Computational modelling of thermo-mechanical and transport properties of carbon nanotubes

Over the recent years, numerical modelling and computer-based simulation of the properties of carbon nanotubes have become the focal points of research in computational nano-science and its associated fields of computational condensed matter physics and materials modelling. Modelling of the mechanical, thermal and transport properties of nanotubes via numerical simulations forms the central part of this research, concerned with the nano-scale mechanics and nano-scale thermodynamics of nanotubes, and nano-scale adsorption, storage and flow properties in nanotubes. A review of these properties, obtained via computational modelling studies, is presented here. We first introduce the physics of carbon nanotubes, and then present the computational simulation tools that are appropriate for conducting a modelling study at the nano-scales. These include the molecular dynamics (MD), the Monte Carlo (MC), and the ab initio MD simulation methods. A complete range of inter-atomic potentials, of two-body and many-body varieties, that underlie all the modelling studies considered in this review is also given. Mechanical models from continuum-based elasticity theory that have been extensively employed in computing the energetics of nanotubes, or interpret the results from atomistic modelling, are presented and discussed. These include models based on the continuum theory of curved plates, shells, vibrating rods and bending beams. The validity of these continuum-based models has also been examined and the conditions under which they are applicable to nanotube modelling have been listed. Pertinent concepts from continuum theories of stress analysis are included, and the relevant methods for conducting the computation of the stress tensor, elastic constants and elastic modulii at the atomic level are also given. We then survey a comprehensive range of modelling studies concerned with the adsorption and storage of gases, and flow of fluids, in carbon nanotubes of various types. This is then followed by an extensive survey of the numerical modelling investigations that have addressed the mechanical and thermal properties of these structures. The survey of modelling studies is supplemented by reviews of experimental investigations, where appropriate, to help clarify their results.

[1]  F. Banhart,et al.  Irradiation effects in carbon nanostructures , 1999 .

[2]  A. Rubio,et al.  Elastic properties of single-wall nanotubes , 1999 .

[3]  Gary G. Tibbetts,et al.  Why are carbon filaments tubular , 1984 .

[4]  S. Stuart,et al.  A reactive potential for hydrocarbons with intermolecular interactions , 2000 .

[5]  T. Ichihashi,et al.  Single-shell carbon nanotubes of 1-nm diameter , 1993, Nature.

[6]  D. Sholl,et al.  Diffusivities of Ar and Ne in Carbon Nanotubes , 2003 .

[7]  Donald W. Brenner,et al.  Molecular dynamics simulations of the nanometer-scale mechanical properties of compressed Buckminsterfullerene , 1991 .

[8]  D. Pettifor,et al.  Electron theory in alloy design , 1992 .

[9]  R. M. Mayer,et al.  Point defects and self‐diffusion in graphite , 1978 .

[10]  William F. Banholzer,et al.  Thermal conductivity of isotopically modified single crystal diamond. , 1993 .

[11]  Sanjay Govindjee,et al.  On the use of continuum mechanics to estimate the properties of nanotubes , 1999 .

[12]  L. Girifalco,et al.  Energy of Cohesion, Compressibility, and the Potential Energy Functions of the Graphite System , 1956 .

[13]  J. Johnson,et al.  An accurate H2–H2 interaction potential from first principles , 2000 .

[14]  W. L. Jorgensen,et al.  Comparison of simple potential functions for simulating liquid water , 1983 .

[15]  David E. Luzzi,et al.  Formation mechanism of fullerene peapods and coaxial tubes: a path to large scale synthesis , 2000 .

[16]  Interpretation of the atomic formulae for stress and stiffness coefficients , 1992 .

[17]  Nicholas Quirke,et al.  FLUID FLOW IN NANOPORES: ACCURATE BOUNDARY CONDITIONS FOR CARBON NANOTUBES , 2002 .

[18]  V. Vítek,et al.  Structural defects in amorphous solids Statistical analysis of a computer model , 1981 .

[19]  R. Feynman,et al.  Quantum Mechanics and Path Integrals , 1965 .

[20]  D. Lévesque,et al.  Monte Carlo simulations of hydrogen storage in carbon nanotubes , 2002 .

[21]  D. C. Rapaport,et al.  The Art of Molecular Dynamics Simulation , 1997 .

[22]  Zhou Jianjun,et al.  STRAIN ENERGY AND YOUNG'S MODULUS OF SINGLE-WALL CARBON NANOTUBES CALCULATED FROM ELECTRONIC ENERGY-BAND THEORY , 2000 .

[23]  A. Leach.  Molecular Modelling: Principles and Applications , 1996 .

[24]  Berry,et al.  Generalization of Nosé's isothermal molecular dynamics. , 1988, Physical review. A, General physics.

[25]  Marc Monthioux,et al.  Carbon nanotube encapsulated fullerenes: a unique class of hybrid materials , 1999 .

[26]  Coil formation in multishell carbon nanotubes: Competition between curvature elasticity and interlayer adhesion , 1997, cond-mat/9704235.

[27]  J. Tersoff,et al.  Empirical interatomic potential for silicon with improved elastic properties. , 1988, Physical review. B, Condensed matter.

[28]  M. Dresselhaus,et al.  Dispersion relations in graphite intercalation compounds: Phonon dispersion curves , 1981 .

[29]  R. T. Yang,et al.  Ab initio molecular orbital study of adsorption of atomic hydrogen on graphite: Insight into hydrogen storage in carbon nanotubes , 2002 .

[30]  R. Ruoff,et al.  Structural properties of a carbon-nanotube crystal. , 1994, Physical review letters.

[31]  T. Chou,et al.  Advances in the science and technology of carbon nanotubes and their composites: a review , 2001 .

[32]  Thomas Frauenheim,et al.  Hydrogen adsorption and storage in carbon nanotubes , 2000 .

[33]  J. Tersoff,et al.  Modeling solid-state chemistry: Interatomic potentials for multicomponent systems. , 1989, Physical review. B, Condensed matter.

[34]  Jordi Martí,et al.  Molecular dynamics calculation of the infrared spectra in liquid H2O-D2O mixtures , 1994 .

[35]  Boris I. Yakobson,et al.  C2F, BN, AND C NANOSHELL ELASTICITY FROM AB INITIO COMPUTATIONS , 2001 .

[36]  C. Bauschlicher.  Hydrogen and fluorine binding to the sidewalls of a (10,0) carbon nanotube , 2000 .

[37]  S. Reich,et al.  Elastic properties of carbon nanotubes under hydrostatic pressure , 2002 .

[38]  Gene Dresselhaus,et al.  Lattice-dynamical model for graphite , 1982 .

[39]  Lisa J. Porter,et al.  Atomistic modeling of finite-temperature properties of β-SiC. I. Lattice vibrations, heat capacity, and thermal expansion , 1997 .

[40]  T. Ebbesen,et al.  Exceptionally high Young's modulus observed for individual carbon nanotubes , 1996, Nature.

[41]  A. Kulik,et al.  Mechanical properties of carbon nanotubes , 1999 .

[42]  A. Rubio,et al.  AB INITIO STRUCTURAL, ELASTIC, AND VIBRATIONAL PROPERTIES OF CARBON NANOTUBES , 1999 .

[43]  J. M. Haile,et al.  Molecular dynamics simulation : elementary methods / J.M. Haile , 1992 .

[44]  L. B. Ebert.  Science of fullerenes and carbon nanotubes , 1996 .

[45]  Mohamed A. Osman,et al.  Temperature dependence of the thermal conductivity of single-wall carbon nanotubes , 2001 .

[46]  T. Belytschko,et al.  Atomistic simulations of nanotube fracture , 2002 .

[47]  Jingqi Li,et al.  Thermal conductivity of multiwalled carbon nanotubes , 2002 .

[48]  Car,et al.  Unified approach for molecular dynamics and density-functional theory. , 1985, Physical review letters.

[49]  W. Steele,et al.  Molecular Potential Structures of Heat-Treated Single-Wall Carbon Nanohorn Assemblies , 2001 .

[50]  Weber,et al.  Gases Do not adsorb on the interstitial channels of closed-ended single-walled carbon nanotube bundles , 2000, Physical review letters.

[51]  Donald E. Williams.  Nonbonded Potential Parameters Derived from Crystalline Hydrocarbons , 1967 .

[52]  Melville S. Green,et al.  Markoff Random Processes and the Statistical Mechanics of Time‐Dependent Phenomena. II. Irreversible Processes in Fluids , 1954 .

[53]  S. Shi,et al.  Molecular dynamic simulations on tensile mechanical properties of single-walled carbon nanotubes with and without hydrogen storage , 2002 .

[54]  Qinyu Wang.  Thermodynamic properties and phase equilibrium of fluid hydrogen from path integral simulations , 1996 .

[55]  J. Karl Johnson,et al.  Optimization of Carbon Nanotube Arrays for Hydrogen Adsorption , 1999 .

[56]  M. Dresselhaus.  Carbon nanotubes , 1995 .

[57]  White,et al.  Detonations at nanometer resolution using molecular dynamics. , 1993, Physical review letters.

[58]  E. Bekyarova,et al.  Oxidation and porosity evaluation of budlike single-wall carbon nanohorn aggregates , 2002 .

[59]  D. Brenner,et al.  Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. , 1990, Physical review. B, Condensed matter.

[60]  Phaedon Avouris,et al.  The effect of structural distortions on the electronic structure of carbon nanotubes , 1998 .

[61]  Nicholas Quirke,et al.  Fluid flow in nanopores: An examination of hydrodynamic boundary conditions , 2001 .

[62]  P. Bernier,et al.  OPENING OF SINGLE-WALLED CARBON NANOTUBES: EVIDENCE GIVEN BY KRYPTON AND XENON ADSORPTION , 2003 .

[63]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[64]  N. Quirke,et al.  Rapid imbibition of fluids in carbon nanotubes. , 2003, Physical review letters.

[65]  Z. Gu,et al.  Defects in arc-discharge-produced single-walled carbon nanotubes , 1999 .

[66]  M. Balkanski,et al.  Elastic properties of crystals of single-walled carbon nanotubes , 2000 .

[67]  R. Ruoff,et al.  Tensile loading of ropes of single wall carbon nanotubes and their mechanical properties , 2000, Physical review letters.

[68]  V. Vítek,et al.  Structural defects in amorphous solids A computer simulation study , 1980 .

[69]  Erik Dujardin,et al.  Young's modulus of single-walled nanotubes , 1998 .

[70]  T. Halicioǧlu,et al.  Stress Calculations for Carbon Nanotubes , 1998 .

[71]  M. S. de Vries,et al.  Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls , 1993, Nature.

[72]  H. Wagner,et al.  Buckling and Collapse of Embedded Carbon Nanotubes , 1998 .

[73]  Boris I. Yakobson,et al.  Atomistic theory of mechanical relaxation in fullerene nanotubes , 2000 .

[74]  Riichiro Saito,et al.  Physics of carbon nanotubes , 1995 .

[75]  Kwon,et al.  Unusually high thermal conductivity of carbon nanotubes , 2000, Physical review letters.

[76]  J. Yates,et al.  Molecular simulation of xenon adsorption on single-walled carbon nanotubes , 2001 .

[77]  O. L. Blakslee,et al.  Elastic Constants of Compression-Annealed Pyrolytic Graphite , 1970 .

[78]  G. A. D. Briggs,et al.  Elastic and shear moduli of single-walled carbon nanotube ropes , 1999 .

[79]  Jacques Lefebvre,et al.  Single-wall carbon nanotube based devices , 2000 .

[80]  J. Bernholc,et al.  Nanomechanics of carbon tubes: Instabilities beyond linear response. , 1996, Physical review letters.

[81]  L. Girifalco.  Molecular properties of fullerene in the gas and solid phases , 1992 .

[82]  R. Superfine,et al.  Bending and buckling of carbon nanotubes under large strain , 1997, Nature.

[83]  Young Hee Lee,et al.  Hydrogen storage in single-walled carbon nanotubes , 2000 .

[84]  I. R. Mcdonald,et al.  Theory of simple liquids , 1998 .

[85]  Bobby G. Sumpter,et al.  The onset of instability in nanostructures: The role of nonlinear resonance , 1995 .

[86]  Sasaki,et al.  Compressibility and polygonization of single-walled carbon nanotubes under hydrostatic pressure , 2000, Physical review letters.

[87]  Dong Qian,et al.  Mechanics of C60 in nanotubes , 2001 .

[88]  M. Nardelli,et al.  Brittle and Ductile Behavior in Carbon Nanotubes , 1998 .

[89]  S. Nosé.  A molecular dynamics method for simulations in the canonical ensemble , 1984 .

[90]  Jijun Zhao,et al.  Gas molecule adsorption in carbon nanotubes and nanotube bundles , 2002 .

[91]  U. Landman,et al.  Nanotribology and the Stability of Nanostructures , 1993 .

[92]  Hansong Cheng,et al.  Influence of carbon curvature on molecular adsorptions in carbon-based materials: a force field approach. , 2002, Physical review letters.

[93]  Jianping Lu,et al.  Elastic properties of single and multilayered nanotubes , 1997 .

[94]  A. Kohlhase,et al.  State resolved rotational excitation in D2+H2 collisions , 1983 .

[95]  Vasyl Harik,et al.  Ranges of applicability for the continuum beam model in the mechanics of carbon nanotubes and nanorods , 2001 .

[96]  Susan B. Sinnott,et al.  Molecular dynamics simulations of the filling and decorating of carbon nanotubules , 1999 .

[97]  Wang,et al.  Stiffness of a solid composed of C60 clusters. , 1991, Physical review. B, Condensed matter.

[98]  J. Martí,et al.  Effects of confinement on the vibrational spectra of liquid water adsorbed in carbon nanotubes , 2001 .

[99]  Xavier Gonze,et al.  Energetics of negatively curved graphitic carbon , 1992, Nature.

[100]  M. Yudasaka,et al.  Revealing properties of single-walled carbon nanotubes under high pressure , 2002 .

[101]  M. Born,et al.  Dynamical Theory of Crystal Lattices , 1954 .

[102]  Hayashi,et al.  Interlayer spacings in carbon nanotubes. , 1993, Physical review. B, Condensed matter.

[103]  P. Kollman,et al.  A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules , 1995 .

[104]  S. Louie,et al.  Analysis of the low-temperature specific heat of multiwalled carbon nanotubes and carbon nanotube ropes , 1999 .

[105]  D. Rooke,et al.  Numerical Fracture Mechanics , 1990 .

[106]  Herman J. C. Berendsen,et al.  MOLECULAR-DYNAMICS SIMULATION OF A BILAYER-MEMBRANE , 1982 .

[107]  G. Seifert,et al.  A hydrogen storage mechanism in single-walled carbon nanotubes. , 2001, Journal of the American Chemical Society.

[108]  Abell.  Empirical chemical pseudopotential theory of molecular and metallic bonding. , 1985, Physical review. B, Condensed matter.

[109]  J. Tersoff,et al.  New empirical approach for the structure and energy of covalent systems. , 1988, Physical review. B, Condensed matter.

[110]  V. Buch,et al.  Path integral simulations of mixed para‐D2 and ortho‐D2 clusters: The orientational effects , 1994 .

[111]  F. Darkrim,et al.  Review of hydrogen storage by adsorption in carbon nanotubes , 2002 .

[112]  Adsorption of He atoms in external grooves of single-wall carbon nanotube bundles , 2002, cond-mat/0211321.

[113]  S. Yip,et al.  Atomistic modeling of finite-temperature properties of crystalline β-SiC: II. Thermal conductivity and effects of point defects , 1998 .

[114]  Z. C. Tu,et al.  Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young's moduli dependent on layer number , 2001, cond-mat/0112454.

[115]  M. Yudasaka,et al.  Adsorption and catalytic properties of single-wall carbon nanohorns , 2000 .

[116]  D. Sholl,et al.  Rapid transport of gases in carbon nanotubes. , 2002, Physical review letters.

[117]  R. Watts,et al.  A spherical potential for hydrogen from solid state and scattering data , 1984 .

[118]  V. Simonyan,et al.  Molecular simulation of hydrogen adsorption in charged single-walled carbon nanotubes , 1999 .

[119]  M. Yudasaka,et al.  Nano-aggregates of single-walled graphitic carbon nano-horns , 1999 .

[120]  Hugh O. Pierson,et al.  Handbook of carbon, graphite, diamond, and fullerenes : properties, processing, and applications , 1993 .

[121]  J. Banavar,et al.  Computer Simulation of Liquids , 1988 .

[122]  J. Johnson,et al.  MOLECULAR SIMULATION OF HYDROGEN ADSORPTION IN SINGLE-WALLED CARBON NANOTUBES AND IDEALIZED CARBON SLIT PORES , 1999 .

[123]  Robertson,et al.  Energetics of nanoscale graphitic tubules. , 1992, Physical review. B, Condensed matter.

[124]  Victor V. Goldman,et al.  The isotropic intermolecular potential for H2 and D2 in the solid and gas phases , 1978 .

[125]  The structure relaxation of carbon nanotube , 2000, cond-mat/0006303.

[126]  M. Monthioux,et al.  Encapsulated C60 in carbon nanotubes , 1998, Nature.

[127]  Sándor Suhai,et al.  Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties , 1998 .

[128]  W. Goddard,et al.  Thermal conductivity of carbon nanotubes , 2000 .

[129]  Yoshinori Ando,et al.  Pentagons, heptagons and negative curvature in graphite microtubule growth , 1992, Nature.

[130]  P. Scharff,et al.  Molecular dynamics simulation of mechanical, vibrational and electronic properties of carbon nanotubes , 2000 .

[131]  R. Parr.  Density-functional theory of atoms and molecules , 1989 .

[132]  S. Nosé.  A unified formulation of the constant temperature molecular dynamics methods , 1984 .

[133]  Peter C. Eklund,et al.  Monte Carlo simulations of H2 physisorption in finite-diameter carbon nanotube ropes , 2000 .

[134]  G. Dresselhaus,et al.  Size Effects in Carbon Nanotubes , 1998 .

[135]  I. S. Sokolnikoff.  Mathematical theory of elasticity , 1946 .

[136]  David Tománek,et al.  Scrolls and nested tubes in multiwall carbon nanotubes , 2002 .

[137]  J. Tersoff,et al.  New empirical model for the structural properties of silicon. , 1986, Physical review letters.

[138]  Phaedon Avouris,et al.  Deformation of carbon nanotubes by surface van der Waals forces , 1998 .

[139]  S. Iijima.  Helical microtubules of graphitic carbon , 1991, Nature.

[140]  V. Crespi,et al.  Plastic Deformations of Carbon Nanotubes , 1998 .

[141]  V. Crespi,et al.  Smallest nanotube: breaking the symmetry of sp(3) bonds in tubular geometries. , 2001, Physical review letters.

[142]  Nino Boccara,et al.  Simple molecular systems at very high density , 1989 .

[143]  M. Hodak,et al.  Carbon nanotubes, buckyballs, ropes, and a universal graphitic potential , 2000 .

[144]  S. Iijima,et al.  Adsorption Mechanism of Supercritical Hydrogen in Internal and Interstitial Nanospaces of Single-Wall Carbon Nanohorn Assembly , 2002 .

[145]  J. Tersoff,et al.  Empirical interatomic potential for carbon, with application to amorphous carbon. , 1988, Physical review letters.

[146]  I. Silvera.  The solid molecular hydrogens in the condensed phase: Fundamentals and static properties , 1980 .

[147]  R. Ruoff,et al.  Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load , 2000, Science.

[148]  C. F. Curtiss,et al.  Molecular Theory Of Gases And Liquids , 1954 .

[149]  S. Ciraci,et al.  Systematic ab initio study of curvature effects in carbon nanotubes , 2002, cond-mat/0203229.

[150]  M. W. Cole,et al.  Interaction between a He atom and a graphite surface , 1980 .

[151]  Hoover,et al.  Canonical dynamics: Equilibrium phase-space distributions. , 1985, Physical review. A, General physics.

[152]  Dresselhaus,et al.  Symmetry properties of chiral carbon nanotubes. , 1993, Physical review. B, Condensed matter.

[153]  A. Walton.  Three phases of matter , 1976 .

[154]  Nan Yao,et al.  Carbon nanotube caps as springs: Molecular dynamics simulations , 1998 .

[155]  M. W. Cole,et al.  Low coverage adsorption in cylindrical pores , 1998 .

[156]  Studies of fullerenes and carbon nanotubes by an extended bond order potential , 1999 .

[157]  V. Popov.  Low-temperature specific heat of nanotube systems , 2002 .

[158]  J. S. Arellano,et al.  Density functional study of adsorption of molecular hydrogen on graphene layers , 2000 .

[159]  S. Louie,et al.  Heat capacity of carbon nanotubes , 1996 .

[160]  M. Dresselhaus,et al.  Electronic structure of double‐layer graphene tubules , 1993 .

[161]  Pulickel M. Ajayan,et al.  Nanometre-size tubes of carbon , 1997 .

[162]  J. Lu,et al.  Elastic Properties of Carbon Nanotubes and Nanoropes , 1997, cond-mat/9704219.

[163]  Charlier,et al.  Energetics of multilayered carbon tubules. , 1993, Physical review letters.

[164]  Chan,et al.  Molecular-dynamics simulation of thermal conductivity in amorphous silicon. , 1991, Physical review. B, Condensed matter.

[165]  Charles M. Lieber,et al.  Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nanorods and Nanotubes , 1997 .

[166]  J. Nye.  Physical Properties of Crystals: Their Representation by Tensors and Matrices , 1957 .