond-order many-body PEF……Page 114
4.3.5 A PEF describing SWCNT–SWCNT interaction……Page 119
4.4 Energetics of SWCNT–C60 and C60–C60 interactions……Page 120
4.4.1 A generalised PEF describing the interactionsbetween graphitic structures……Page 122
4.5.1 A PEF describing the average fluid–SWCNT interaction……Page 123
4.5.3 PEFs describing the Poiseuille flow of methane through an SWCNT……Page 127
4.5.5 PEFs describing the diffusion of Ar and Ne through SWCNTs……Page 129
4.5.6 PEFs describing the flow of oil through an SWCNT……Page 130
4.5.7 PEFs describing water adsorption inside SWCNTs……Page 131
4.6.1 PEFs describing H2–H2 and H2–SWCNT interactions……Page 133
4.6.2 Curvature-dependent force-field describing H2 adsorption in SWCNT……Page 137
4.6.3 PEFs describing the interaction of rare gases with SWCNTs……Page 140
4.6.4 PEF describing the Xe–SWCNT interaction……Page 145
4.6.5 A PEF describing the interaction of N2 molecules with SWCNHs……Page 146
5.1.1 Hooke’s laws in isotropic elastic materials……Page 149
5.1.2 Principal stresses and strains……Page 151
5.1.3 Hydrostatic and deviatoric stresses and strains……Page 153
5.1.4 Displacement tensor……Page 155
5.1.6 Plane strain and plane stress……Page 157
5.1.7 Stress equations of equilibrium……Page 159
5.1.8 Hooke’s laws in anisotropic elastic materials……Page 160
5.1.9 Stored elastic strain energy……Page 161
5.2 Nonlinear thin-shell theories……Page 166
5.2.1 Donnell’s shallow-shell theory……Page 167
5.2.2 The Sanders–Koiter theory……Page 172
5.3.1 Continuum-based theory……Page 173
5.3.2 Atomistic-based continuum theory……Page 178
5.4 Theories of vibration, bending and buckling of beams……Page 180
5.4.1 Flexural vibrations in an Euler–Bernoulli beam……Page 182
5.4.2 Longitudinal vibrations in an Euler–Bernoulli beam……Page 184
5.4.3 Torsional vibrations of an Euler–Bernoulli beam……Page 188
5.4.4 Bending and buckling in an Euler–Bernoulli beam……Page 189
5.4.5 Flexural vibrations in the Timoshenko beam……Page 194
6.1 Atomic-level stress tensor……Page 200
6.2 Elastic constants from atomistic dynamics……Page 204
6.3 Bulk and Young’s moduli……Page 206
7.1 Thermal conductivity……Page 209
7.1.1 Temperature gradient direct method……Page 210
7.1.2 The Green–Kubo time-correlation method……Page 212
7.2.1 Electronic specific heat……Page 216
7.2.2 Phonon specific heat……Page 217
7.2.3 SWCNT specific heat……Page 219
7.2.4 MWCNT specific heat……Page 221
7.2.5 Specific heat of SWCNT ropes……Page 222
Part II……Page 223
8 Modelling fluid flow in nanotubes……Page 225
8.1 Modelling the influence of a nanotube’s dynamics andlength on the fluid flow……Page 226
8.2 Modelling the flow of CH4 through SWCNTs……Page 229
8.3 Modelling self- and collective diffusivities of fluids in SWCNTs……Page 231
8.4 Modelling the capillary flow in an SWCNT……Page 233
8.5 Modelling the confinement and flow of liquid water inside SWCNTs……Page 235
8.6 Modelling the dynamics of C60@nanotubes……Page 237
9.1 Atomic and molecular hydrogen in nanotubes……Page 239
9.1.1 Modelling the adsorption of molecular hydrogen in isolated SWCNTs and SWCNT arrays……Page 240
9.1.2 The interplay between hydrogen adsorption and the SWCNT geometry and conduction properties……Page 252
9.1.3 Adsorption of H2 in charged SWCNTs……Page 255
9.1.4 Ab initio modelling of hydrogen adsorption in nanotubes……Page 256
9.1.6 Comparison of hydrogen adsorption in graphite and nanotubes……Page 262
9.2.1 Determination of adsorption sites for gases in a bundle of SWCNTs……Page 265
9.2.2 Adsorption of He, Xe, Kr and Ne in SWCNT bundles……Page 267
9.2.3 Ab initio modelling of nonhydrogen gas molecules in nanotubes……Page 271
9.2.4 The interplay between nonhydrogen-gas adsorption and the SWCNT geometry and conduction properties……Page 275
9.3.1 Further notes on the structural properties of SWCNHs……Page 278
9.3.2 Adsorption of N2 in SWCNH assemblies……Page 280
9.3.3 Classification of the pore structure in SWCNHs……Page 283
9.3.5 Adsorption of supercritical hydrogen in SWCNHs……Page 285
10 Modelling the mechanical properties of carbon nanotubes……Page 291
10.1.1 Applications of nonlinear shell theories……Page 293
10.1.2 Applications of beam theories……Page 315
10.1.3.1 Structural deformations of SWCNTs……Page 340
10.1.3.2 Structural deformations of MWCNTs……Page 343
10.1.4.1 Mechanism of strain release in SWCNTs: Stone–Wales topological defect……Page 346
10.1.4.2 Mechanisms of generating brittle and ductile behaviour in SWCNTs……Page 349
10.1.4.3 Dynamics of crack propagation in nanotubes……Page 353
10.1.4.4 Structural deformation of nanotubes……Page 365
10.2.1 A short survey of the experimental results……Page 397
10.2.2.1 Elastic constants……Page 400
10.2.2.2 Young’s modulus and stiffness constants……Page 407
10.2.2.3 Young’s modulus of SWCNT-based fibres……Page 418
10.2.2.4 In-plane and bending stiffness……Page 421
10.2.2.5 Vibrational frequencies of SWCNTs and their bundles……Page 423
10.2.3.1 Structural properties……Page 425
10.2.3.2 Young’s modulus and Poisson’s ratio……Page 428
10.2.3.3 Elastic properties of SWCNT bundles……Page 429
10.3.1 SWCNT bundles and MWCNTs……Page 430
10.3.1.1 The discrete model……Page 431
10.3.1.2 The continuous model……Page 433
10.3.2 Computation of atomic-level stress in strained SWCNTs……Page 435
10.3.3 Computation of transverse properties for nanotube crystals……Page 437
10.3.4 Computation of tangential and radial stresses in MWCNTs……Page 439
10.3.5.1 Stress–strain variations……Page 441
10.3.5.2 Interfacial properties……Page 443
10.3.5.3 Weight and volume fractions of SWCNTs in a composite……Page 450
10.4 Validity of application of continuum-based theories to model the mechanical properties of nanotubes……Page 453
10.4.1 Applicability of the Euler–Bernoulli beam theory……Page 455
10.4.2 Applicability of the curved plate theory……Page 459
11 Modelling the thermal properties of carbon nanotubes……Page 464
11.1.1 Variations of thermal conductivity of nanotubes with temperature, length, vacancies and defects……Page 465
11.1.2 Temperature dependence of thermal conductivity due to electron modes……Page 470
11.1.3 Chirality and radius dependence of thermal conductivity of nanotubes……Page 472
11.1.4 Measurement of thermal conductivity of SWCNT bundles……Page 477
11.1.5 Measurement of thermal conductivity of MWCNTs……Page 479
11.1.6 Thermal conductivity of decorated SWCNTs……Page 481
11.2.1 Measurement of temperature-dependent specific heat of SWCNTs……Page 482
11.2.2 Measurement of temperature-dependent specific heat of MWCNT bundles……Page 484
11.2.3 Computation of low-temperature specific heat of SWCNTs and MWCNTs……Page 485
11.2.4 Low-temperature behaviour of the specific heat of SWCNT ropes and MWCNTs……Page 487
References……Page 491
Index……Page 501