nanomechanics of carbon nanotubes and composites · nanomechanics of carbon nanotubes and...

en-la-rgee

dpererse-

ng

m-more0%ictwithinesiteloadbes.ngc-

-ex-do

de-artod-

Nanomechanics of carbon nanotubes and composites

Deepak Srivastava and Chenyu WeiComputational Nanotechnology, NASA Ames Research Center, Moffett Field, California 94035-1000;[email protected]

Kyeongjae ChoDepartment of Mechanical Engineering, Stanford University, Stanford, California 93045

Computer simulation and modeling results for the nanomechanics of carbon nanotubes andcarbon nanotube-polyethylene composite materials are described and compared with experi-mental observations. Young’s modulus of individual single-wall nanotubes is found to be inthe range of 1 TPa within the elastic limit. At room temperature and experimentally realizablestrain rates, the tubes typically yield at about 5–10% axial strain; bending and torsional stiff-ness and different mechanisms of plastic yielding of individual single-wall nanotubes are dis-cussed in detail. For nanotube-polyethylene composites, we find that thermal expansion anddiffusion coefficients increase significantly, over their bulk polyethylene values, above glasstransition temperature, and Young’s modulus of the composite is found to increase throughvan der Waals interaction. This review article cites 54 references.@DOI: 10.1115/1.1538625#

1 INTRODUCTION

Since the discovery of multi-wall carbon nanotubes in 1991by Iijima @1#, and subsequent synthesis of single-wall carbonnanotubes by others@2,3# there are numerous experimentaland theoretical studies of their electronic, chemical, and me-chanical properties. Chemical stability, diverse electronicproperties~ranging from 1eV band gap semiconductors tometals!, and predicted extreme strength of the nanotubeshave placed them as fundamental building blocks in the rap-idly growing field of nanotechnology. Diverse nanoscale de-vice concepts have been proposed to develop nanoscale elec-tronic devices, chemical sensors, and also high strengthnanotube composite materials with sensing and actuating ca-pacity. To realize the proposed devices and materials con-cepts, it is crucial to gain detailed understanding on the fun-damental limits of nanotubes’ diverse properties. Atomisticsimulations are a very promising approach to achieve thisgoal since one can investigate a large range of possibilitieswhich are often very difficult to access through experimentalstudies. The insights and detailed mechanistic understandingprovide valuable guiding principles to optimize and developnovel nanoscale devices and materials concepts.

In this review article, we focus our discussion on the me-chanical properties of carbon nanotubes in the context ofhigh-strength nanotube composite materials. For this pur-pose, it is crucial to gain a detailed understanding of thenanotubes’ intrinsic mechanical properties, as well as theirinteraction with the polymer matrix in nanotube-polymercomposite materials. First, we will examine the elastic and

failure properties of nanotubes to understand their fundamtal strength and stiffness behavior. Initial atomistic simutions of nanotube mechanics have predicted unusually laYoung’s moduli ~of up to 5TPa or 5 times larger than thmodulus of diamond! and elastic limits~of up to 20–30%strain before failure!. These predictions immediately raisethe intriguing possibility of applying the nanotubes as sustrong reinforcing fibers with orders of magnitude highstrength and stiffness than any other known material. Subquently, more accurate simulations employing tight-bindimolecular dynamics methods andab-initio density functionaltotal energy calculations involving realistic strain rate, teperature dependence, and nanotube sizes have providedrealistic values of 1TPa as Young’s modulus and 5–1elastic limit of the strain before failure. These values pred50 GPa as the nanotube strength, in good agreementrecent experimental observations. Second, we will examthe mechanical properties of nanotube-polymer compomaterials to understand the mechanisms of mechanicaltransfer between a polymer matrix and embedded nanotuThis research area is rapidly moving forward with excitipossibilities of designing and developing very small strutures~eg, MEMS devices! with tailored mechanical properties. Near-term practical applications of nanotubes arepected to emerge from the composite materials, as theynot require a precise control of nanotube positioning forvice applications. For the future development of smnanotube-polymer composite materials, computational m

ASME Reprint No AMR345 $16.00Appl Mech Rev vol 56, no 2, March 2003 21

Transmitted by Associate Editor AK Noor

© 2003 American Society of Mechanical Engineers5

s

n

T

u

boa

e

ongdno-

lso

bess ofali-

ng

tic-

lowsofheetorof

extc,

mi-

216 Srivastava et al: Nanomechanics of carbon nanotubes and composites Appl Mech Rev vol 56, no 2, March 2003

eling will play a catalytic role in facilitating and acceleratinthe design and fabrication of composite materials with seing, actuation, and self-healing capabilities.

2 CARBON NANOTUBES:STRUCTURE AND PROPERTIES

A single-wall carbon nanotube~SWNT! is best described aa rolled-up tubular shell of graphene sheet@Fig. 1a# @4,5#.The body of the tubular shell is mainly made of hexagorings ~in a sheet! of carbon atoms, where as the ends acapped by a dome-shaped half-fullerene molecules.natural curvature in the sidewalls is due to the rolling of tsheet into the tubular structure, whereas the curvature inend caps is due to the presence of topological~pentagonalring! defects in the otherwise hexagonal structure of thederlying lattice. The role of the pentagonal ring defect isgive a positive~convex! curvature to the surface, which helpin closing of the tube at the two ends. A multi-wall nanotu~MWNT! is a rolled-up stack of graphene sheets into ccentric SWNTs, with the ends again either capped by hfullerenes or kept open. A nomenclature~n,m! used to iden-tify each single-wall nanotube, in the literature, refersinteger indices of two graphene unit lattice vectors corsponding to the chiral vector of a nanotube@4#. Chiral vec-tors determine the directions along which the graphsheets are rolled to form tubular shell structures and perp

er-is

ais

ali-a

po-

ctsse-ionro-the

les,areauelas

eryno-

ili-h-usiesghttingouldus

n

gns-

alrehe

hethe

n-tosen-lf-

tore-

neen-

dicular to the tube axis vectors, as explained in Ref.@4#. Thenanotubes of type~n, n!, as shown in Fig. 1b, are commonlycalled armchair nanotubes because of the\ / \ / shape,perpendicular to the tube axis, and have a symmetry althe axis with a short unit cell~0.25 nm! that can be repeateto make the entire section of a long nanotube. Other natubes of type~n, 0! are known as zigzag nanotubes~Fig. 1c!because of the /\/\/ shape perpendicular to the axis, and ahave a short unit cell~0.43 nm! along the axis. All the re-maining nanotubes are known as chiral or helical nanotuand have longer unit cell sizes along the tube axis. Detailthe symmetry properties of the nanotubes of different chirties are explained in detail in Refs.@4# and @5#.

The single- and multi-wall nanotubes are interestinanoscale materials for the following four reasons:1! Single- and multi-wall nanotubes have very good elas

mechanical properties because the two-dimensional~2D!arrangement of carbon atoms in a graphene sheet allarge out-of-plane distortions, while the strengthcarbon-carbon in-plane bonds keeps the graphene sexceptionally strong against any in-plane distortionfracture. These structural and material characteristicsnanotubes point towards their possible use in making ngeneration of extremely lightweight, but highly elastiand very strong composite materials.

2! A single-wall nanotube can be either conducting or seconducting, depending on its chiral vector~n, m!, where nand m are two integers. The rule is that when the diffence n-m is a multiple of three, a conducting nanotubeobtained. If the difference is not a multiple of three,semiconducting nanotube is obtained. In addition, italso possible to connect nanotubes with different chirties creating nanotube hetero-junctions, which can formvariety of nanoscale molecular electronic device comnents.

3! Nanotubes, by structure, are high aspect-ratio objewith good electronic and mechanical properties. Conquently, the applications of nanotubes in field-emissdisplays or scanning probe microscopic tips for metlogical purposes, have started to materialize even incommercial sector.

4! Since nanotubes are hollow, tubular, caged molecuthey have been proposed as lightweight large surfacepacking material for gas-storage and hydrocarbon fstorage devices, and gas or liquid filtration devices,well as nanoscale containers for molecular drug-delivand casting structures for making nanowires and nacapsulates.

A broad interest in nanotubes derives from the possibties of a variety of applications in all of the above four tecnologically interesting areas. In this review, we mainly focon the exceptionally stiff and strong mechanical propertthat can be used for making future generation of lightweistructural composite materials. The other three intereselectrical, surface area, and aspect-ratio characteristics cbe used to impart specific functional behavior to the thprepared composite materials.

ersg

Fig. 1 a) A graphene sheet made of C atoms placed at the corof hexagons forming the lattice with arrows AA and ZZ denotinthe rolling direction of the sheet to makeb) an ~5,5! armchair and~c! a ~10,0! zigzag nanotubes, respectively.

i

r

r

c

l

lo

y

ojf

a

a

s

na

rend

icaltionns,

-

a-llowtheap-e-

at-

ndof

an-ur-

silyi-ral

ide-

ll as

findder

ms

ip-tiesly

the

s am

s an-c-

eion

Appl Mech Rev vol 56, no 2, March 2003 Srivastava et al: Nanomechanics of carbon nanotubes and composites 217

3 SIMULATION TECHNIQUES

In earlier days, the structural, mechanical, and thermal prerties of interacting, bulk condensed matter systems wstudied with analytical approximation methods for infinsystems. Numerical computer simulations of the finsample systems have become more common recentlycause powerful computers to simulate nanoscale systemfull complexity are readily available. Atomistic moleculadynamics~MD! refers most commonly to the situation whethe motion of atoms or molecules is treated in approximfinite difference equations of Newtonian mechanics. Excwhen dealing with very light atoms and very low tempetures, the use of the classical MD method is well justifieIn MD simulation, the dynamic evolution of the systemgoverned by Newton’s classical equation of motiod2RI /dt25FI52dV/dRI , which is derived from theclassical Hamiltonian of the system, H5SPI

2/2MI

1V( $RI%). The atomic forces are derived as analytic derivtives of the interaction energy functions,FI($RI%)52dV/dRI , and are used to construct Newton’s classiequations of motion which are second-order ordinary diffential equations. In its global structure, a general MD cotypically implements an algorithm to find a numerical sotion of a set of coupled first-order ordinary differential equtions given by the Hamiltonian formulation of Newton’s seond law @6#. The equations of motion are numericalintegrated forward in finite time steps using a predictcorrector method.

The MD code for carbon based systems involves analmany-body force field functions such as Tersoff-Brenner@7#potentials for C-C and C-H interactions@8#. The Tersoff-Brenner potential is specially suited for carbon-based stems, such as diamond, graphite, fullerenes, and nanotuand has been used in a wide variety of scenarios with resin agreement with experimental observations. Currenthere is no universal analytic many-body force field functithat works for all materials and in all scenarios. A madistinguishing feature of the Tersoff-Brenner potentialcarbon-based systems is that short-range bonded interacare reactive, so that chemical bonds can form and breaking the course of a simulation. Therefore, compared to soother molecular dynamics codes, the neighbor list describthe environment of each atom includes only a few atomsneeds to be updated more frequently. The computationalof the many-body bonded interactions is relatively high copared to the cost of similar methods with non-reactive intactions with simpler functional forms. As a result, the overcomputational costs of both short-range interactionslong-range, non-bonding van der Waals~Lennard-Jones6–12! interactions are roughly comparable. For large-scatomistic modeling (105– 108 atoms), multiple processorare used for MD simulations, and the MD codes are genally parallelized@9#.

In recent years, several more accurate quantum molecdynamics schemes have been developed in which the fobetween atoms are computed at each time step via quamechanical calculations within the Born-Oppenheimer

op-ereteitebe-s inreateepta-d.isn,

a-

aler-deu-a-c-yr-

tic

ys-bes,ultstly,n

orortionsdur-meingnd

costm-er-allnd

ale

er-

ularrcestump-

proximation. The dynamic motions for ionic positions astill governed by Newtonian or Hamiltonian mechanics adescribed by molecular dynamics.

In the most general approach of full quantum mechandescription of materials, atoms are described as a collecof quantum mechanical particles, nuclei, and electrogoverned by the Schro¨dinger equation, HF@$RI ,r%#5EtotF@$RI ,r%#, with the full quantum many-body Hamiltonian operator H5SPI

2/2MI1SZIZJe2/RIJ1Sp2/2me

1Se2/r2SZIe2/uRI2r u, whereRI andr are nuclei and elec-

tron coordinates. Using the Born-Oppenheimer approximtion, the electronic degrees of freedom are assumed to foadiabatically the corresponding nuclear positions, andnuclei coordinates become classical variables. With thisproximation, the full quantum many-body problem is rduced to a quantum many-electron problemH@RI#C@r #5EelC@r #, whereH5SPI

2/2MI1H@RI#.In the intermediate regimes, for up to few thousand

oms, the tight-binding@10# molecular dynamics~TBMD! ap-proach provides very good accuracy for both structural amechanical characteristics. The computational efficiencythe tight-binding method derives from the fact that the qutum Hamiltonian of the system can be parameterized. Fthermore, the electronic structure information can be eaextracted from the tight-binding Hamiltonian, which in addtion also contains the effects of angular forces in a natuway. In a generalizednon-orthogonaltight-binding molecu-lar dynamics ~TBMD! scheme, Menon and Subbaswamhave used a minimal number of adjustable parameters tovelop a transferable scheme applicable to clusters as webulk systems containing Si, C, B, N, and H@11,12#. Themain advantage of this approach is that it can be used toan energy-minimized structure of a nanoscale system unconsideration without symmetry constraints.

Additionally, theab-initio or first principles method is asimulation method to directly solve the complex quantumany-body Schro¨dinger equation using numerical algorithm@13#. An ab-initio method provides a more accurate descrtion of quantum mechanical behavior of materials propereven though the system size is currently limited to onabout few hundred atoms. Currentab-initio simulation meth-ods are based on a rigorous mathematical foundation ofdensity functional theory~DFT! @14,15#. This is derived fromthe fact that the ground state total electronic energy ifunctional of the density of the system. Kohn and Sha@14,15# have shown that the DFT can be reformulated asingle-electron problem with self-consistent effective potetial, including all the exchange-correlation effects of eletronics interactions:

H15p2/2me1VH~r !1XXC@r~r !#1Vion-el~r !,

H1c~r !5«c~r !, for all atoms

r~r !5Suc~r !u2.

This single-electron Schro¨dinger equation is known as thKohn-Sham equation, and the local density approximat

dvn

t

rttro

hh

nsa

in

,

eo

uchiffertor-ith

tu-a

ongs ofneth,aneulary the

ls

ced

stedth is.

-toall

.

a,ent

nalbeite,

all

ng’sndr in

ioas-enesed

ry

ofom-

of-


~LDA ! has been introduced to approximate the unknownfective exchange-correlation potential. This DFT-LDmethod has been very successful in predicting materproperties without using any experimental inputs other ththe identity of the constituent atoms.

For practical applications, the DFT-LDA method has beimplemented with a pseudopotential approximation anplane wave~PW! basis expansion of single-electron wafunctions @13#. These approximations reduce the electrostructure problem to a self-consistent matrix diagonalizatproblem. One of the popular DFT simulation programs isVienna Ab intio Simulation Package~VASP!, which is avail-able through a license agreement~VASP! @16#.

The three simulation methods described above each htheir own advantages and are suitable for studies for a vaof properties of material systems. MD simulations haveleast computational cost, followed by Tight Bending meods. Ab initio methods are the most costly among the thMD simulations can study systems with up to millionsatoms. With well-fitted empirical potentials, MD simulationare quite suitable for studies of dynamical propertieslarge-scale systems, where the detailed electronic propeof systems are not always necessary. While DFT methcan provide highly accurate, self-consistent electronic strtures, the high computational cost limits them to systemsto hundreds of atoms currently. To take the full capacityDFT methods, a careful choice of an appropriate sized stem is recommended. Tight Binding methods lay in betweMD simulations and DFT methods, as to the computatiocost and accuracy, and are applicable for systems up to tsands of atoms. For moderate-sized systems, TB metcan provide quite accurate mechanical and electronic chateristics.

For computational nanomechanics of nanotubes, all thsimulation methods can be used in a complementary mato improve the computational accuracy and efficiency. Baon experimental observations or theoretical dynamicstructure simulations, the atomic structure of a nanosyscan first be investigated. After the nanoscale system confirations have been finalized, the functional behaviors ofsystem are investigated through staticab-initio electronic en-ergy minimization schemes. We have covered this in detaa recent review article focusing exclusively on computationanotechnology@17#.

In the following, we describe nanomechanics of nantubes and nanotube-polymer composites, and comparesimulation results, where ever possible, with experimenobservations.

4 MODULUS OF NANOTUBES

The modulus of the nanotube is a measure of the stiffnagainst small axial stretching and compression strainswell as non-axial bending and torsion strains on the natubes. The simulation results mainly pertain to the stiffnof SWNTs, where as most of the experimental observatiavailable so far are either on MWNTs or ropes/bundlesnanotubes. For axial strains, SWNTs are expected tostiffer than the MWNTs because of smaller radii of curvatu

ef-Aialsan

ena

eic

ionhe

aveietyheh-ee.fsof

rtiesodsuc-upofys-en

nalou-odsrac-

reenerednd

temgu-the

l inal

o-the

tal

essas

no-ssnsofbe

re

and relatively defect free structure. For non-axial strains sas bending and torsion, the MWNTs are expected to be stthan the SWNTs. In this section, the axial, bending, andsion moduli of SWNTs are described and compared wexperimental observations known so far.

4.1 Young’s modulus for axial deformations

As described above, single-wall carbon nanotubes havebular structure that can be conceptualized by takinggraphene sheet, made of C atoms, and rolling into a ltubular shape. Contributions to the strength and stiffnesSWNTs come mainly from the strength of graphene in-placovalent C—C bonds. It is expected that modulus, strengand stiffness of SWNTs should be comparable to the in-plmodulus and strength of the graphene sheet. In the tubshape, however, the elastic strain energies are affected bintrinsic curvature of C—C bonds. Robertsonet al @18# havefound ~using both Tersoff and Tersoff-Brenner potentia!that the elastic energy of a SWNT scales as 1/R2 , where R isthe radius of the tube. This is similar to the results dedufrom continuum elastic theory@19#. The elastic energy ofCNTs responding to a tensile stress in their study suggethat SWNTs are very strong materials and that the strengmainly due to the strong C—C sp2 bonds on the nanotubeThe Young’s modulus of a SWNT isY5 (1/V)]2E/]«2 ,where E is the strain energy andV is the volume of thenanotube.

Initial computational studies@20#, using the same TersoffBrenner potential, reported the value of Young’s modulusbe as high as 5.5TPa. This was mainly due to a very smvalue of wall thickness~h;0.06 nm! used in these studiesUsing an empirical force constant model, Lu@21# found thatthe Young’s modulus of a SWNT is approximately 970GPwhich is close to that of a graphite plane, and is independof tube diameter and chirality. Rubioet al @22# used a betterdescription for interatomic forces through a non-orthogotight-binding method and found the Young’s modulus toapproximately 1.2TPa, which is larger than that of graphand is slightly dependent on the tube size especially for smdiameter nanotubes~D,1.2 nm!. High surface curvature forsmall diameter nanotubes tends to decrease the Youmodulus. The Young’s modulus of a variety of carbon aalso non-carbon nanotubes as a function of tube diameteRabioet al’s studies is shown in Fig. 2. In both Lu and Rabet al’s studies, the thickness of the nanotube wall wassumed to be 0.34 nm~the van der Waals distances betwegraphite planes!, and the computed modulus is within thrange of experimental observations, as will be discusbelow.

Later on, using the ab initio density functional theo~with pseudopotentials!, Rubioet al @23# have found that thestiffness of SWNTs is close to that of the in-plane stiffnessgraphite, and SWNTs made of carbon are strongest as cpared with other non-carbon, such as boron-nitride~BN! orBxCyNz, nanotubes known so far. The Young’s modulusMulti Wall CNTs ~MWCNTs! and SWCNT ropes as a function of tube diameters from Rabioet al’s study is shown inFig. 3.

y as to

f

andal-rynt

s aso-2,

alltionree-nd-a


The calculated Young’s modulus from the above variousmethods is summarized in Table 1. The different results fromvarious theoretical studies arise from the use of differentdefinitions of carbon nanotube wall thickness, or differentatomic interaction potentials. It has been suggested that byinvestigating the value]2E/]«2 instead of Young’s modulus,the ambiguity of thickness of the CNT wall can be avoided.

Using a non-orthogonal tight-binding molecular dynamicsmethod and the DFT method, we recently carried out@24#axial compression of single-wall C and BN nanotubes andhave found Young’s modulus to be about 1.2TPa, we alsofound that the modulus of a similar BN nanotube is about80% of that of the carbon nanotube. These results are inqualitative and quantitative agreement with Rubio’s DFT re-sults and the general experimental observations known sofar.

There have been attempts to use continuum theory to de-scribe the elastic properties of carbon nanotubes. The valueof the wall thickness of CNTs is usually required when ap-plying a continuum elastic model, and it is not immediately

clear how to define wall-thickness for syssingle layer of atoms form the wall. Thereuse the value of]2E/]«2 instead of Youavoid the ambiguity in the definition ofnanotubes. Some recent studies by Ru@25# apendent variable as bending stiffnessavoid the parameter corresponding to thtogether. However, in the description ofbased models, the definition of wall thicand needs to be consistent with the expwell as those obtained from atomistic olecular dynamics simulations. Accordingthe value of 0.34 nm for the wall thicknecarbon nanotube gives experimental andbased results of Young’s modulus thatment with each other. This means that 0ing to the van der Waal radius of a si

Table 1. Summary of Young’s modulus from various theoretical calculations used in different studies„where Y is Young’s modulus andh is the wall thickness…

Method

Molecular dynamicsTersoff-Brenner forcefield †20‡

Empirical forceconstant model†21‡

Non-orthogonaltight-binding †22‡ Ab initio DFT †23‡

h 0.06 nm 0.34 nm 0.34 nm 0.34 nmY SWCNT: 5.5 TPa SWCNT: 0.97 TPa SWCNT: 1.2 TPa SWCNT rope:

0.8 TPa; MWCNT:0.95 TPa

Table 2. Summary of Young’s modulus from various experimental studies„with definitions of Y and hsame as in Table 1…

MethodThermal vibrations

†28‡Restoring forceof bending †29‡

Thermalvibrations†30‡

Deflection forces†31‡

h 0.34 nm 0.34 nm 0.34 nm 0.34 nmY MWCNT: 1.861.4 TPa MWCNT: 1.2860.59 TPa SWCNT: 1.7 TPa SWCNT: 1.0 TPa

la-lidn.lso

tems where onlare suggestion

ng’s modulus towall thickness olso use an inde-for nanotubese wall thicknesscontinuum theo

kness is importaerimental result

r tight-binding mto Tables 1 andss of a single-watomistic simulaare in broad ag.34 nm, correspongle C atom, is

C,

Fig. 3 Young’s modulus verse tube diameter for ab initio simution. Open symbols for the multiwall CNT geometry and sosymbols for the single wall tube with crystalline-rope configuratioThe experimental value of the elastic modulus of graphite is ashown@23#.

Fig. 2 Young’s modulus as a function of the tube diameter forBN, BC3, BC2N from tight binding simulation@22#.

t

s

i

onofdi-sestomsin

g

ngin

e

ys


reasonable assumption for the wall thickness for evaluaof Young’s modulus. A recent work by Harik@26# examinesthe validity of using continuum elastic theory for nanotuband suggests that small and large diameter nanotubes nebe described with different approaches, such as a solidmodel for small diameter nanotubes and a shell modellarge diameter nanotubes. This is also consistent with u0.34 nm for wall thickness, because thin nanotubes wabout 0.7 nm diameter would behave as a solid rod andas a cylindrical shell, as was assumed earlier.

4.2 Bending stiffness and modulus

Besides axial strains discussed above, SWNTs havebeen subjected to bending and torsional strains. The benstiffness of a SWNT is (1/L)d2E/dC2 , whereE is the totalstrain energy,L is the length, andC is the curvature of thebent nanotube, which is related with the bending angleu asC5 u/L . From the elastic theory of bending of beams, tstrain energy of a bent nanotube can be expressed aE50.5YhLrt2C2dl, whereY is the Young’s modulus of theSWNT, andh is the thickness of the wall@27#. The integral istaken around the circumference of the nanotube, andt is thedistance of atoms from the central line~or the bent axis! ofthe tube. From this expression, the bending stiffnessK isfound to be equal toYh(pr 3), and scales as the cube of thradius of the tube. Results from molecular dynamics simutions with the Tersoff-Brenner potential show that the stness K scales asR2.93 ~Fig. 4!, which is in good agreemenwith scaling predicted by the continuum elastic theory. Tcorresponding bending Young’s modulus (YB) of SWNTswith varied diameters can be calculated from the above eqtion. For a small diameter SWNTYB is found to be about 0.9TPa, smaller than the stretching Young’s modulus calculafrom the tight-binding method or first principle theory. Th

lid-

ion

ion

esed torodforingithnot

alsoding

hes

ela-ff-the

ua-

tede

computed smaller value is also similar to what Robertset al @18# showed in their study of the elastic energySWNTs. The qualitative agreement is rather good. An adtional feature is that the bending Young’s modulus decreawith the increase in tube diameter. This is mainly duemore favorable out-of-plane displacements of carbon atoon a larger diameter tube, resulting in flattening of the tubethe middle section.

Poncharalet al @28# experimentally studied the bendinYoung’s modulus of MWNTs~diameter.10 nm! using elec-trically induced force and have found that the bendiYoung’s modulus is decreased sharply with the increasetube diameter~Fig. 5a!, which they have attributed to thwave-like distortion of the MWNT, as shown in Fig. 5b.

At a large bending angle, SWNTs can buckle sidewa

Das

rom

Fig. 5 a) Bending modulus as a function of tube diameter. Socircles are from Poncharalet al @28#; others are from other experiments as referred in Poncharalet al’s paper. The dropping in thebending modulus is attributed to the onset of a wavelike distortin lateral direction as shown inb). b) High–resolution TEM imageof a bent nanotube~Radius of curvature'400 nm), showing thewavelike distortion and the magnified views@28#.

Fig. 4 Bending stiffness as a function of tube diameter from Msimulation with Tersoff-Brenner potential. The stiffness is scaledD2.93, closed to the cubic dependence on diameter D predicted fcontinuum elastic theory.

ni

h

d

n

a.ods

uc-

ibithend,

oc-, orsticm-eld-etich aininthe

les

foren-er-ntssed

hin

col-

-ck-sse


similarly to a macroscopic rod, at which non-uniform strais induced. Shown in Fig. 6a and b are the images from ahigh-resolution electron microscope@29# of the buckling ofbent SWNT and MWCNT. Such buckling at large bendiangle is closely related with buckling of a SWNT under axcompression stress, which will be discussed later.

4.3 Torsion stiffness and modulus

The torsion stiffness of a CNT isK5 (1/L)d2E/du2 , whereE is the total strain energy andu is the torsion angle. Theshear strain is related with torsion angle as«5 Ru/L , whereR is the radius of tube, andL is its length. From continuumelastic theory, the total strain energy of a cylinder canwritten asE5 1

2G***«2dV, whereG is shear modulus ofthe tube. The torsion stiffness thus is related withG as K5 (1/L)d2E/du2 5G(2ph) R3/L2 , whereh is the thicknessof the wall of the nanotube. In Fig. 7, we show our recencomputed values of the torsion stiffness of several armcand zigzag carbon nanotubes using the Tersoff-Brennertential. The dependence of the torsion stiffness on the raof tube is found to be asK}R3.01 ~for tube diameter.0.8 nm!. This is in excellent agreement with the predictioof cubic dependence from the continuum elastic theory.

The shear modulus of CNTs is found to be around 0.3 Tand is not strongly dependent on diameters~for D.0.8 nm). This value is smaller than that of about 0.45 Tin Lu’s study @20#, calculated with an empirical force constant model. For small diameter tubes, such as a~5,5! nano-tube, the shear modulus deviates away from the continuelastic theory description.

4.4 Experimental status: Modulus of carbon nanotubes

The high stiffness of carbon nanotubes has been verifiedseveral experiments. Treacyet al @30# studied Young’smodulus of MWNTs by measuring the thermal vibrationand Y was found to be about~1.861.4! TPa. Later researchby Wonget al @31# on MWCNT found the Young’s modulusto be about~1.2860.59! TPa by measuring the restorinforce of bent nanotubes. Krishnanet al @32# conducted astudy on the stiffness of single-walled nanotubes and hfound the average Young’s modulus to be approximat1.25 TPa. This is close to the experimental results obtai

in

gal

be

tlyairpo-ius

n

Pa

Pa-

um

by

s,

g

aveelyed

by Salvetatet al @33#, where Y was found to be about 1 TPThe measured Young’s modulus from these various methis summarized in Table 2.

5 PLASTIC DEFORMATIONAND YIELDING OF NANOTUBES

Nanotubes under large strain go through two kinds of strtural changes. First, early simulations by Yakobsonet al @20#showed that under axial compression, nanotubes exhstructural instabilities resulting in sideways buckling, but tdeformed structure remains within the elastic limit. Secounder large strains, bonding rearrangements or transitionscur giving rise to permanent damage, plastic deformationyielding of nanotubes. In this section, we discuss the pladeformation and failure of nanotubes under large axial copression and tensile strains. Critical dependence of the yiing of nanotubes on the applied strain-rate and the kintemperature of the simulation are also discussed througmodel that shows that nanotubes may typically yield with5–10% tensile strain at room temperature. This finding isgood agreement with the experimental observations onbreaking and yielding of MWNTs and the ropes or bundof SWNTs.

5.1 Plastic deformation under compressive strain

Yakobsonet al @20# found that SWNTs form non-uniformfin-like structures under large compressive strain~Fig. 8!.The sideways displacement or buckling of tubes occurs,larger strain, and contributes towards the relief of strainergy from the fin-like structures, but the tubes remain supelastic for more than 20% compressive strains. Experimehave observed a sideways buckling feature in compresmulti-wall nanotubes in polymer composite materials@34#.Another mode of plastic deformation of compressed tnanotubes is also observed in the same experiments@34#; ie,the tubes remain essentially straight, but the structurelapses locally as shown in Fig. 9.

Srivastavaet al @35# used a tight-binding molecular dynamics method and have found that within the Euler buling length limitation, an~8,0! carbon nanotube collapselocally at 12% compressive strain. The local plastic collapis due to a graphitic (sp2) to diamond-like (sp3) bonding

Shown
Fig. 6 HREM images of kink structures formed in bent CNTs. Shown on left is a single kink in a SWCNT with diameter 1.2nm;on right is a kink on a MWCNT with diameter 8 nm@29#.

c

o

o


transition at the location of the collapse and the releaseexcess strain in the remaining uncollapsed section. Theleased strain in the uncollapsed section drives the locallapse with a compressive pressure as high as 150 GPa alocation of the collapse~Fig. 10!.

Srivastavaet al @24# have also studied the influencechanges in the chemical nature on the nanomechanicsthe plasticity of nanotubes. For example, this is doneconsidering the structure, stiffness, and plasticity of bornitride ~BN! nanotubes. The results for Young’s modulusBN nanotubes have been discussed above. It turns outBN nanotubes are only slightly less stiff~80–90%! as com-pared to their carbon equivalent. The tight-binding MD a

o

ing

ed

ofre-ol-t the

fandbyn-

ofthat

nd

Fig. 8 a) The strain energy of a compressed 6 nm long~7,7! CNT,from Tersoff-Brenner potential, has four singularities correspondto the buckled structures with shapes shown inb to e. The CNT iselastic up to 15% compression strain despite of the highly deformstructures. The MD study was conducted at T50 K @20#.

r aff-

Fig. 7 The torsion stiffness as a function of tube diameter foseries of zigzag and armchair SWNTs calculated with TersBrenner potential. The stiffness is scaled asD3.01 for D.0.8 nm, inagreement with the prediction from continuum elastic theory.

s shown
Fig. 9 TEM image of fractured multiwalled carbon nanotubes under compression within a polymeric film. The enlarged image ion right @52#.

wB

d

no-of

-e-the


ab-initio total energy simulations further show that, dueBN bond rotation effect, BN nanotubes show anisotropicsponse to axial strains. For example, Fig. 11 shows sponeous anisotropic plastic collapse of a BN nanotube thatbeen compressed at both ends, but strain release is shobe more favorable towards nitride atoms in the rotatedbonds. This results in the anisotropic buckling of the tutowards one end when uniformly compressed at both en

ion.ctsno-

tore-nta-hasn toN

bes.

5.2 Plastic deformation under tensile strain

For the case of tensile strain, Nardelliet al @36,37# havestudied the formation of Stone-Wales~SW! bond rotationinduced defects as causing the plastic deformation of natubes. This mechanism is explained by formationheptagon-pentagon pair~5775! defects in the walls of nanotubes~Fig. 12!. The formation energy of such defects is dcreased with the applied strain and is also dependent ondiameter and chirality of the nanotube under consideratAt high temperatures, plastic flow of the thus-formed defeoccurs, and that can even change the chirality of the na

25%r

-ube,e-oss

rm-ano-

Fig. 10 Shown on top froma to d are four stages of spontaneouplastic collapse of the 12% compressed~8,0! CNT, with diamondlike structures formed at the location of the collapse@35#.

s

Fig. 11 Five stages of spontaneous plastic collapse of the 14.compressed~8,0! BN nanotube.a! Nucleation of deformations neathe two ends,b-d anisotropic strain release in the central compressed section and plastic buckling near the right end of the tande) the final anisotropically buckled structure where all the dformation is transferred toward the right end of the tube. The crsection of each structure is shown on right.

Fig. 12 The Stone Wales bond rotation on a zigzag and an achair CNT, resulting pentagon-heptagon pairs, can lengthen a ntube, with the greatest lengthening for an armchair tube@38#.

h

t

t

s

tlyto

areena,rtedso

orypen-c-a

as

si-

re,rrier


tube ~Fig. 13!. On further stretch, the plastic flow and increased formations of more such defects continue until neing and breaking of the nanotube occurs. Zhanget al @38#have also examined the plastic deformations of SWNTsduced by the Stone-Wales dislocations under tensile strand they have found that the SW defects can releasestrain energy in the system. Zhanget al note that SW defectsform more favorably on an armchair nanotube than onzigzag nanotube because the rotation of the C—C bond cancompensate for more tensile strain along the axis informer case.

5.3 Strain-rate and temperature dependenceof yielding of nanotubes

In all the simulations and discussions so far, no dependeof the failure or yielding of nanotubes on the rate of tapplied strain or the temperature of the system has been mtioned. This is because all of the above results were eiobtained with theab-initio or tight binding static total energycalculations or with MD simulations at much higher strarates~than ever possible in experiments!, where barriers tocollapse would beartificially higher. In reality, it is expectedthat barrier to collapse and yielding strain at experimentarealizable strain rates and at room temperatures maylower than the simulation values reported so far. The yieldor failure of SWNTs in different scenarios depends onformation of defects discussed above. Classical MD simutions report values@39# as high as 30% yielding strain undetensile stretch and above 20% under compression@20#. Dueto the limitations in the time scales of the phenomenon tcan be simulated with MD, the nanotubes were typicastrained at 1/ns at 300–600 K. The experiments so far sgest much lower yielding strains for nanotubes. Walterset al@40# have studied the SWNT rope under large tensile strand observed the maximum strain to be 5.860.9% beforeyielding occurs. Yuet al @41# have found similar breakingstrain ~5.3% or lower! for different SWNT rope sample~Fig. 14a!. Similar measurements on MWNTs by Yuet al@42# show breaking strain to be about 12% or lower~the

theheak-

usund

-ck-

in-ain,the

a

the

nceeen-

her

in

llybe

inghela-r

hatllyug-

ain

lowest one is 2% in their experiments! ~Fig. 14b!. The re-ported lower yielding strains in experiments could be pardue to tube defects in the SWNT ropes, or could be duethe much~orders of magnitude! lower rates at which straincan be applied in experiments.

The breaking, collapse, or yielding of the nanotubesclearly a temperature and strain-rate-dependent phenomand a model needs to be developed to relate the repomuch higher yielding strain from simulation studies to thefar observed lower yielding strain in experiments. Weiet al@43,44# have recently developed a transition-state thebased model for deducing strain rate and temperature dedence of the yielding strain as simulated in MD studies. Acording to the Arrhenius formula, the transition time forsystem to go from the pre-yielding state to another~postyielding! state is dependent on the temperaturest5 (1/n)eEn /KT, whereEn is the activation energy andn isthe effective vibration frequency or attempts for the trantion. For a system with strain«, the activation barrier islowered asEn5En

02kV«, wherek is the force constant, andV is the activation volume. At higher temperatures, therefoa system has larger kinetic energy to overcome the ba

sile

the

Fig. 14 Top: Eight stress versus strain curves obtained fromtensile-loading experiments on individual SWCNT ropes. TYoung’s modulus is ranged from 320GPa to 1470 GPa. The breing strain was found at 5.3% or lower@41#. Bottom: Plot of stressversus strain curves for individual MWCNTs. The Young’s modulis ranged form 270GPa to 950 GPa, with breaking strain aro12% ~one sample showed a 3% breaking strain! @42#.

Fig. 13 A heptagon-pentagon pair appeared on a 10% tenstrain ~10,10! CNT at T52000 K. Plastic flow behavior of thePentagon-heptagon pairs after 2.5ns at T53000 K on a 3% tensilestrained CNT. The shaded region indicates the migration path of~5-7! edge dislocation@36#.

,

be,end

the

gnceof

ove

va-


between the pre-yielding and post-yielding states, andtransition time is shortened. Similarly, the lower strain rateeach step allows the system to find an alternative minimenergy path, thus again lowering the effective barrier heiseparating the pre- and post-yielding states.

For example, yielding strain of a 6nm long~10,0! SWNTat several temperature and strain rate varying betw1023/ps to 1025/ps, is shown in Fig. 15. Yielding strainsare found to be 15% at low temperature and 5% at htemperature at about 1025/ps strain rate. Stone-Wales rotations are found to first appear before necking, resultingheptagon and pentagon pairs, which provide the coresformations of larger rings, and further resulting in the breaings of the nanotubes~Fig. 16!. Detailed analysis@43,44#shows that the complex dependence on the temperature

nghein

ibedha-of

noc-

l in-othighsand

theat

umght

een

igh-infork-

and

the strain rate with transition state theory~TST! can be ex-pressed as«Y5En /YV1(KT/YV)ln(N«/nsite«0), where En

is the averaged barrier for the yielding initiating defect,N isthe number of processes involved in the breaking of the tuY is Young’s modulus,nsite is the number of sites availablfor yielding, which is dependent on the structural details, a«0 is a constant related to the vibration frequency of C—Cbonds. For a more realistic strain rate such as 1%/hr,yielding strain of the 6nm long~10,0! CNT can be estimatedto be around 11%. A longer CNT will have a smaller yieldinstrain, as more sites are available for defects. The differebetween the yielding strain of a nanometer long CNT anda micron long CNT can be around 2% according to the abexpression for the yielding strain@43,44#. The advantage ofsuch a model is that one could directly compute the actition energy for yielding defect formation and get the yieldistrain from the developed model. Within error bars on tknown activation energies computed so far, our model isvery good agreement with experimental observations.

On the other hand, under compressive strain, as descrabove, Srivastavaet al @35# showed that CNT collapses witthe graphitic to diamond-like bonding transition at the loction of the collapse. Another observation is the formationnon-uniform fin-like structures by Yakobsonet al @20# thatleads to sideways Euler buckling of the tube, anddiamond-like bonds or defects would form within the struture. Recent MD studies at finite temperatures@45# give dif-ferent results. Using the same Tersoff-Brenner potentiaMD studies, Weiet al @45# show that, with thermal activation, nanotubes under compressive strain can form bdiamond-like bonds and SW-like dislocation defects at htemperatures~Fig. 17! @45#. Similar analysis of nanotubeunder compressive strain therefore, is more complicated

Fig. 15 The yielding strain of a 6nm long~10,0! CNT is plotted asfunctions of strain rate and temperature. Stone-Wales bond rotatappear first resulting in heptagon and pentagon ring; then largerings generated around such defects followed by the necking ofCNT; and the CNT is broken shortly after~from MD simulationswith Tersoff-Brenner potential!.

f the

ionsr Cthe

ing

00K,Fig. 17 A 12% compressed~10,0! CNT at T51600 K. A Stone-Wales dislocation defect can be seen at the upper section oCNT. Several sp3 bonds formed in the buckled region~from MDsimulation with Tersoff-Brenner potential!.

Fig. 16 Left: A 9% tensile strained~5,5! CNT with numerousStone-Wales bond rotation defects at 2400K, and the followbreaking of the tube. Right: An 11.5% tensile strained~10,0! with agroup of pentagon and heptagon centered by an octagon at 16and the following breaking of the tube~from MD simulation withTersoff-Brenner potential!.

ie

a

c

ag

i

i

l

r

ib

o

n

e

eh

tiesTwoitionto aec-, theasepure

her-to ame

em12

y--ere tomem-l-e its

iffu-ubebet-e.

f are-mether

ne-

raleneon


currently underway because sideways buckling can occurfore tube yields with SW dislocation or diamond-like defeformation.

6 STRUCTURE AND MECHANICSOF NANOTUBE COMPOSITES

As discussed above, the strong in-plane graphitic C—Cbonds make defect free SWNTs and MWNTs exceptionastrong and stiff against axial strains and very flexible aganon-axial strains. Additionally, nanotubes also have vgood electrical and thermal conduction capabilities. Maapplications, therefore, are proposed for nanotubes as ative fibers in lightweight multi-functional composite materals. Several recent experiments on the preparation and cacterization of nanotube-polymer composite materials halso appeared@46–48#. These measurements suggest modenhancement in strength characteristics of CNT-embedpolymer matrices as compared to the bare polymer matriVigolo et al @46# have been able to condense nanotubesthe flow of a polymer solution to form nanotube ribbonswell. These ribbons can be greatly bent without breakinghave Young’s modulus that is an order of magnitude larthan that of the bucky paper. In the following, we discustructural, thermal and mechanical implications of addSWNTs to polyethylene polymer samples.

6.1 Structural and thermal behaviorof nanotube-polyethylene composites

Thermal properties of polymeric materials are importafrom both processing and applications perspectives. Afunction of temperature, polymeric materials go throustructural transformation from solid to rubbery to liqustates. Many intermediate processing steps are done inliquid or rubber-like state before the materials are coodown to below glass transition temperature for the finaneeded structural application. Besides the melting proceshigh temperatureTm , like other solid materials, the structural and dynamic behavior of polymeric material above abelow glass transition temperatureTg is important to inves-tigate. BelowTg the conformations of polymer chains afrozen, when polymers are in a solid glassy state, andbetweenTg andTm polymers are in a rubber-like state witviscous behavior. Preliminary experimental and simulatstudies on the thermal properties of individual nanotushow very high thermal conductivity of SWNTs@49#. It isexpected, therefore, that nanotube reinforcement in pmeric materials may also significantly change the thermand structural properties as well.

Atomistic MD simulation studies of the thermal and strutural properties of a nanotube-polyethylene composites hbeen attempted recently@50#. Polyethylene is a linear chaimolecule withCH2 as the repeating unit in the chain. Thdensity as a function of temperature for a pure polyethylsystem~a short chain system with 10 repeating units in eapolymer, with 50 polyethylene chains in the simulatiosample! and a nanotube-polyethylene composite system wabout 8% volume ratio capped nanotubes in the mixturshown in Fig. 18. Both systems show discontinuity in t

be-ct

llynstrynyddi-i-har-ve

estdedes.in

asnder

ssng

nts aghdthe

edllys at-nd

ein

hones

ly-al

c-ave

enechnithise

slope of the density-temperature curve. The discontinuirepresent glass transition temperatures in the two cases.features are apparent in the figure. First, the glass transtemperature of the composite system has increasedhigher value than that of the pure polyethylene system. Sond, above glass transition temperature in both the casesdensity as a function of temperature in the composite cdecreases at a much faster rate than the decrease in thepolyethylene system case. This means that the volume tmal expansion coefficient of the composite has increasedlarger value above glass transition temperature. The voluthermal expansion coefficient for the composite systabove glass transition temperature is found to be31024 K21, which is 40% larger than that of the pure polethylene system aboveTg . The increase in the thermal expansion coefficient due to mixing of SWNTs in the polymsample is attributed to the increased excluded volume duthermal motions of the nanotubes in the sample. In the sasimulations, we also found that, above glass transition teperature, the self-diffusion coefficient of the polymer moecules in the composite increases as much as 30% abovpure polyethylene sample values. The increase in the dsion coefficient is larger along the axis of the added nanotfibers and could help during the processing steps due toter flow of the material above glass transition temperatur

6.2 Mechanical behavior ofnanotube-polyethylene composites

Using fibers to improve the mechanical performance ocomposite material is a very common practice, and thelated technology has been commercialized for quite sotime. Commonly used fibers are glass, carbon black, or o

Fig. 18 Density as a function of temperature for a polyethylesystem~50 chains withNp510), and a CNT-polyethylene composite ~2nm long capped~10,0! CNT! The CNT composite has anincrease of thermal expansion aboveTg . ~From a MD simulationwith Van der Waals potential between CNT and matrix. Dihedangle potential and torsion potential were used for the polyethylmatrix, and Tersoff-Brenner potential was used for carbon atomthe CNT.!

slh

luo

y

nr

t

f

i

te

p

oronso

bes, a

lingin-

fortrixberep-nd

mere-

aintalitheenrhatong-in-e-

e

ialstalical

ave

c

hat

ene

ite


ceramics. These not only can add structural strength toterials but also can add desired functionality in thermal aelectrical behavior. The structural strength characteristicsuch composite materials depend on the mechanicaltransfer from the matrix to the fiber and the yielding of tcoupling between the two. Mechanical load from matrixthe fibers in a composite is transferred through the coupbetween the two. In some cases, the coupling is throchemical interfacial bonds, which can be covalent or ncovalent in nature, while in other one the coupling couldpurely physical in nature through non-bonded van der Wa~VDW! interactions. Covalently coupled matrix and fibeare strongly interacting systems, while VDW coupled stems are weakly interacting systems, but occur in a wvariety of cases. The aspect ratio of fiber, which is definedL/D ~L is the length of the fiber, and D is the diameter!, isalso an important parameter for the efficiency of load trafers because the larger surface area of the fiber is bettelarger load transfer. It is expected, therefore, that the embded fibers would reach their maximum strength under tenload only when the aspect ratio is large. The limiting valuethe aspect ratio is found to be related to the interfacial shstresst as L/D.smax/2t, where smax is the maximumstrength of the fiber. Recent experiments on MWNTsSWNT ropes@41,42# have reported the strength of the nantubes to be in the range of 50GPa. With a typical value50MPa for the interfacial shear stress between the nanoand the polymer matrix, the limiting value of the aspect rais 500:1. Therefore, for an optimum load transfer withMWNT of 10nm diameter, the nanotube should be at le5mm long, which is in the range of the length of nanotubtypically investigated in experiments on nanotube reinforccomposites.

Earlier studies of the mechanical properties of the coposites with macroscopic fibers are usually based on ctinuum media theory. Young’s modulus of a compositeexpected to be within a lower bound of 1/Ycomp

5 Vf iber /Yf iber 1 Vmatrix /Ymatrix and an upper bound oYcomp5Vf iberYf iber1VmatrixYmatrix , where Vf iber andVmatrix are the volume ratios of the fiber and the matrrespectively. The upper bound obeys the linear mixing ruwhich is followed when the fibers are continuous andbonding between fibers and the matrix is perfect, ie, thebedded fibers are strained by the same amount as the mmolecules. The lower bound is reached for the case ofticulate filler composites because the aspect ratio is closone. For a nanotube fiber composite, therefore, an uplimit can be reached if the nanotubes are long enough andbonding with the matrix is perfect. Additionally, short nantubes, with Poisson’s ratio of about 0.1–0.2, are much hamaterial as compared to the polymer molecules with Pson’s ratio of about 0.44. Therefore, as a nanotube contaipolymer matrix is stretched under tensile strain there iresistance to the compression pressure perpendicularly taxis of the tube. For the short, but hard nanotubes andpolymer matrix mixture, this provides additional mechanisof load transfer that is not possible in other systems.

An MD simulation of the mechanical properties of a com

ma-ndof

oadetoingghn-

bealsrss-

ideas

s-for

ed-sileofear

oro-ofubetioa

astesed

m-on-is

x,le,hem-atrixar-

e toperthe-deris-ingathe

softm

-

posite sample was recently performed with short nanotuembedded in a short-chained polyethylene system at 50Ktemperature below glass transition temperature. The coupat the interface was through non-bonded van der Waalsteractions. Shown in Fig. 19 is the stress-strain curveboth the composite system and the pure polyethylene masystem. Young’s modulus of the composite is found to1900MPa, which is about 30% larger than that of the pupolymer matrix system. This enhancement is within the uper and lower bound limits discussed above. We have fouthat further enhancement of Young’s modulus of the sasample can be achieved by carrying the system throughpeated cycles of the loading-unloading of the tensile stron the composite matrix. In agreement with the experimenobservation, this tends to align the polymer molecules wthe nanotube fibers, causing a better load transfer betwthe two. Franklandet al @51# have studied the load transfebetween polymer matrix and SWNTs and have found tthere was no permanent stress transfer for 100nm l~10,10! CNTs within polyethylene if only van der Waals interaction is present. In the study, they estimated that theterfacial stress could be 70MPa with chemical bonding btween SWCNT and polymer matrix, while only 5MP for thnonbonding case.

6.3 Experimental status

Using nanotubes as reinforcing fibers in composite materis still a developing field from theoretical and experimenperspectives. Several experiments regarding the mechanproperties of nanotube-polymer composite materials hbeen reported recently. Wagneret al @52# experimentallystudied the fragmentation of MWNTs within thin polymerifilms ~urethane/diacrylate oligomer EBECRYL 4858! undercompressive and tensile strain. They have found tnanotube-polymer interfacial shear stresst is of the order of

Fig. 19 Plot of the stress versus strain curve for pure polyethylmatrix and CNT composite~8 vol%! at small strain region (T550 K). Young’s modulus is increased 30% for the compos~from MD simulation!.

i

g

a

f

ieN

-

no

se

e

an

b

u

ciu

on-tedain

ingu-

ndites

sticsthe

d forals.ta-ys-oly-

alepeds

tnd

2-Dr

ezes

ns

r

ture,

l-

ity,

s,

JDy

g


500MPa, which is much larger than that of conventionalbers with polymer matrix. This has suggested the possibof chemical bonding between the nanotubes and the polyin their composites. The nature of the bonding, howevernot clearly known. Later, Lourieet al @53# examined thefragmentation of single-walled CNT within the epoxy resunder tensile stress. Their experiment also suggested abonding between the nanotube and the polymer insample. Schadleret al @47# have studied the mechanicproperties of 5%~by weight! MWNTs within epoxy matrixby measuring the Raman peak shift when the compositesunder compression and under tension. A large Raman pshift is observed for the case of compression, while the sin the case of tension is not significant. The tensile moduof the composites is found to enhance much less as cpared to the enhancement in the compression modulus osimilar system. Schadleret al have attributed the differencesbetween the tensile and compression strain cases, to theing of inner shells of the MWNTs when a tensile stressapplied. In cases of SWNT polymer composites, the posssliding of individual tubes in the SWCNT rope, whichbonded by van der Waals forces, may also reduce theciency of load transfer. It is suggested that for the SWrope case, interlocking using polymer molecules might bothe SWCNT rope more strongly. Andrewset al @48# havealso studied composites of 5%~by weight! of SWNT embed-ded in petroleum pitch matrix, and their measurements shan enhancement of Young’s modulus of the composite untensile stress. Qianet al’s measurement of a 1% MWNTpolystyrene composite under tensile stress also shows aincrease of Young’s modulus compared with the pure pomer system@54#. The possible sliding of inner shells iMWCT and of individual tubes in a SWNT rope was ndiscussed in these later two studies. There are, at presenexperiments available for SWNT-polymer compositescompare our simulated values with the experimental obvations. However, if it is assumed that polymer matrix esstially bonds only to the outer shell of a MWNT embeddeda matrix, the above simulation findings could be qualitativcompared with experiments. This issue needs to be conered in more detail before any direct comparison is mbetween theory/simulations and experimental observatio

7 COMMENTS

Nanomechanics of single-wall carbon nanotubes have bdiscussed from a perspective of their applications in carnanotube reinforced composite materials. It is clear thatsingle-wall carbon nanotubes, a general convergencestarted to emerge between the simulated Young’s modvalues and the values observed in experiments soYoung’s modulus is slightly larger than 1TPa, and tubeswithstand about 5–10% axial strains before yielding, whcorresponds to a stress of about 50 GPa before nanotyield. Bending and torsional modulus and stiffness have abeen computed, but no comparison with experimentsavailable so far. Real progress is made in coming up wittransition state theory based model of the yielding of SWNunder tensile stress. The yielding is identified as a bar

fi-litymer, is

inood

thel

areeakhiftlusom-the,slid-is

iblesffi-T

nd

owder

36%ly-

tt, notoer-n-

inlysid-des.

eenonforhaslusfar.anchbeslsois

h aTsrier

dependent transition between the pre- and post-yielding cfigurations. The model, within the error bars of the compuactivation barrier, correctly predicts that under tensile strat realistic~experimentally realizable! strain rates, yield oc-curs at about 5–10% applied strain, but not at high yieldstrains of 20–30% as was predicted in the earlier MD simlations. Preliminary results of the structural, thermal, amechanical characterization of nanotube polymer composhave been obtained and show that important characterisuch as thermal expansion and diffusion coefficients fromprocessing and applications perspective can be simulatecomputational design of nanotube composite materiThese simulations illustrate the large potential of computional nanotechnology based investigations. For larger stem sizes and realistic interface between nanotubes and pmer, the simulation techniques and underlying multi-scsimulations and modeling algorithms need to be develoand improved significantly before high fidelity simulationcan be attempted in the near future.

ACKNOWLEDGMENTS

Part of this work~DS! was supported by NASA contrac704-40-32 to CSC at NASA’s Ames Research center. KC aCW acknowledge a support from NASA contract NCC5400-2. KC acknowledges many helpful discussions withM Baskes and Prof R Ruoff on nanotube mechanics.

REFERENCES

@1# Iijima S ~1991!, Helical microtubules of graphitic carbon,Nature(London)354, 56–58.

@2# Iijima S ~1993!, Single-shell carbon nanotubes of 1-nm diameter,Na-ture (London)363, 603–605.

@3# Bethune DS, Kiang CH, Devries MS, Gorman G, Savoy R, VazquJ, and Beyers R~1993!, Cobalt-catalyzed growth of carbon nanotubwith single-atomic-layerwalls,Nature (London)363, 605–607.

@4# Saito R, Dresselhaus G, and Dresselhaus MS~1998!, Physical Prop-erties of Carbon Nanotubes, Imperial College Press, London.

@5# Dresselhaus MS, Dresselhaus G, and Avouris Ph~eds! ~2001!, Car-bon Nanotubes: Synthesis, Structure, Properties, and Applicatio,Springer-Verlag Berlin, Heidelberg.

@6# Allen MP and Tildsley DJ~1987!, Computer Simulations of Liquids,Oxford Science Publications, Oxford.

@7# Brenner DW, Shenderova OA, and Areshkin DA~1998!, Reviews inComputational Chemistry, KB Lipkowitz and DB Boyd~eds!, 213,VCH Publishers, New York.

@8# Garrison BJ and Srivastava D~1995!, Potential-energy surfaces fochemical-reactions at solid-surfaces,Ann. Rev. Phys. Chem.46, 373–394.

@9# Srivastava D and Barnard S~1997!, Molecular dynamics simulationof large scale carbon nanotubes on a shared memory archi-tecProc. IEEE Supercomputing97 ~SC’97 cd-rom!, @Published#. @PaperNo#.

@10# Harrison WA~1980!, Electronic Structure and the Properties of Soids, Freeman, San Francisco.

@11# Menon M and Subbaswamy KR~1997!, Nonorthogonal tight-bindingmolecular dynamics scheme for silicon with improved transferabilPhys. Rev. B55, 9231–9234.

@12# Menon M ~2001!, Generalized tight-binding molecular dynamicscheme for heteroatomic systems: Application to SimCn clustersJ.Chem. Phys.114, 7731–7735.

@13# Payne MC, Teter MP, Allan DC, Arias TA, and Joannopoulos~1992!, Iterative minimization techniques for abinitio total-energcalculations: molecular-dynamics and conjugate gradients,Rev. Mod.Phys.68, 1045–1097.

@14# Hohenberg P and Kohn W~1964!, Inhomogeneous electron gas,Phys.Rev. B136, B864.

@15# Kohn W and Sham LJ~1965!, Self-consistent equations includinexchange and correlation effects,Phys. Rev.140, A1133.

--

p

c

m

-

-

,ll

per-

on

tp://

e

nderes

C,f

Y,s,

e

imu-

er

yxial

om-


@16# Check the web site for the details, http://cms.mpi.univie.ac.at/vas@17# Srivastava D, Menon M, and Cho K~2001!, Computational nanotech

nology with carbon nanotubes and fullerenes,~Special Issue on Nanotechnology! Computing in Engineering and Sciences3~4!, 42–55.

@18# Robertson DH, Brenner DW, and Mintmire JW~1992!, Energetics ofnanoscale graphitic tubules,Phys. Rev. B45, 12592–12595.

@19# Tibbetts GG ~1984!, Why are carbon filaments tubular,J. Cryst.Growth 66, 632–638.

@20# Yakobson BI, Brabec CJ, and Bernholc J~1996!, Nanomechanics ofcarbon tubes: Instabilities beyond linear response,Phys. Rev. Lett.76,2511–2514.

@21# Lu JP~1997!, Elastic properties of carbon nanotubes and nanoroPhys. Rev. Lett.79, 1297–1300.

@22# Hernandez E, Goze C, Bernier P, and Rubio A~1998!, Elastic prop-erties of C and BxCyNz composite nanotubes,Phys. Rev. Lett.80,4502–4505.

@23# Sanchez-Portal D, Artacho E, Solar JM, Rubio A, and Ordejon~1999!, Ab initio structural, elastic, and vibrational properties of cabon nanotubes,Phys. Rev. B59, 12678–12688.

@24# Srivastava D, Menon M, and Cho K~2001!, Anisotropic nanome-chanics of boron nitride nanotubes: Nanostructured ‘‘skin’’ effePhys. Rev. B63, 195413–195416.

@25# Ru CQ~2000!, Effective bending stiffness of carbon nanotubes,Phys.Rev. B62, 9973–9976.

@26# Harik VM ~2001!, Ranges of applicability for the continuum beamodel in the mechanics of carbon nanotubes and nanorods,SolidState Commun.120, 331–335.

@27# Landau LD and Lifschitz EM~1986!, Theory of Elasticity, 3rd Edi-tion, Pergamon Press, Oxford@Oxfordshire#, New York.

@28# Poncharal P, Wang ZL, Ugarte D, and deHeer WA~1999!, Electro-static deflections and electromechanical resonances of carbon ntubes,Science283, 1513–1516.

@29# Iijima S, Brabec C, Maiti A, and Bernholc J~1996!, Structural flex-ibility of carbon nanotubes,J. Chem. Phys.104, 2089–2092.

@30# Treacy MMJ, Ebbesen TW, and Gibson JM~1996!, Exceptionallyhigh Young’s modulus observed for individual carbon nanotubes,Na-ture (London)381, 678–680.

@31# Wong EW, Sheehan PE, and Lieber CM~1997!, Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotuScience277, 1971–1975.

@32# Krishnan A, Dujardin E, Ebbesen TW, Yianilos PN, and Treacy MM~1998!, Young’s modulus of single-walled nanotubes,Phys. Rev. B58,14013–14019.

@33# Salvetat JP, Briggs GAD, Bonard JM, Bacsa RR, Kulik AJ, StockliBurnham NA, and Forro L~1999!, Elastic and shear moduli of singlewalled carbon nanotube ropes,Phys. Rev. Lett.82, 944–947.

@34# Lourie O, Cox DM, and Wagner HD~1998!, Buckling and collapse ofembedded carbon nanotubes,Phys. Rev. Lett.81, 1638–1641.

@35# Srivastava D, Menon M, and Cho K~1999!, Nanoplasticity of single-wall carbon nanotubes under uniaxial compression,Phys. Rev. Lett.83, 2973–2676.

@36# Nardelli MB, Yakobson BI, and Bernholc J~1998!, Brittle and ductilebehavior in carbon nanotubes,Phys. Rev. Lett.81, 4656–4659.

p/.

es,

Pr-

t,

ano-

bes,

J

T,

@37# Nardelli MB, Yakobson BI, and Bernholc J~1998!, Mechanism ofstrain release in carbon nanotubesPhys. Rev. B57, 4277–4280.

@38# Zhang PH, Lammert PE, and Crespi VH~1998!, Plastic deformationsof carbon nanotubes,Phys. Rev. Lett.81, 5346–5349.

@39# Yakobson BI, Campbell MP, Brabec CJ, and Bernholc J~1997!, Highstrain rate fracture and C-chain unraveling in carbon nanotubes,Com-put. Mater. Sci.8, 341–348.

@40# Walters DA, Ericson LM, Casavant MJ, Liu J, Colbert DT, Smith KAand Smalley RE~1999!, Elastic strain of freely suspended single-wacarbon nanotube ropes,Appl. Phys. Lett.74, 3803–3805.

@41# Yu MF, Files BS, Arepalli S, and Ruoff RS~2000!, Tensile loading ofropes of single wall carbon nanotubes and their mechanical proties,Phys. Rev. Lett.84, 5552–5555.

@42# Yu MF, Lourie O, Dyer MJ, Moloni K, Kelly TF, and Ruoff RS~2000!, Strength and breaking mechanism of multiwalled carbnanotubes under tensile load,Science287, 637–640.

@43# Wei CY, Cho K, and Srivastava D~2002!, Tensile strength of carbonnanotubes under realistic temperature and strain rate, htxxx.lanl.gov/abs/condmat/0202513.

@44# Wei C, Cho K, and Srivastava D~2001!, Temperature and strain-ratdependent plastic deformation of carbon nanotube,MRS Symp Proc677, AA6.5.

@45# Wei CY, Srivastava D, and Cho K~2002!, Molecular dynamics studyof temperature dependent plastic collapse of carbon nanotubes uaxial compression,Computer Modeling in Engineering and Scienc3, 255–261.

@46# Vigolo B, Penicaud A, Coulon C, Sauder C, Pailler R, JournetBernier P, and Poulin P~2000!, Macroscopic fibers and ribbons ooriented carbon nanotubes,Science290, 1331–1334.

@47# Schadler LS, Giannaris SC, and Ajayan PM~1998!, Load transfer incarbon nanotube epoxy composites,Appl. Phys. Lett.73, 3842–3844.

@48# Andrews R, Jacques D, Rao AM, Rantell T, Derbyshire F, ChenChen J, and Haddon RC~1999!, Nanotube composite carbon fiberAppl. Phys. Lett.75, 1329–1331.

@49# Osman MA, and Srivastava D~2001!, Temperature dependence of ththermal conductivity of single-wall carbon nanotubes,Nanotechnol-ogy 12, 21–24.

@50# Wei CY, Srivastava D, and Cho K~2002!, Thermal expansion anddiffusion coefficients of carbon nanotube-polymer composites,NanoLett. 2, 647–650.

@51# Frankland SJV, Caglar A, Brenner DW, and Griebe M~2000, Fall!,Reinforcement mechanisms in polymer nanotube composites: slated non-bonded and cross-linked systemsMRS Symp Proc, A14.17.

@52# Wagner HD, Lourie O, Feldman Y, and Tenne R~1998!, Stress-induced fragmentation of multiwall carbon nanotubes in a polymmatrix, Appl. Phys. Lett.72, 188–190.

@53# Lourier O, and Wagner HD~1998!, Transmission electron microscopobservations of fracture of single-wall carbon nanotubes under atension,Appl. Phys. Lett.73, 3527–3529.

@54# Qian Q, Dickey EC, Andrews R, and Rantell T~2000!, Load transferand deformation mechanisms in carbon nanotube-polystyrene cposites,Appl. Phys. Lett.76, 2868–2870.

in-ntationsynmanapersenerstava

viewed

Deepak Srivastavais a senior scientist and technical leader of computational nanotechnologyvestigations at NASA Ames Research Center. In recent years, he has given about 100 preseon carbon nanotubes and nanotechnology. His accomplishments include co-winner of FePrize (Theory) in 1997, NASA Ames Contractor Council Award (1998), two Veridian Medal PAuthorship awards (1999), NASA Group Excellence Award to IPT (2000), and The Eric ReisMedal (2002) for distinguished contributions to nanoscience and carbon nanotubes. Srivaserves as associate editor of two nanotechnology and computer modeling related peer rejournals, and sits on the advisory board of New York based Nano-Business Alliance.

a PhDin theiver-tronic.

y, inhD in

MITional

prop-ms,y. Theange of


Chenyu Weiis a research scientist at NASA Ames Research Center since 2002. She receivedin physics from the University of Pennsylvania in 1999 and worked as a postdoctoral fellowLaboratory for Advanced Materials and Department of Mechanical Engineering at Stanford Unsity from 2000-2002. Her research area is in nanoscience, including mechanical and elecproperties of nanotubes and their composites, nanotube-polymer interfaces, and biomaterials

Kyepongjae (KJ) Chois an Assistant Professor of Mechanical Engineering and, by courtesMaterials Science and Engineering at Stanford University (1997-present). He received his PPhysics at MIT in 1994. During 1994-1997, he worked as a research staff member at theResearch Laboratory of Electronics and Harvard University. His research area is computatmaterials modeling using high-performance supercomputers. To investigate complex materialserties with true predictive power, his group is applying efficient atomistic simulation prograwhich enable one to study increasingly larger complex materials systems with more accuraccomplex materials systems he has studied using the atomistic method encompass a wide rdifferent chemical systems, biomolecules, and electronic materials.

nanomechanics of carbon nanotubes and composites · nanomechanics of carbon nanotubes and...

Documents