## Nanomechanics and Nanostructured Multifunctional Materials: Experiments, Theories, and Simulations

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Ling Liu, Guoxin Cao, Xi Chen, "Mechanisms of Nanoindentation on Multiwalled Carbon Nanotube and Nanotube Cluster", *Journal of Nanomaterials*, vol. 2008, Article ID 271763, 12 pages, 2008. https://doi.org/10.1155/2008/271763

# Mechanisms of Nanoindentation on Multiwalled Carbon Nanotube and Nanotube Cluster

**Academic Editor:**Junlan Wang

#### Abstract

Nanoindentation is a promising technique for deducing the elastic property of carbon nanotubes (CNTs). The paper presents an atomistic study on the nanoindentation mechanisms of single-walled and multiwalled CNTs and CNT clusters, through which the deformation characteristics are linked with CNT elastic stiffness. The assembly of individual single-walled CNTs (SWCNTs) into multiwalled CNTs (MWCNTs) and CNT clusters would significantly increase the buckling resistance in terms of withstanding the indentation load. Reverse analysis algorithms are proposed to extract the CNT stiffness by utilizing the indentation force-depth data measured from the prebuckling regimes. The numerical studies carried out in this paper may be used to guide the nanoindentation experiments, explain and extract useful data from the test, as well as stimulate new experiments.

#### 1. Introduction

##### 1.1. Carbon Nanotube Elastic Property and Available Experimental Techniques

Carbon nanotubes (CNTs) [1] are perceived to receive a wide range of potential applications thanks to their unique combinations of mechanical and electrical properties: (i) as structural components with extraordinary mechanical performances [2]; (ii) as conducting or semiconducting wires in nanoelectronic components [3]; (iii) as probes in scanning-probe microscopy with the added advantage of a chemically functionalized tip [4]; (iv) as high-sensitivity microbalances [5]; (v) as gas detectors [6]; (vi) as hydrogen storage devices by utilizing its large specific area [7]; (vii) as field-emission-type displays [8]; (viii) as electrodes in organic light-emitting diodes [9], and (ix) as tiny tweezers for nanoscale manipulation [10], among others.

The mechanical properties of the CNTs must be fully understood in order to fulfill their
promises. Perhaps the most fundamental phenomenological mechanical property of CNTs
is its Young's modulus *E* upon elastic
deformation. A variety of experimental attempts have been put together to
measure Young's modulus of carbon nanotubes: Treacy et al. [2]
have pioneered the measurement of thermally induced vibration amplitudes of multiwalled carbon
nanotube (MWCNT) cantilevers and have reported a range of Young's modulus from
0.40 TPa to 4.15 TPa. Young's moduli of single-walled carbon nanotubes (SWCNTs)
were measured by the same technique, varying from 0.9 TPa to 1.9 TPa [11]. Alternatively, by using an
atomic force microscope (AFM) tip to impose lateral forces to bend an MWCNT
cantilever deposited on a low-friction substrate, Young's moduli of MWCNTs were
found to be 1.28 0.59 TPa [12]. The elasticity of CNTs was
also found to be size-dependent; by measuring the electromechanical resonances
of CNTs, Poncharal et al. have discovered that the stiffness of MWCNTs decreases
quickly when their diameter exceeds about 10 nm [5]. Such phenomenon was not
observed in the static bending experiments with AFM [12]. These diverse experimental
measurements suggest that when viewed as a structure, the mechanical properties
of CNTs vary with different radius, length, and the number of walls, which are
affected by various manufacturing techniques adopted by different research
groups; and it was speculated that the measurement of elastic modulus may also
be affected by the magnitude and type of loading [13]. In addition, all of these
experimental techniques mentioned above were very challenging at the nanoscale,
in particular the sample preparation, mounting, and testing setup.

New experimental methods need to be developed to quickly and effectively measure the mechanical properties of CNTs. Among them, nanoindentation is an ultralow-load indentation technique that has been widely used to measure the constitutive relationships of material structures at very small scales [14]. It is arguably the simplest and most direct way of probing the mechanical properties of materials of very small volumes, thus suitable and attractive for CNTs [15]. In nanoindentation experiments, an indenter tip is driven into and then withdrawn from a specimen. High-resolution depth-sensing instruments are used to continuously control and monitor the penetration and reaction force on the indenter. In some commercial systems, indentation forces as small as several nN and displacements less than 1 nm can be accurately measured. One of the great advantages of nanoindentation is that the test can be performed quickly on any specimen and does not require the removal of the specimen from its substrate. This simplifies specimen preparation and makes indentation measurements easier on CNTs compared with other methods.

Recently, we carried out a preliminary study on the buckling mechanisms induced by
nanoindentation on an isolated, vertically aligned SWCNT [16]. An important finding is that
the buckling behavior of an SWCNT is shell-like if its dimensionless ratios *R*/*L*
and *R*/*t* are large (with *L*, *R*, and *t* the length, radius, and effective thickness, resp.),
beam-like when the ratios are small, and show an interesting transition
behavior in between. For nanoindentation experiment on CNTs, it is important to
note that first, MWCNTs are easier to fabricate since they do not require
stringent catalyst particle preparation [17], and thus they are more
popular than SWCNTs. The equilibrium spacing between neighboring layers of
MWCNT is approximately 3.4 Å, and they interact with each other through van der
Waals forces when the tube is deformed. Second, most CNTs prepared using
chemical vapor deposition (CVD), and plasma-enhanced CVD can take the form of a
cluster, or sometimes referred to as vertically aligned carbon nanotube forests
[17]. In some cases when seeds are
used [18], the CNTs are separated far
apart during deposition and the nanoindentation experiment can be regarded as
that carried out on an isolated tube—the indentation
behavior of SWCNT was covered by [16] while the mechanism of MWCNT
was yet to be explored. More frequently, the equilibrium spacing between the
neighboring tubes is also several Å in the cluster and the CNTs are *closely packed*, forming a
dense forest [19]. In fact, the van der Waals
interactions between nanotubes help to keep the tubes aligned [17]. Thus, as the indentation
depth increases, the bending and buckling behaviors of the tube are strongly
influenced by its neighbors.

In order to take advantage of nanoindentation with minimum sample preparation, an experiment carried out directly on the vertically aligned MWCNT and/or CNT cluster is more desired than the isolated SWCNT. Therefore, it is critical to extend out previous study [16] to the nanoindentation mechanisms of both MWCNT and CNT clusters, where the interactions between tube layers and neighboring tubes play an essential role [15]. For example, it is expected that due to the nonbonded interactions, all tube layers in MWCNT and all tubes in a CNT cluster are forced to deform at the same time, which would make them become more resistant to buckling—such hypothesis will be verified quantitatively through the atomistic studies in this paper. We note that the detailed characteristics of the nonbonded interactions may be revealed through molecular mechanics simulations, which serve as the main vehicle in this study. Since the mechanical properties of CNTs are only implicitly related with the indentation response, the establishment of such relationship must be also based on a thorough understanding of the mechanism of nanoindentation—this is the focus of the present paper, and the numerical study may be used to guide nanoindentation experiments, explain and extract useful data (such as the effective stretching stiffness of CNT) from the tests, explore the strengthening mechanism (e.g., determine whether concentric assembly or array assembly is more efficient for increasing the buckling resistance), as well as stimulate new experiments.

##### 1.2. Modeling and Simulation of Carbon Nanotubes

With the development of better force field and numeric algorithms, molecular mechanics (MM) simulations have been shown to play an important role in revealing precise constitutive mechanisms of CNTs. In fact, MM simulations have been widely used to study tension, bending, and torsion behaviors of CNTs [20–22] as well as buckling caused by uniaxial compression, torsion, and bending [20, 23–29]. It should be noted that buckling initiated from uniaxial compression is radically different from buckling induced by indentation in the present study. In uniaxial compression, the lateral displacements of atoms in the end layers are constrained and they are only allowed to move along the axis of CNT. In nanoindentation, however, the CNT atoms in the top layer are initially free, and their subsequent interactions with indenter atoms are dominated by the van der Waals force. In this paper, our previous work on SWCNT [16] will be extended to explore the nanoindentation mechanisms of MWCNT and CNT clusters through atomic detailed MM simulation, where the interactions among tube layers of MWCNT and neighbors in tube cluster are also taken into account.

One of the important
goals of nanoindentation test is to measure Young's modulus of nanotube—since *E* is a phenomenological parameter, it
must be established via a continuum approach. Perhaps the simplest and most
convenient model is to roll an SWCNT from a planar graphite sheet, and by
comparing the rolling and stretching energies obtained from both the thin plate
theory and atomistic simulations [20, 30–32],
an effective Young's modulus *E* = 3.9–5.5 TPa and effective nanotube
thickness *t* = 0.066–0.089 nm were
fitted. In general, the SWCNT can then be effectively modeled as a cylindrical elastic
thin shell. By comparing critical buckling loads and total strain energy for
SWNTs under axial compression and bending obtained from atomistic simulation
and finite element method, Pantano et al. [33] obtained *E* = 4.84 TPa and *t* = 0.075 nm
for the equivalent continuum shell. A comprehensive study of SWCNTs at small
deformation was carried out in our previous work [21], where by fitting the MM simulations
of uniaxial tension, bending, and torsion of SWCNTs of various chirality, *E* = 6.85 TPa and *t* = 0.08 nm were derived and also validated from the lateral and
axial thermal vibration frequencies.

The variation
of CNT Young's moduli obtained from previous theoretical and experimental
studies is partly due to the following reasons. First, *E* is closely associated with the effective shell thickness used in
different approaches, which is required to provide both the required stretching
and bending rigidity in the continuum shell model. In some approaches in the
literature, *t* = 0.34 nm is taken according
to the interlayer spacing of graphite and thus the resulting Young's moduli of 0.9–1.9 TPa [34–37] is much smaller than those
obtained from other theoretical studies [20, 21, 33]. However, when *E* and *t* are combined, the stretching stiffness *Et* obtained from most previous studies are reasonably close. In
addition, a recent analytical study by Huang et al. [13] verified that *t*, and therefore *E*, is dependent on the type and magnitude of loading, nanotube
radius *R*, and also chirality when *R* < 1 nm. As a result, it may be
inappropriate to take the results of *t* and *E* derived from a particular SWCNT
under certain loading conditions as universal parameters. For example, the *t* and *E* derived from our recent work [21] may be regarded as the
averaged parameters when the SWCNTs (with varying chirality) are subjected to several
basic small deformation modes. Based on such consideration, the present paper
aims to use nanoindentation to derive the nanotube stretching stiffness *Et* for both MWCNT and CNT clusters, from
the established relationships among indentation force, displacement, and
intrinsic tube deformation using MM simulations. The numerical studies not only
underpin the indentation mechanics of CNTs and serve as the basis of measuring
their mechanical properties, but also help to advance understanding on the
mechanical behavior of CNTs.

#### 2. Computational Method

The molecular mechanics (MM) simulations may be readily employed to explore CNTs containing hundreds of thousands of atoms, by ignoring the electron motions and expressing the system potential energy as a function of the nuclear positions of atoms. The COMPASS force field, which is the first ab initio force field that enables an accurate and simultaneous prediction of various gas-phase and condensed-phase properties of organic and inorganic materials [38] (including carbon nanotubes [16, 23–25, 29, 39, 40, 41, 42, 43, 44, 45]), is used in this paper. Simulations are performed at 0 K, so as to not involve the kinetic energy term and to obtain more intrinsic buckling behavior of the tube, since buckling would be otherwise very sensitive to thermal fluctuations [20, 24–28]. The CNT radius is much smaller than the radius of any commercial diamond indenter tip (100 nm); moreover, the diamond bulk is much stiffer than the nanotube. Therefore, the indenter is modeled as a flat plane consists of rigid diamond atoms, and during the simulation, such plane continuously moves down with a displacement increment of 0.05 Å, so as to simulate a displacement-controlled experiment with a prescribed rate. The tube (or tube cluster) is aligned perpendicular to the indenter plane (e.g., see Figure 1). All degrees of freedom of atoms in bottom layers of the CNTs are fixed to simulate clamped end conditions in mechanical analyses; the atoms in the top of CNT interact with the diamond atoms via nonbonded forces, and there is no displacement boundary condition imposed on those atoms. Breakage of C–C bonds in CNT is not considered in the present study because the strain induced by nanoindentation loading is not that large (see below).

**(a)**

**(b)**

Tubes with various chirality and structures (multiwalled and cluster) are used in the simulation. The long, beam-like tubes are chosen in this study because they are more practical. The initial atomic structure is first optimized such that the total potential energy is minimized and all atoms are located in their equilibrium states. Due to the complicated bonded and nonbonded interactions, the carbon atoms are not exactly at their ideal positions after optimization. For instance, all SWCNT atoms at the same height may not align precisely along a circle even though the deviation from ideal positions is very small. Such small perturbation will be accumulated during the indentation process and contribute to the instability of CNTs [16, 25].

The initial separation between the top layer
of CNT atoms and the indenter plane is set such that no net force acts on the
tube at the starting point. The magnitude of slightly depends on the tube chirality. As the indenter
plane moves down with a displacement *δ*,
normally termed as indentation depth, the separation *d* falls below ,
and the overall van der Waals repulsion between indenter and carbon atoms acts
as an indentation force (*F*), to
compress and deform the nanotube. The deformed configuration of CNT(s) at the current
indentation depth is obtained by structural optimization in search for the minimum
system potential. The total potential energy of the atomic system, , varies
during the indentation process. Since no heat exchange is considered in the MM
studies, the work done by indentation force (*F*) is the only external work, which is responsible for the variation
of system potential energy with respect to its reference state where can be interpreted as the
deformation energy of the whole system, which includes the strain energy of the
nanotube , and the van
der Waals interaction energy (arises from that between the indenter layer and CNT, between neighboring CNT
layers of MWCNT, and between neighboring CNTs of a cluster):

With a
state-of-the-art commercial nanoindenter, both indentation force *F* and indentation depth *δ* can be readily measured, and *U* may be integrated following (1). *F*-*δ* and *U*-*δ* curves
will be explored extensively in this study to obtain valuable insights, such as
their links to the deformation modes and elastic constants of CNTs. Moreover, since
CNTs are slender structures having the possibility of buckling under indentation
load, so *F* may attain the peak value termed
as the critical indentation force, ,
when the buckling occurs at the critical indentation depth, . will
be used as an index to evaluate the specimen's
capability of withstanding the indentation load. Such buckling resistance will
be compared among CNT clusters, MWCNTs, and their subunit SWCNTs.

#### 3. Nanoindentation on MWCNT

##### 3.1. Deformation Mechanisms

Although the SWCNT is more fundamental, MWCNTs are easier to make and they can be used as an AFM tip or as a reinforcement phase in nanocomposites. As a slender structure in nature, MWCNT may buckle under indentation load and/or axial compression, although one would typically suspect that the MWCNTs should have higher resistance to buckle compared with their single-walled counterparts, thanks to the van der Waals interactions between neighboring layers.

The indentation response of a representative (5,5)-(10,10)-(15,15) MWCNT is analyzed
using MM simulation. The length of MWCNT is relatively long, *L* = 216.77 Å, such that it would exhibit
beam-like buckling mechanisms, elaborated below; the long CNTs are more
practical and their buckling behaviors are easier to analyze (as opposed to
shell-like characteristics of short tubes) [16]. Figure 1 shows the sequential snap shots of the deformed
configurations as the indentation depth *δ* (or equivalently, the *nominal* axial strain *ε* = *δ*/*L*) is increased; both side and top views
are given. The relationships between *F* and *δ*, and that between *U*
and *ε*, are given in Figures 1(a) and 1(b), respectively. The loading process may be divided
into three mechanism zones based on system responses that correspond to
different deformation modes.

During regime (I)
when *δ* is sufficiently small (*ε* smaller than about 1.6% for the
current case), the MWCNT undergoes uniaxial compression. However, the *F*-*δ* curve is in fact not linear in this stage although the MWCNT deformation is
supposed to be linear elastic at small strain. Note that *δ* is the displacement of the indenter tip and
thus it includes contributions from both the compression of nanotube and the
distance change between indenter and nanotube's top layer due to nonbonded
interactions—the
force-displacement relationship of the nonbonded component is nonlinear, which
will be analyzed in detail in Section 3.3.

When the axial compressive strain becomes critical, the buckling regime (II) happens, which is marked by the sudden slide buckling of MWCNT, a characteristic of continuum beam. At this critical point, = 73.5 nN. Due to the slide buckling, most of the compression energy is relieved, leading to 85% drop of indentation force (Figure 1(a)) and substantial reduction of system energy (Figure 1(b)); the system configuration changes from compression-dominated before buckling to bending-dominated after buckling, and such structural change is mainly responsible for the sudden decrease of indentation force. After the nanotube bounces out, the van der Waals interaction between MWCNT and indenter tip exerts a bending moment on the MWCNT, which slides the MWCNT with respect to the indenter plane as indentation depth is increased. This is the postbuckling regime (III). The applied force is observed to maintain almost a constant value when the MWCNT remains in its buckled (bending) state, which is consistent with the classical beam theory (when the buckle mode is fixed).

The three subunit SWCNTs are of the same length but have different aspect ratios (*L*/*R*)
of 63.6, 31.8, and 21.2, respectively, and their indentation force-depth curves
are also plotted in Figure 1(a). It is readily seen that the curves for (5,5) and
(10,10) SWCNTs exhibit the same trend (sharp reduction of load after buckle) as
the MWCNT, while the *F*-*δ* curve of (15,15) SWCNT is characterized
with two peaks. This difference in indentation response curves can be explained
by the postbuckling configurations shown in Figure 2, where the examined three SWCNTs are
found to have different deformation modes: beam-like slide buckling deformation for (5,5) and
(10,10) SWCNTs and shell-like snap buckling deformation for (15,15) SWCNT, due
to their different *L*/*R* and *R*/*t* ratios [16].

Note that at
the present length, the MWCNT contains one subunit SWCNT having shell-like
buckling characteristic if individually indented, yet the overall buckling
behavior of MWCNT is still beam-like. This observation indicates that the
interlayer van der Waals interactions must have coordinated the subunit
deformation and thus *strengthened* the MWCNT during the indentation
process, without which the outer SWCNT should have experienced snap buckling
before the MWCNT collapses. In view of the system energy, considerable van der
Waals energy between MWCNT layers would be generated if the deformation of each
subunit SWCNT does not conform to those of the others, which is unfavorable to
stabilize the system. In other words, the individual walls of MWCNTs are integrated
by the interlayer van der Waals interactions, which make the MWCNT a reinforced
structure analogous to a composite beam (the property of the intermediate *material* that accounts for the interlayer van der Waals interactions is anisotropic,
which is much stronger in the radial direction than in the hoop direction).

##### 3.2. Strengthening of Buckling Resistance

The three armchair
SWCNTs examined above can also construct two double-walled CNTs (DWCNTs): (5,5)-(10,10)
DWCNT and (10,10)-(15,15) DWCNT. In order to compare the buckling resistance of
MWCNT with its subunits, the following groups of combination are considered,
where in each group, the performance of an MWCNT is compared with its subunit
SWCNTs and/or DWCNTs:(i)(5,5)
SWCNT, (10,10) SWCNT, and (5,5)-(10,10) DWCNT;(ii)(10,10)
SWCNT, (15,15) SWCNT, and (10,10)-(15,15) DWCNT;(iii)(15,15)
SWCNT, (5,5)-(10,10) DWCNT, and (5,5)-(10,10)-(15,15) MWCNT;(iv)(5,5)
SWCNT, (10,10)-(15,15) DWCNT, and (5,5)-(10,10)-(15,15) MWCNT. The first two
groups represent the assembly of two SWCNTs into a DWCNT, and the last two
groups are examples of assembly of a SWCNT and a DWCNT into a three-layered
MWCNT. Nanoindentation experiments are carried out on all these nanotubes,
generating a series of *F*-*δ* curves for both the subunit and
assembled structures, shown in Figure 3 with respect to these four groups. In addition, the
summation of the *F*-*δ* curves of subunits is also shown,
referred to as the superposition curve.

**(a)**

**(b)**

**(c)**

**(d)**

For the
assembled structures, the two DWCNTs are found to have beam-like buckling
behaviors (Figure 2), and their *F*-*δ* curves also show a sharp declination
of the load after is reached (Figure 3). This is consistent with the assembled 3-layer
MWCNT. In other words, among the selected assembled CNTs and their subunits, only
the (15,15) SWCNT undergoes snap buckling because its low *L*/*R* aspect ratio of 21.2 falls into the shell-like region [16].

When *δ* is sufficiently small such that the assembled CNTs
and their subunits are under uniaxial compression, the linear *F*-*δ* response of a MWCNT or DWCNT equals to the superposition of that of its
subunits—this also
indicates that the concentric assembly of CNTs (or interlayer van der Waals
interactions) has essentially no influence on the elastic properties of CNTs
during the axial compression regime, and the interlayer distance of MWCNT is
essentially unchanged.

When *δ* is increased to a critical value, one of the subunit
SWCNTs buckles first (usually the shell-like SWCNT, or the more slender member
of SWCNTs in the beam-buckling region), leading to a sharp reduction of the
load on the superposition curve; meanwhile, the other subunit can still hold
the load and *F* keeps rising on the
superposition curve. Note that at this instant, the assembled multiwalled structure
shows no sign of buckle. As *δ* is further increased, the other subunit CNT is
observed to buckle at its critical force, making the force on superposition
curve suddenly decrease again, yet the assembled DWCNT or MWCNT
structure still holds. For all groups, the critical force of the assembled tube
is observed to be 30%–90% larger than
the peak value of the superposition curve. This significant enhancement of implies that
the concentric assembly of CNTs could dramatically strengthen the system in resisting buckling
when subjected to indentation load. This conclusion is also supported by the
increase of critical nominal strain of the assembled structure.

Such strengthening effect of MWCNT may be attributed to the van der Waals
interactions between subunits. Consider any atom in the inner or outer tube
layer of a DWCNT, the net van der Waals force acting on this atom should be small
enough in the pure compression regime so as to keep the tube straight. However,
just before buckling occurs when the atom attempts to move radially, the
corresponding van der Waals repulsion or attraction will be developed in the
normal (radial) direction; in other words, the van der Waals interactions between
neighboring tube layers serve as an invisible nonlinear spring-like *material* which makes the assembled structure behave like a sandwich tube. The sandwich
tube, with an equivalent tube thickness larger than the combined thicknesses of
subunits, has a higher bending stiffness and therefore a higher critical buckling
force compared to the superposition of subunits (where such van der Waals
strengthening effect is absent).

The strengthening
efficiency may be evaluated by the increase of critical force per reference
tube, divided by the reference number of strengthening tubes. Note that subunit
SWCNTs in any MWCNT must have distinct chirality with different numbers of C
atoms, so in order to ensure a fair comparison, the MWCNT can be regarded as an
assembly of several reference SWCNTs with same total number of atoms. For
example, if we take the (10,10) SWCNT as the reference tube, a (10,10)-(15,15)
DWCNT has 2.5 reference tubes in terms of equivalent number of C atoms (the
reference number of (10,10) is 1 and that of (15,15) is 1.5). In this case, the
critical buckling force per reference tube is increased by 28%: from 15 nN for an
isolated (10,10) SWCNT to 19.2 nN (= 48 nN/2.5) per reference tube for the
(10,10)-(15,15) DWCNT. In other words, 28% is the increase of buckling
resistance for the (10,10) SWCNT when a (15,15) SWCNT is used to *strengthen* it by forming a DWCNT. For that matter, a strengthening efficiency for the
reference subunit SWCNT can be defined, which is the percentage of increase of
buckling resistance divided by the total reference number of other subunit
tubes used for strengthening the MWCNT (i.e., the *cost* required for
promoting the buckling resistance of the reference tube). For the (10,10)
reference SWCNT under consideration, its strengthening efficiency is 18.7% (28% divided by 1.5).
Following this procedure, the strengthening efficiency of all
MWCNTs examined above is calculated to vary from 18.7% to 75% by choosing
different reference tubes.

##### 3.3. Reverse Analysis to Deduce the Elastic Stiffness

As shown in Figure 1, the MWCNT is kept straight in the pure compression
regime (I), which makes it possible to deduce its elastic stiffness from
nanoindentation test before the tube buckles (when the nominal axial strain is
less than about 1.6%). However, due to the van der Waals interaction, the
displacement of the indenter tip, *δ*, which
is a measurable quantity from experiment, does not equal exactly to the intrinsic
axial compression of SWCNT, *u*, which
is defined as the downward displacement of carbon atoms in the top layer. In
fact, the separation between indenter plane and top layer of MWCNT atoms
changes nonlinearly with respect to the indentation force, sketched in Figure 4, where the van der Waals interaction between MWCNT
and indenter is simplified as a nonlinear spring. The force acting on the
nonlinear spring and MWCNT equals to the indentation force *F* in this regime, and the indentation
depth is where with *d* defined as the
deformed *spring* length, and the undeformed length depends on
the chirality of tube ( Å
for armchair tubes). Since the *F*-*δ* curve is a measurable
characteristic system response, the contribution of *s* needs to be subtracted off from *δ* in order to obtain the intrinsic elastic
deformation of MWCNT.

Without losing generality, *F* can be regarded as the
summation of normal component of nonbonded interaction forces between the indenter
plane and every C atom in CNT, which may be effectively modeled by the pairwise
relationship where Å
represents the equilibrium separation between a C atom and indenter plane, Δ* *denotes the separation after indenter
penetration, and *C*, *m*, and *n* are constants to be determined. It should be emphasized that *f*(Δ) is constructed to characterize the interaction
between any single C atom and the indenter tip, which is independent of the CNT
chirality and the lateral position of the C atom with respect to the indenter
tip. In other words, Δ* *is the distance between a C atom and the
indenter plane, which is different from the variable *d* used before, since *d* represents the separation between the CNT top layer consisting a set of C atoms
and the indenter plane, and thus *d* is
chirality-dependent. Note that is different from , since is
the equilibrium separation
between the top layer of CNT (a set of C atoms) and indenter plane. By moving an
isolated carbon atom with respect to the indenter tip, the net van der Waals
force *f* acting on the carbon atom can
be obtained and fitting of (4) leads to *C* = 0.79 nN, *m* = 8, and *n* = 5. Therefore,
the nonbonded force per carbon atom may be calibrated as where the
dimension of Δ* *is Å.

For any *N*-walled armchair MWCNT assembly with
chirality () (),
assuming that the top layers of subunit SWCNTs are aligned in the same plane, one could obtain
its total nonbonded force as a summation of the interaction forces between
indenter and carbon atoms in different layers near the top of MWCNT. Note that
for the topmost layer, *d* is its
distance to indenter, for other layers, their distances to the indenter plane also
include relevant projections of bond length in the axial direction. Therefore,
the total interaction force between indenter and MWCNT can be written as where yields the number of atoms in each layer, *b* denotes the length projection of a C–C bond in the
axial direction (shown in Figure 4), and *M* is the
number of layers interacting with the indenter plane. Note that the nonbonded interaction
(c.f. (5)) decays quickly as Δ* *gets above , thus is ensured to generate converged solution; the results are also
insensitive to the lateral alignment of CNT with respect to indenter lattice. Relationships
similar to (6)
may be developed for other chirality of CNTs, by taking into account different
positions of carbon atoms near the top of subunit SWCNTs (see Section 4.3 for
the example of zigzag tubes).

From (3) and (6), the compressive displacement of examined MWCNT can be written as Based on the *F*-*δ* curve measured from either numerical or experimental indentation study, the
constitutive relationship for MWCNT can be established at small strain. For the
numerical example of (5,5)-(10,10)-(15,15) MWCNT presented in Figure 1, after removing the contribution of the MWCNT-indenter
interaction using (7),
the predicted *F*-*u* curve is shown in Figure 5, which is in excellent agreement with that measured directly
from MM simulation.

Since the resulting *F*-*u* relationship is
almost linear, the MWCNT mechanical property is linear elastic when the nominal
axial strain is smaller than about 1.6%. If each subunit SWCNT is modeled as a
continuum thin shell, the linear relationship between *F* and *u* at small axial strain
can be written as where *L* is
the tube length, *E* Young's
modulus, *t* the wall thickness, the radius of the *i*th subunit SWCNT, and *A* denotes the total area of MWCNT cross
section. Thus, the elastic stiffness of SWCNT, *Et*, can be measured from the slope (*k*) of *F*-*u* curve as For the present (5,5)-(10,10)-(15,15)
SWCNT, *k* = 231.89 N/m, *L* = 216.77 Å, = 3.41 Å, = 2, and = 3,
which leads to *Et* = 391.35 Pa m (the
same approach can also be applied to either (5,5)-(10,10) or (10,10)-(15,15)
DWCNT studied in this paper and the results are almost the same). This value is
fairly close to that reported by [20], where Pa m, and by [33], where Pa m.

#### 4. Nanoindentation on SWCNT Cluster

##### 4.1. Deformation Mechanisms

Besides forming the concentric MWCNT, SWCNTs of the same chirality and length may
assemble into clusters. The initial (undeformed) cluster configuration, that
is, the arrangement of SWCNTs in lateral directions, is obtained by minimizing the
system potential energy of the unit cell. Upon nanoindentation on an *n*-tube cluster, the indentation force per
tube = *F*/*n* is compared with that of isolated
tubes. For a representative 7-tube
(8,0) cluster, the -*δ* curve is presented in Figure 6 whose characteristic is very similar to its component
SWCNT (with *L*/*R* = 24.4), as well as being similar to other long
SWCNTs and MWCNTs shown in Figure 3. Based on both -*δ* curve and sequential snap shots of the deformed cluster
in Figure 6, the indentation process can also be divided into
three main regimes associated with distinct deformation modes: (I) uniaxial
compression where all the tubes remain straight as the nominal axial stress is
smaller than 1.4% for the cluster under investigation; (II) slide buckling where
all subunit SWCNTs buckle toward the same direction after is attained; (III) postbuckling where the top layers
of subunit SWCNTs are pushed to slide with respect to the indenter plane after
the indentation force is significantly reduced.

In Figure 6, the -*δ* curve of the 7-tube cluster is compared with that of an isolated (8,0) component, as well as
that of 12-tube and 19-tube clusters made by the same subunit. The buckled
configurations of the clusters are shown in Figure 7, where all tubes seem to buckle into the same shape. From
Figure 6, the -*δ* curve of an
isolated tube is very different than that of a subunit tube in a cluster—as a part of a
cluster, the same tube could sustain much higher load before it buckles
(compare with the isolated tube), and only when the nominal axial strain is smaller
than 1.4% does the subunit tube has the same constitutive relationship as its
isolated counterpart. Such increased buckling resistance per tube is attributed
to the van der Waals interactions among neighboring tubes, elaborated
below.

**(a)**

**(b)**

**(c)**

##### 4.2. Strengthening of Buckling Resistance

Figure 6 clearly indicates that with the increase of tube
numbers in a cluster, each tube could sustain a higher force. Before buckling,
the intertube distance remains essentially unchanged and thus the van der Waals
interaction among neighboring tubes is not activated in the pure compression
regime. With the help of both normal and lateral nonbonded interactions, the
coordinated deformation of subunit tubes increases the bending stiffness of the
cluster. With reference to Figure 6, with the assistance of the normal van der Waals interaction, the deformed shape of all
buckled subunit tubes is almost identical and therefore in order to overcome
the relative sliding between the neighboring tubes (mainly along the axial
direction), the indentation force must be increased. In other words, when a
SWCNT slides with respect to a nearby counterpart, any atom in the tube has to
overcome a *frictional* axial van der Waals force.

The strengthening
efficiency can still be computed by the same definition in Section 3.2, that
is, the increased buckling resistance divided by the *cost* of additional
reference tube number. Take the 7-tube (8,0) cluster as an example, the
increase of critical force per tube is 14% (from 7 nN to 8 nN), and therefore the
strengthening efficiency turns out to be 2.3% (= 14%/6). Following the same
procedure, the strengthening efficiency of all clusters examined above is found
to be in the range between 2.3% and 5%, which is less significant than that of
MWCNTs (18.7% 75%). The lower level of strengthening efficiency indicates that
the concentric assembly of SWCNTs into MWCNTs is able to provide higher promotion
of buckling resistance than the assembly leading to clusters, since the
effective area of the nonbonded interaction in a MWCNT is larger.

The intertube cohesive
forces can also explain some unique features in the indentation response of
SWCNT clusters. Compared with an isolated SWCNT and MWCNTs for which the indentation
force maintains almost a constant value during postbuckling region (c.f. Figures 1(a)
and 3), the SWCNT clusters are observed in Figure 6 to have a decreasing indentation force after buckle.
This phenomenon may be attributed to the different degrees of lateral
interactions between subunit tubes as the bending curvature is varied: the slide motion during each
loading step is slowed down with the increase of indentation depth and
curvature of subunit SWCNTs, leading to a gradual reduction of
intertube *friction* and thus smaller bending resistance.

In order to
investigate the influence of tube length on the strengthening effect of SWCNT
clusters, (8,0) SWCNTs with *L*/*R* = 12.2 are
compared with their counterparts discussed above (with doubled aspect ratio). As
shown in Figure 8, it is much harder to buckle the shorter specimens, which
matches well with the beam theory where shorter beams always have higher
buckling resistance than longer beams. Although the longer cluster should have
more atoms involved in interaction and therefore higher intertube cohesion is expected, it is observed from Figure 8 that the percentage of increase of critical force per
tube is nearly the same for both clusters. Here, the shorter specimens have
larger critical nominal strain and therefore larger axial slide motion is initiated
during the buckling process, which leverages the smaller *interaction area* and leads to almost the same strengthening efficiency when compared with the
longer cluster.

The effect of
chirality is also investigated. The armchair (5,5) tube with *L*/*R* = 25.7 is selected to compare with
zigzag (8,0) tube with similar aspect ratio. Nanoindentation responses of both
SWCNTs and their 7-tube bundles are shown in Figure 9. It is found that compared to the results for (8,0)
SWCNTs, the assembly of (5,5) SWCNTs into clusters does not significantly enhance
the buckling resistance per tube. This phenomenon indicates that allocation of
atoms in tube surface has a profound effect on the magnitude of *frictional* axial van der Waals force. This conclusion is also supported by MM simulations
showing that more force is required to slide a zigzag SWCNT along the axial
direction (with respect to a nearby identical neighbor) than an armchair SWCNT.

##### 4.3. Reverse Analysis to Deduce the Elastic Constants

A procedure similar to that outlined in Section 4.3 is used to derive CNT elastic stiffness *Et* based on the prebuckling
indentation response of zigzag SWCNT clusters. The pairwise C–C interaction
law, (5), holds for any chirality; nevertheless, analogous to (6) for
armchair CNT, a new geometrical relationship needs to be established for zigzag
tubes. For any *n*-tube zigzag SWCNT
cluster with chirality (*m*, 0), assuming
the top layers of subunit SWCNTs are aligned in the same plane, and denote the
separation between this plane and the indenter as *d*, one could obtain the force per tube as where *m* is
also the number of atoms per layer per tube, denotes the distance between the *j*th layer and the top layer, and *M* is the number of layers to be considered. The zigzag SWCNTs have two types
of layer separations in the axial direction as illustrated in Figure 4: one is associated with the inclined C–C bonds whoselength projection in the axial
direction is denoted as ,
and the other one is due to the vertical (axial) C–C bonds having the length of . Consequently, one can derive , , , , and so forth. Again, due
to the fading nature of van der Waals interaction, *M* = 4 is adequate to obtain a converged CNT cluster-indenter
relationship with reasonable accuracy. By the virtue of (7) and (10) with = 3.27 Å for zigzag tubes,
the constitutive relationship for SWCNT cluster can be established at small
strain: for the numerical example of 7-tube (8,0) cluster presented in Figure 6, after removing the variation of cluster-indenter van
der Waals distance, the curve can be predicted
in Figure 10. Again, the predicted relationship is almost linear
and in excellent agreement with that measured from MM simulation. Denote *k* as the slope of relationship, the
elastic stiffness of CNT can be extracted as For the present cluster, *k* = 100.63 N/m, *L* = 76.97 Å, and *R* = 3.16 Å, which
leads to *Et* = 390.29 Pa m (the same
result is obtained for clusters containing different number of tubes). This
value is very close to the result reported in Section 3.3 (*Et* = 391.35 Pa m) based on nanoindentaion experiment on the MWCNTs,
and also very close to those reported by Yakobson et al. and Pantano et al. [20, 33].

#### 5. Conclusion

In this study,
MM is used to simulate nanoindentation experiments on SWCNTs, MWCNTs, and SWCNT
clusters. In both cases, the buckling deformations are beam-like despite of
some of the very small overall aspect ratios of MWCNTs or CNT clusters; in
other words, while their subunits may individually snap buckle like a shell,
when assembled into a structure, even short MWCNTs or SWCNT clusters slide
buckle like beams due to the nonbonded reinforcement among subunits. In terms
of both critical buckling load and critical buckling strain, the assembled structures
(MWCNT and cluster) are found to have higher buckling resistance than their
subunit SWCNTs, thanks to the van der Waals interactions between either
neighboring layers or neighboring tubes. For MWCNTs, the normal (radial) van
der Waals interaction between neighboring layers significantly enhances the
bending stiffness; whereas for SWCNT clusters, the resistance to the sliding
motion between nearby SWCNTs (required for buckling) is dominated by the *frictional* axial van der Waals interaction—both mechanisms
give rise to higher load required for buckle. Consequently, both MWCNTs and
SWCNT clusters are found to be sturdier than their subunit components in
withstanding buckling, although the strengthening efficiency is significantly
higher for MWCNTs than clusters.

With the penetration
of indenter, both MWCNTs and SWCNT clusters composed of SWCNTs undergo three sequential
deformation regimes: uniaxial compression, slide buckling, and postbuckling. The
indentation responses in the first regime are utilized to deduce the elastic
constant *Et*. A nonlinear spring model
generalized from MM simulation is proposed to simulate the van der Waals
interaction between indenter and MWCNT/cluster. By utilizing the proposed
model, contribution of the van der Waals interaction is excluded from the
measured indentation force-depth curve, leading to a linear relationship
between the applied compression and intrinsic sample deformation. The obtained elastic
stiffnesses for both
MWCNT and SWCNT clusters are in good agreement with each other, as
well as with that in the literature, which demonstrates the potential of using
the proposed nonlinear spring model and algorithm to deduce elastic constants
of CNTs via nanoindentation experiment.

#### Acknowledgments

The work is supported in part by NSF CMS-0407743 and CMMI-0643726, and in part by the Department of Civil Engineering and Engineering Mechanics, Columbia University.

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#### Copyright

Copyright © 2008 Ling Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.