Table Of ContentMechanics of Carbon Nanotubes
Mechanics of Carbon Nanotubes
Fundamentals, Modeling and Safety
Vasyl Harik
Nanodesigns Consulting, Wilmington, DE, United States
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Foreword
This monograph on mechanics of carbon nanotubes presents fundamentals about material
and structural properties of carbon nanotubes, classification of their atomic lattice structures
into 20 classes of distinct nanostructures, nanoscale laws of motion and friction, atomic
potentials, nanoscale expressions for stress, nanoscale homogenization criteria for the
atomic-continuum representation and their effect on Cauchy(cid:1)Born approximation, scaling
analysis of atomic lattice structures and scaling inequalities, and ranges of applicability of
the continuum beam and shell models and their limitations at nanoscale. This monograph
also examines molecular dynamics (MD) simulations of carbon nanotubes, nanoscale
interfacial structures and nanocomposites, and includes examples of such simulations and
their results for the stress(cid:1)strain dependence in nanocomposites, the pullout process of a
carbon nanotube, and its interfacial sliding. Nanoscale averaging of MD simulations,
criteria for nanoscale homogenization of MD models and their results have been discussed,
along with applicability of various continuum models and macroscale laws at nanoscale.
University, graduate and postgraduate students and academic and industrial researchers will
learn about a rigorous definition of nanomaterials and under what conditions their material
properties attain the unique and well-defined values independent of a nanoscale
phenomenon of the size-dependency of material properties of atomic structures.
Postdoctoral students will be fascinated by the so-called “effective thickness paradox” for
carbon nanotubes associated with the multiple estimates for the effective thickness of a
single layer of atoms, and its resolution based on the balance between the in-plane elastic
interactions and transverse van der Waals forces. A solution of the thickness paradox for
carbon nanotubes has been presented with a review of different estimates for their
thickness. Academic researchers and scholars will find out about the recent discovery of a
nanoscale analog of Pauli’s principle associated with the spatial exclusion of electrons
between atomic interfaces and its role in the interfacial friction and sliding.
A new matrix analysis for the atomic registry between adjacent atomic lattices has been
developed for the hexagonal lattices of carbon nanotubes and graphene. That is the atomic
registry matrix analysis, which includes the matrix-based description for the commensurate
and noncommensurate registry between adjacent carbon rings with different matrices for
distinct registries and the interregistry transition matrices. The value of coefficients in each
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xiv Foreword
registry matrix depends on local interatomic interactions and atomic potentials, while those
in the interregistry transition matrices reflect more complex interactions associated with the
changes from each atomic registry. Atomic registry matrix analysis allows one to describe
the interfacial interactions and sliding between adjacent atomic lattices by taking into
account local interactions. Interfacial sliding between graphene sheets can be described by
nanoscale analogs of Newton’s friction laws.
An updated classification of carbon nanotubes includes 20 classes of single-wall carbon
nanotube (SWCNT) and multi-wall carbon nanotube (MWCNT) with up to 10 classes of
the long beam-like atomic structures, four classes of thin and thick lattice shells with
cylindrical symmetry, and several classes of the carbon nanotube nanobeams and
nanocrystal rods with polygonal symmetry. Different classes of carbon nanotubes are
defined by the separation criteria based on scaling analysis of their atomic lattice structure.
It is shown that MWCNTs may also behave as thin shells for a certain range of geometric
parameters, which is represented in the materials map. Degeneration of the cylindrical
symmetry, the circumferential buckling and corrugation of carbon rings, and increase in the
bond length and internal van der Waals forces characterize the unique structural and
material properties of the carbon-nanotube nanobeams and nanocrystal rods. A topological
transition separates the hollow carbon nanotube nanobeams with the polygonal symmetry
and the nanocrystal rods having a tighter atomic lattice structure.
Considering a wide variety of carbon nanotube geometries and their atomic structures, it is
understandable that some carbon nanotube lattice shells require the thin shell theories for
their analysis and the data reduction of MD simulations results, while other polygonal
lattices are better represented by atomic-continuum beam models. Classification of
SWCNTs and the appropriate models are represented in a parametric map, which is useful
for the design of nanoscale probes (e.g., for atomic force microscopy) and predicting their
deformation modes. This monograph presents chiral formulation of several formulae for the
critical buckling strain of carbon nanotube lattice shells and nanobeams in the atomistic-
continuum form. A new lattice-sensitive formula for the critical strain of chiral SWCNTs is
evaluated, along with its predictions for nanoscale buckling of SWCNTs for different
estimates for their effective thickness. Numerical results compare well with the available
data of MD simulations. Equivalent-continuum shell models for carbon nanotubes have
been reviewed, along with the ranges of applicability of the continuum shell theories to the
mechanics of carbon nanotubes.
Unique material properties of MWCNTs have been analyzed to demonstrate how their
structure changes between different classes and influences Poisson’s affects in some cases.
A nanoscale phenomenon of simultaneous beam- and shell-like deformation modes
occurring during a complex buckling of short MWCNTs is described. Timoshenko beam
model is presented for the deformation of MWCNTs. Interfacial sliding in MWCNTs and in
Foreword xv
the carbon nanotube-reinforced nanocomposites is also analyzed. Interfacial friction models
based on nanoscale friction laws have been presented for MWCNTs and nanocomposites. A
friction model is introduced for the characterization of the strength of interfacial adhesion
during the pullout of a carbon nanotube from the polymer matrix. Interfacial structure in
nanocomposites is examined by the scaling analysis, atomic registry analysis and reflected
in the modified rules of mixture for the evaluation of elastic properties of the carbon
nanotube-reinforced nanocomposites. Tsai(cid:1)Halpin relation and its modified nanoscale form
are also presented for the approximation of elastic properties of nanocomposites.
Constitutive models of multifunctional nanocomposites have been presented to describe the
electromechanical, thermomechanical, and elastomagnetic effects. Recent trends in
publications on mechanics of carbon nanotubes and their safety have been reviewed.
The author gratefully acknowledges resources of Nanodesigns Consulting, color images of
the shutterstock database, along with the help and support of Elsevier staff and Nanodesigns
Press.
Vasyl Harik
Preface
Carbon nanotubes have superior mechanical and electrical properties which can be used for
a variety of nanotechnology applications. Carbon nanotubes have been some of the most
popular nanomaterials including graphene. Atomic lattice structures of carbon nanotubes
and graphene have many similarities and differences associated with the effects of
curvature. Mechanical behavior and material properties of carbon nanotubes, graphene and
other nanomaterials, which are subjected to various physical forces, are studied by
nanomechanics and nanoscale mechanics, molecular mechanics, and molecular dynamics,
as well as in other disciplines. Nanoscale mechanics may include micromechanics with
some nanoscale effects. Foundations of nanomechanics are based on different branches of
nanoscale physics, nanoscale chemistry, and the areas of solid state physics and continuum
mechanics models, which are capable of describing the nanoscale effects through new
methods, new experimental techniques and devices, as well as appropriate extensions of
existing methodologies and instrumentation. The mechanics of carbon nanotubes is a part of
nanomechanics and nanoscale mechanics of nanomaterials, nanocomposites, and
nanostructured materials. The first course on mechanics of carbon nanotubes was developed
by the author in 2001 at NASA Langley Research Center in Hampton, Virginia, and
presented later in the year at Annual Congress of American Society of Mechanical
Engineers in New York.
Carbon nanotubes are being used in computer industry (e.g., 9nm size transistor). The
carbon nanotube-based technology has surpassed the silicon-based technology in the
computer chip industry. Carbon nanotubes can be also used for different nanoscale devices,
sensors, and actuators, nanoscale probes and microelectronics. The use of carbon nanotubes
and their bundles in nanocomposites and the carbon fiber-reinforced composite materials is
still being optimized in order to improve the materials properties of nanocomposites, the
strength of interfaces, and safety. Long high aspect ratio multi-wall carbon nanotubes may
have toxic effects, depending on the exposure and the dose. Reducing the health risks
associated with the use of carbon nanotubes is very important for stable development of
nanotechnology and the nanotechnology driven industrial revolution. This monograph
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xviii Preface
addresses both the mechanical aspects of the carbon nanotube technology and reduction of
health risks by controlling the size and structure of carbon nanotubes and their use in
nanocomposites.
Vasyl Harik
CHAPTER 1
Nanotechnology of Carbon Nanotubes:
Sensors, Transistors and Nanocomposites
Chapter Outline
1.1 Properties of Carbon Nanotubes 1
1.1.1 DiscoveryofCarbonNanotubesinthe1950sand1990s 3
1.1.2 AtomicStructureofCarbonNanotubes:ChiralityandItsEffects 5
1.1.3 GeometryofCarbonNanotubesandTheirProperties 7
1.1.4 EffectiveThicknessParadoxforCarbonNanotubes 8
1.1.5 MaterialPropertiesofCarbonNanotubes 11
1.2 Applicationsof Carbon Nanotubes in Nanotechnology 13
1.2.1 CarbonNanotubeProbesforAtomicForceMicroscopes 14
1.2.2 CarbonNanotube-BasedSensorsandActuators 16
1.2.3 CarbonNanotube-BasedTransistorsforComputers 17
1.2.4 CarbonNanotubesinNanocomposites 19
1.2.5 CarbonNanotubesinBiomedicalApplications 20
1.2.6 NanotechnologyLiteracyProblems 21
Sample Problems 22
References 23
1.1 Properties of Carbon Nanotubes
The discovery of carbon nanotubes in 1991 by S. Iijima made these nanostructures the most
popular type of nanomaterial and nanoparticles used in novel applications; they have since
been studied by academic and industrial researchers, and college students (Iijima, 1991).
The name “carbon nanotubes” is derived from the material named carbon (C) and the
hollow, tubular structure of the wall, formed by the one-atom-thick layer of carbon lattice
(Dresselhaus et al., 1996). The single-wall carbon nanotubes (SWCNTs) have one layer of
carbon atoms, while the multiwall carbon nanotubes (MWCNTs) have multiple layers of
carbon lattice (Iijima and Ichihashi, 1993). Carbon nanotubes have extraordinary material
properties, such as high stiffness and strength, and exceptional electrical and thermal
conductivities when compared to other promising carbon materials (Gogotsi, 2006, 2017).
Their elastic modulus is much higher than that of steel and carbon fiber-reinforced
MechanicsofCarbonNanotubes. 1
DOI:https://doi.org/10.1016/B978-0-12-811071-3.00001-9
©2018ElsevierInc.Allrightsreserved.
