Mechanical Strength of Polymer-Based Composites Using Carbon...

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14 Enhancement of the Mechanical Strengthof Polymer-Based Composites Using Carbon Nanotubes

Kin-tak Lau, Jagannathan Sankar, and David Hui

CONTENTS

14.1 Introduction.................................................................................................32314.2 Properties of Carbon Nanotubes ................................................................324

14.2.1 Experimental Measurements ........................................................32414.2.2 Theoretical Study and Molecular Dynamics Simulation ............32614.2.3 Finite Element Modeling .............................................................331

14.3 Fabrication Processes of Nanotube/Polymer Composites .........................33214.4 Interfacial Bonding Properties of Nanotube/Polymer Composites ...........334

14.4.1 Experimental Investigation...........................................................33414.4.2 Theoretical Study and Molecular Dynamics Simulation ............336

14.5 Concluding Remarks ..................................................................................339Problems ................................................................................................................339Acknowledgments..................................................................................................340References..............................................................................................................340

14.1 INTRODUCTION

Since the discovery of carbon nanotubes (hereafter called nanotubes) a decadeago,1 many important studies and results related to these nanostructural materialsin different scientific and engineering fields have emerged. The extraordinarymechanical, electrical, and thermal properties of the nanotubes are governed by theiratomic architecture, commonly called their chiral arrangement. Ideally all carbonatoms in the nanotubes are covalently bonded and form repeated close-packedhexagonal structures in each layer or shell. Due to these chemically formed atomicarrangements, the carbon nanotubes possess superior mechanical properties, and the

1653_C14.fm Page 323 Tuesday, December 21, 2004 5:27 PM

324 Nanoengineering of Structural, Functional, and Smart Materials

nanotubes are stronger than any known metallic materials. Many critical results havebeen reported recently by using nanotubes as an atomic force microscope (AFM)probe, conductive devices in artificial muscles, and nanothermometers, and to storehydrogen for fuel cells.2–5 In the United States, the investment in the developmentof fuel cells by storing hydrogen atoms inside the cavity of nanotubes to supplyelectricity to microelectromechanical (MEM) or even nanoelectromechanical (NEM)devices has been increasing. Although all this work is still at the research stage, itdemonstrates a high potential for the development of nanotube-related products andcomponents for real-world applications.

In the past few years, many researchers and engineers from the advancedcomposites community have attempted to use these tiny structural materials toenhance the properties of conventional advanced composite structures by alteringtheir mechanical, thermal, and electrostatic behavior for space and infrastructureapplications. However, in order to achieve these goals, more work is needed for:(1) understanding the mechanical properties of both singlewalled and multiwallednanotubes, (2) investigating appropriate fabricating processes of nanotube/polymercomposites, (3) clarifying the interfacial bonding properties between the nanotubesand surrounding matrix, and (4) justifying the benefit based on the strength improve-ment of composites after mixing with the nanotubes. It is understandable thatresearch related to the aforementioned issues still has a long way to go due to manyuncertainties in the nanotubes properties themselves and their structural integrity innanotube/polymer composites.

In this chapter, a critical review of the aforementioned aspects is given basedon recent research by the authors and other researchers. All these aspects cannot beconsidered individually in the development of nanotube/polymer composites. There-fore, a discussion of how these aspects affect each other, as well as a detaileddiscussion of each one, is given. Since this chapter is mainly focused on the mechan-ical properties of nanotube/polymer composites, the fundamental physics of nano-tubes is briefly explained here.

14.2 PROPERTIES OF CARBON NANOTUBES

14.2.1 EXPERIMENTAL MEASUREMENTS

Carbon nanotubes are similar to a flat graphene layer rolled up to form a tube whereboth ends of the tube are sealed with semihemisphere caps. All the carbon atoms aretightly bonded to each other to form a close-packed hexagonal structure in the formof a circular tube, as shown in Fig. 14.1. If more than one layer of graphene layersare rolled together to form a coaxial tube, this nanotube is called a multiwallednanotube (MWNT). The mechanical and electrical properties of nanotubes are gov-erned by their atomic structure. In the past few years, many researchers haveattempted to measure the mechanical properties of nanotubes. However, due to theirsmall size, it is difficult to directly measure the strength of the nanotubes by usingtraditional testing methods. Yu et al.6 mounted MWNTs on two AFM tips andconducted a tensile test. The whole stretching process was captured inside a scanningelectron microscope (SEM) as shown in Fig. 14.2. Since different sizes and types


Enhancement of the Mechanical Strength of Polymer-Based Composites 325

of multiwalled nanotubes were tested, the results were quite diverse. The Young’smoduli of the nanotubes ranged from 0.32 TPa to 1.47 TPa. In strength testing, almostall the outer layers of the nanotubes broke first, and then pulling out of inner layersoccurred. According to the results in the literature, the maximum stress applied to thenanotubes occurred in the outer layer. The larger diameter nanotubes could withstanda larger tensile force. This conclusion may not be satisfied based on some estimatesof strength from continuum models and molecular dynamics (MD) simulations.

An indirect tensile test on nanotubes by Demczyk et al.7 demonstrated that thetensile strength of MWNTs was about 0.15 TPa, which is far below theoreticalestimates and the known properties of a graphene layer. A telescoping action wasseen in some fractured nanotubes. This finding agreed with the work done in

FIGURE 14.1 Schematic illustration of carbon nanotubes.

FIGURE 14.2 SEM images of a nanotube linked between two opposing AFM tips beforetensile loading.6

Two-dimensionalgraphene sheet

Chiral tubule (4,3)

Armchair tubule (4,4)

Zigzag tubule (7,0)

Seamless nanotube

a1

a2

O

CCh

θ

1653_C14.fm 25 Tuesday, December 21, 2004 12:35 PM


Reference 8, wherein a weak van der Waals attractive force between individual shellsof the MWNTs resulted in generating an extremely low frictional force among theshells, and all the inner shells slid freely in their longitudinal direction (Fig. 14.3).The principal result is that all the inner shells cannot contribute any strength to thenanotubes while they are subjected to tension. Qi et al.9 also measured the Young’smodulus of MWNTs using a nanoindentation method. An indentor was pressed ontovertical aligned nanotubes and the bending stiffness of the nanotubes was measuredand also calculated using classical bending theory. The effective bending and axialelastic moduli ranged 0.91 ∼ 1.24 TPa and 0.90 ∼ 1.23 TPa, respectively. Since theineffectiveness of the stress transfer among different shells of the nanotubes was notconsidered in the calculation, particularly in the bending case, the moduli estimatedby this method do not represent the true tensile modulus of the nanotubes.

14.2.2 THEORETICAL STUDY AND MOLECULAR

DYNAMICS SIMULATION

Because the size of nanotubes is extremely small, direct measurement of the mechan-ical properties of nanotubes is extremely difficult. In the past few years, much efforthas been spent on using MD simulations to predict the properties of nanotubes.Since the accuracy of the simulated results is highly dependent on the size of themodels as well as the capacity (memory size) and running speed of computers, theresults from different papers and approaches are therefore different. Tu and Ou-Yang10 have used the local density approximation method associated with elasticshell theory to estimate the mechanical properties of single-walled nanotubes(SWNTs) and MWNTs. They found that the Young’s modulus of the multiwallednanotubes decreases with an increase in the number of shell layers. The Young’smodulus of the nanotubes can be determined by the following equation:

(14.1)

where Em and Es represent the Young’s moduli of MWNTs and SWNTs, respectively,and n, t, and d denote the number of shell layers, average thickness of the shell(≈0.75 Å), and spacing between the shells (≈0.34 Å), respectively. They have alsoreduced Equation 14.1 to the continuum limit form, and the classic shell theory canbe used to describe the deformation of the nanotubes. Lau et al.11 used the Tersoff-Brenner bond order potential to represent the interaction between carbon atoms to

FIGURE 14.3 A telescoping action in a multiwalled nanotube.7

En

n t dtd

Em s=− +

⋅ ⋅1 /



study the stretching motion of MWNTs. In their study, it was found that the outershell of the nanotubes took all the applied loads when the nanotubes were subjectedto tensile and torsion motions (Fig. 14.4). Since only a weak van der Waals inter-action exists between the shells, external motions of the outer shell cannot be easilytransferred to inner shells. Eventually fracture at the outer shell was initiated(Fig. 14.5) and this phenomena was similar to that observed in Reference 7, where

FIGURE 14.4 MD-simulated results of three-layer MWNTs. The inner two layers would notbe affected by the surface layer when the MWNTs are subjected to tensile (top) and torsional(bottom) loads.

FIGURE 14.5 Example of an outer shell of a MWNT pulling out of a matrix.



pullout of nanotubes occurred. Lau12 has also pointed out that the size and Young’smodulus of nanotubes are highly dependent on their chiral arrangement. The diam-eter of a zigzag nanotube is generally smaller than that of an armchair type. Theradius of the first layer (an inner layer) of nanotubes can be determined by usingthe rolled graphene sheet model:

(14.2)

where ro, m, n, and ao are the nonrelaxed radius and indices of the SWNTs and C-Cbond distance (1.42 Å), respectively.

The MD simulation has been popularly adopted in recent years to predictthe properties of nanostructural materials at the atomic scale with high accuracy.At the early stage, several studies have used the empirical-force-potential molec-ular dynamic simulations to estimate the Young’s modulus of nanotubes andfound that its value was four times that of the diamond. However, those calcu-lations were based on the SWNTs of several Å in radius. To determine themechanical properties of nanotubes, the details of their atomic arrangementshave to be clearly understood. In general, there are three types of nanotubestructures: zigzag (n, 0), armchair (n, n), and chiral (n, m where n ≠ m). Lauand Hui13 have given a comprehensive review on the structures of nanotubes.To investigate the mechanical properties of materials at the atomic scale by usingMD simulations, the interactions between neighboring atoms must be accuratelycalculated.

Two common approaches based on quantum mechanics and molecular mechan-ics are used to simulate these interactions. Both approaches attempt to capture thevariation of system energy associated with the change in atomic positions by fol-lowing Newton’s second law, F = ma. In carbon nanotubes, the mutual interactionsare described by force potentials from both bonding and nonbonding interactions.The nonbonding interactions are either due to the van der Waals force (which canbe attractive or repulsive depending on the distance between atoms), or to electro-static interactions. The van der Waals force FVDW is most often modeled using theLennard-Jones potential function,14 originally derived for inert gases. The generalform of this potential is

(14.3)

For van der Waals forces arising from dipole-dipole interactions, the attractivepart corresponds to m = 6. The most common form of this potential is the so called(6-12) form:

(14.4)

rpo o

m n mna= + + ⋅

2 2

3

Φ( )rr r

nn

mm= −l l

Φ( )rr r

=

−

4

12 6

e s s



The minimum of Φ(r) is determined by equating to zero the first-order derivativeof Φ(r) versus r. The van der Waals force between two carbon atoms can be estimatedfrom

(14.5)

The two parameters, s and e can be estimated from experimental data such as theequilibrium bond length (lattice parameters at equilibrium), equilibrium bond energy(cohesive energy), and bulk modulus at equilibrium. The bonding energy (Ebond) isthe sum of four different interactions among atoms, namely bond stretching (Ur),angle variation (Uq), inversion (Uw) and torsion (Ut ),15 written as

(14.6)

A schematic illustration of each energy term and corresponding bond structurefor a graphene cell is shown in Fig. 14.6. The most commonly used functionalforms are

(14.7a–d)

where dRi is the elongation of the bond identified by the label i, Ki is the forceconstant associated with the stretching of the i bond, and dqj and dwk are the varianceof bond angle j and inversion angle k, respectively. Cj and Bk are force constantsassociated with angle variance and inversion, respectively. Ai is the barrier height torotation of the bond i; ni is the multiplicity that gives the number of minimums asthe bond is rotated through 2p.15

To determine the tensile modulus of a SWNT subjected to uniaxial loading, itis useful to observe that at small strains the torsion, the inversion, the van der Waals,and the electrostatic interaction energy terms are small compared with the bond

Fddr r r rVDW = − =

−

f e s s242

12 6

E U U U UBond = + + +r q w t

U K dRi

i

ir = ∑12

2( )

U C dj

i

jq q= ∑12

2( )

U B dk

k

kw w= ∑12

2( )

U A ni

i

i i it t f= + −∑12

1[ cos( ) ]



stretching and the angle variation terms. Thus the total energy of the single-wallnanotube can be reduced to

. (14.8)

The force constants Ki and Ci can be obtained from quantum mechanics (ab initio).The average macroscopic elastic modulus and Poisson’s ratio were estimated to beabout 1.347 TPa and 0.261, respectively.15 Such calculations may be performed usingeither the force or the energy approach, by measuring the mechanical forces developedbetween carbon atoms in nanotubes with different chiral arrangements. Lu16 has usedthe empirical-force-potential molecular dynamic simulation to investigate the proper-ties of nanotubes. The structure of the nanotubes was obtained by the conformationalmapping of a graphene sheet onto a cylindrical surface. The nanotube radius wasestimated by Equation 14.2. The average estimated tensile modulus of SWNTs andMWNTs is about 1 TPa. The elastic properties are the same for all nanotubes with aradius larger than 1 nm. Zhou et al.17 have used the first principles cluster model withinthe local density approximation to evaluate the mechanical properties of a SWNT.The estimated values for tensile modulus, tensile strength, and Poisson’s ratio were0.764 TPa, 6.248 GPa, and 0.32,17 respectively. The binding energy of the nanotube isless than that of graphite due to the curvature effect. Lier et al.18 calculated the tensilemodulus of zigzag and armchair SWNTs using the ab initio multiplicative integralapproach (MIA), which is based on the energy of elongation of nanotubes in a simpletension (which is not constrained laterally or in any other way). They found that themodulus of SWNTs or MWNTs was larger than that of a graphene sheet. The MDsimulations show that the fracture behavior of zigzag nanotubes is more brittle thanthe fracture behavior of armchair nanotubes.19 The formation of a local Stone-Walesdefect (5-7-7-5) in the deformed armchair nanotube induced ductile deformation.

FIGURE 14.6 Bond structures and corresponding energy terms of a graphene cell.

Torsion

Non-bonded

Bond stretching

Bond-anglevariation

Carbon atom

Inversion

E K dR C dTotal i

i

i j

j

j= +∑ ∑12

12

2 2( ) ( )q



14.2.3 FINITE ELEMENT MODELING

In the past few years, the demand for the development of faster methods to computethe mechanical properties of nanostructures has been increasing. The classicalshell theory has been judged as too simple and less accurate because it is limitedby some unrealistic boundary conditions. The finite element modeling (FEM)method associated with the molecular dynamics (MD) or equivalent-continuum(EC) model has been recently adopted to calculate the mechanical properties ofnanotubes. Odegard et al.20 have developed an equivalent-continuum tube model todetermine the effective geometry and effective bending rigidity of a graphene struc-ture. Molecular mechanics considerations (see Equation 14.6 and Equation 14.7)were first used to determine linking forces between individual carbon atoms. Thismolecular force field was simulated by using a pin-joint truss model; i.e., eachtruss member represents the force between two atoms, as shown in Fig. 14.7.Therefore, the truss model allows the accurate simulation of the mechanicalbehavior of nanotubes in terms of atom displacements. As the nanotube wassubjected only to a uniaxial load,20 the bond stretching and bond-angle variationenergies (see Equation 14.8) were considered. The strain energy of the whole systemwas used in the FEM computation to estimate the effective thickness of the nanotubelayer. It was found that the effective thickness of the nanotubes (0.69 Å and 0.57 Å)was significantly larger than the interlayer spacing of graphite, estimated to about∼0.34 Å.

Li and Chou21–22 have worked out the contributions of van der Waals interactionsbetween individual carbon atoms within nanotubes within the FEM truss model.The relationships between the structural mechanics parameters EA, EI, and GJ and

FIGURE 14.7 Truss model of a carbon nanotube’s structure.

Carbon atoms

Truss members

Single-walled carbon nanotube



the molecular mechanics parameters Kr , Cq , and At , as shown in Equations 14.7a, b,and d for each truss member:

(14.9a–c)

In Fig. 14.8, the dependence of the van der Waals force versus the distance betweentwo carbon atoms is plotted. Nonlinear truss elements were used in simulations, asthe force between two carbon atoms is also nonlinear. A uniaxial load was applieduniformly at the end of the nanotubes;21–22 the effects due to the end caps wereneglected. It was found that the Young’s modulus of the nanotubes increases as thenanotubes’ diameter is increased. The Young’s and shear moduli of MWNTs are inthe range of 1.05 ± 0.05 and 0.4 ± 0.05 TPa, respectively. For SWNTs, the Young’smodulus is almost constant when the nanotubes’ diameters are larger than 1.0 nm.The average Young’s modulus of the zigzag nanotubes is slightly higher than thearmchair type. Also the Young’s modulus of MWNTs is generally higher (∼7%) thanthat of SWNTs.

14.3 FABRICATION PROCESSESOF NANOTUBE/POLYMER COMPOSITES

To achieve desirable properties of nanotube polymer-based composites, control ofthe fabrication process in order to produce well-dispersed composites is an essentialissue. Several parameters such as the selection of dispersion solvents, sonicationtime, stirring speed, and ambient temperature are the keys to disperse the nanotubes

FIGURE 14.8 Van der Waals force versus the distance between two carbon atoms.

−0.05

−0.03

−0.01

0.01

0.03

0.05

0.07

0.09

4.4 5.46.4 7.4

8.4

Distance between two atoms (armstrong)

van

der

Waa

ls fo

rce

betw

een

inte

rcat

ing

atom

s(k

Cal

/mol

e.nm

)

3.4 9.4

EAL

KEIL

CGJL

A= = =r q t; and



uniformly in the resin. Another crucial factor that influences the strength of the com-posites is the alignment of the nanotubes. Sonication is one of the most popular waysto disperse nanotubes into polymer-based resin. Park et al.23 first demonstrated the useof in situ polymerization to disperse SWNT bundles in the polymer matrix. A diluteSWNT solution, typically around 0.05 wt% of nanotubes in dimethylformamide (DMF)subjected to sonication for 1 1/2 hours in an ultrasonic bath (40 kHz) was followed bymixing with the hardener. The long sonication time may cause the nanotubes to entangleand form bundles, which may result in losing the benefit of their high-strength propertiesin composite materials. Mukhopadhyay et al.24 have reported that if the sonication timeis greater than 4 hours, it may destroy most of the graphene layers of the nanotubes,and possibly cause the formation of junctions in the nanotubes.

Lu et al.25 have found that the use of different chemical solvents as dilute solutionswould greatly influence the integrity of composites. Since DMF has a high boilingtemperature compared with ethanol and acetone, the solvent remains inside the resinand lowers the chemical reaction process between the resin and hardener. The boilingpoint of acetone, ethanol, and DMF are 56°C, 78°C, and 130°C, respectively. Even-tually the overall mechanical and chemical properties of the composite decrease. InTable 14.1, the Vicker’s hardness of different nanotube/epoxy composites mixed byusing different types of solvent is shown. Obviously the mechanical properties of thecomposites are directly affected by the melting temperatures of the solvent.

In Fig. 14.9, FTIR spectra of different types of composite are shown. Spectrab, c, and d represent the solvents acetone, ethanol, and DMF, respectively, whichwere used to disperse the nanotubes in an epoxy matrix. Spectrum a is a pure epoxy.The most prominent feature in the spectra b, c, and d is the appearance of a newabsorption band located at ∼1650 cm−1. Considering epoxide and hydroxyl groupsare the only two reaction groups in the epoxy molecule, the ∼1650 cm−1 band canbe assigned to an amino group formed by the intermolecular nucleophilic substitutionof hydroxyl at the amide functionality, which can therefore be used to estimate therelative amount of the product of the cure reaction. It is also noticed that the bandlocations are different in the spectra b, c, and d, with a value of 1645 cm–1, 1649 cm–1,and 1664 cm–1, respectively. This indicates that there are functional and curing differ-ences among epoxy resins with different solvent treatments, which could be relatedto the variation of the mechanical properties discussed previously. Blanchet et al.26

TABLE 14.1Vicker’s Hardness Values of Different Samples Determined from a Load Force of 100 g and a Dwell Time of 15 S

Composition Solvent Vicker’s Hardness No.

CNTs/Epoxy Acetone 18.0 ± 0.11CNTs/Epoxy Ethanol 14.4 ± 0.08CNTs/Epoxy DMF 7.7 ± 0.10Pure Epoxy 17.8 ± 0.06



directly dispersed nanotubes into polyaniline (PANI) using a sonication method.The nanotubes were first sonicated in xylene and that dispersion afterward wassonicated in the DNNSA-PANI solution. It was found that all the nanotubes werewell dispersed into the PANI and enhancement of the electrical conductivity wasachieved. Tang et al.27 have also introduced the use of a melt processing techniquefor making nanotube/high-density polyethylene (HDPE) composites. The nano-tubes and HDPE were premelted together to form pellets, followed by feedingthem into a twin-screw extruder to make composites (Fig. 14.10). In their work,a difficulty was to control the uniformity of the nanotubes in the HDPE. Once thepellets were melted, some of the nanotubes were trapped inside the injection headof the extruder and entangled forming bundles. Although research related to nan-otube/polymer composites has been done for more than a half decade, the detailedstudy on how to fabricate well-dispersed and -aligned nanotube/polymer compositesis still a critical aspect.

14.4 INTERFACIAL BONDING PROPERTIESOF NANOTUBE/POLYMER COMPOSITES

14.4.1 EXPERIMENTAL INVESTIGATION

It has been recognized that high mechanical strength of nanotube/polymer compos-ites could be achieved when all nanotubes are aligned parallel to the load direction,and an external stress should be effectively transferred to the nanotubes. Jin et al.28 andWood et al.29 have studied the control of the alignment of nanotubes in polymermatrices for making high-strength advanced nanocomposite structures. However, the

FIGURE 14.9 The representative FTIR spectra measured from the samples a to d.

4000 3500 3000 2500 2000 1500 1000

a

b

c

d

Wavenumbers (cm−1)

Abs

orba

nce



investigation on the mass production and control of the alignment of such nanotubestoward the predetermined direction is still under investigation. The stress transfermechanisms of different types of nanotubes in nanotube/polymer composites havenot been comprehensively discussed elsewhere. Wagner et al.30 and Qian et al.31

have studied the load transfer properties of the nanotube/polymer composites byconducting many experimental investigations. They found the interfacial shearstrength between the nanotube and polymer to be as high as 43.3 MPa whenmeasured in a fragmentation test. However, in several experimental investigations,it is obvious that a poor adhesion property in the nanotube/polymer composites wasfound because of the pulling out of the nanotubes.32 In addition, a weak attractionbetween different shells inside the multiwall nanotube also causes failure that isinitiated only at an outer shell of the nanotubes. The latest experimental studyconducted by Cooper et al.33 demonstrated that the shear strength between a nano-tube and epoxy matrix varied with the diameter, the embedding length, and thenumber of shells of the nanotubes. Two sections of an epoxy matrix were formedwith a nanotube bridging across the sections. The two sections were pulled apartusing the tip of a scanning probe microscope. The maximum shear strength ranging

FIGURE 14.10 Process of preparing MWNT/HDPE nanocomposite films.27

CNTsHDPEpellets

(a)

(d)

(g)

(b) (c)

(e) (f)

(h)

Heat Dough

Film Compositepellets



from 38 MPa to 376 MPa was recorded, and a high shear strength was reported forSWNT ropes. A pull out of nanotubes from a solid polyethylene matrix was dem-onstrated using an AFM,34 and the average interfacial stress required to remove aSWNT from the matrix measures was about 47 MPa. This experiment was closerto a realistic situation since all the nanotubes were physically bonded to the matrixand a typical pullout test was actually conducted. However, as described in theprevious section, the mechanical properties of different kinds of nanotubes couldinfluence their bonding strength to the matrix. To better understand the interfacialbonding behavior between the nanotube and matrix, an improved experimental setupto perform pullout tests for different types and geometries of nanotubes is needed.

14.4.2 THEORETICAL STUDY AND MOLECULAR DYNAMICS SIMULATION

The MD simulation is a powerful tool to predict the interfacial bonding propertiesamong different shells in nanotubes, as mentioned in Section 14.2.2, and alsobetween the nanotubes and surrounding polymer molecules. Liao and Li35 simulateda pullout action of nanotubes in polystyrene (PS) matrix using the commercialsoftware Hyerchem®. The adhesion strength between PS molecules and a graphenesheet was studied using a molecular mechanics model employing an empirical MM+force field. A random coil of PS [(-CH2CHC6H6-)n] molecules with n = 2, 4, 10, 20,40, and 80 was constructed and located near the surface layer of the nanotubes(Fig. 14.11). The nanotubes were then subjected to a pullout force to measure theinterfacial shear strength. In their study, the shear stress between the nanotubes andpolymer estimated from the simulation was about 160 MPa. An experiment was thensubsequently carried out by one of the coauthors of Reference 35 and a stronginterfacial adhesion between nanotubes and matrix was observed from electronmicroscopy.36 Lordi and Yao37 used a force field–based molecular mechanics calcu-lation to determine binding energies and sliding frictional stresses between pristinenanotubes and a range of polymer substrates. They found that all of the substratesstudied showed more sliding friction on a nanotube surface than that between twographene sheets and much more than that between MWNTs. The key factor informing a strong bond between the polymer and nanotube lies in the polymer’smorphology, specifically its ability to form large-diameter helices around individualnanotubes. Frankland et al.38 studied the effect of chemical cross-links on theinterfacial bonding strength between a SWNT and polymer matrix using MD sim-ulations. Their models were composed of single-wall (10,10) armchair nanotubesembedded into either a crystalline or amorphous matrix. The nonbond interactionswithin the polyethylene matrix and between the matrix and nanotubes were modeledwith Lennard-Jones 6-12 potentials. They found that even a relatively low densityof cross-links can have a large influence on the properties of the nanotube-polymerinterface. Ren and Liao39 also found that pullout energies are affected by the thermalcondition of composites. However, in those simulated studies, due to the reductionof complexity, chemical interactions between the nanotubes and matrix were gen-erally ignored. Only nonbond interactions, and electrostatic and van der Waals forceswere assumed.



In the early stages of nanotube-related research, many diverse results on theinterfacial-bonding characteristic between nanotubes and polymer-based matriceswere found. Xu et al.40 noted that high interfacial shear stress between a MWNTand an epoxy matrix was observed from a fractured sample. Wagner41 first used theKelly-Tyson model, which has been widely used to study the matrix-fiber stresstransfer mechanism in micron-size fiber composites to study the interfacial shearstrength between the nanotube and polymer matrix. Since it was found that thebinding force between inner layers is very low, and sliding failure always occurs,only a single-wall system is of interest in his work. In his study, it was assumedthat an externally applied stress to a nanotube/polymer composite is fully transferredto the nanotubes via a nanotube-matrix interfacial shear mechanism at the molecularlevel; i.e., the length of the nanotubes (l) is larger than the critical length (lc). Asingle nanotube cylindrical model was used to study the stress transfer propertiesof the composite shown in Fig. 14.12. To consider the force balance in the compositesystem, the following equation is formed:

(14.10)

FIGURE 14.11 A molecular model of a double-walled carbon nanotube in a PS matrix.35

t s s sNT i NT NTo i

NTo id dx d

d d d d= +

−

−−

( )2 2 2 2

4 4



where tNT is the interfacial shear strength between the nanotube and matrix, sNT isthe tensile strength of a nanotube segment of length dx, and do and di are outer andinner diameters of the nanotube, respectively.

After integrating Equation 14.10 and considering the critical length of a typicalshort fiber system in composite structures, the interfacial shear strength can be written as

(14.11)

where lc /do is the critical aspect ratio of the nanotube and di /do is the diameter ratio.As an external applied stress of 50 GPa was used in the literature, the interfacialstrength was calculated for critical length values of 100, 200, and 500 nm. It isconcluded that the interfacial shear stress is affected by several factors: (1) the criticallength, and (2) the outer diameter of the nanotubes. Increasing the diameter of thenanotube results in increasing the interfacial shear strength at the bond interface.

Recently Lau12 conducted an analytical study of the interfacial bonding proper-ties of nanotube/polymer composites by using the well-developed local densityapproximation model described in Section 14.2.2,10 and classical elastic shell theoryand the conventional fiber-pullout model. In his study, several important parameterssuch as the nanotube wall thickness, Young’s modulus, volume fraction, and chiralvectors of the nanotubes were considered. It was found that decrease of the maximumshear stress occurs by increasing the size of the nanotubes. Increasing the numberof the walls of the nanotubes will cause (1) a decrease of Young’s modulus of thenanotubes, (2) an increase of the effective cross-sectional area, and (3) an increasein the total contact surface area at the bond interface and the allowable pulloutforce of the nanotube/polymer system. In Fig. 14.13 a plot of the interfacial shearstress of SWNTs with different chiralities is shown. In the figure, it shows that themaximum shear stress of a zigzag nanotube (5,0) is comparatively higher than thoseof chiral (5,3) and armchair (5,5) nanotubes.

FIGURE 14.12 Stress transfer study using Kelly-Tyson model.

t sNT NTc

o

i

o

l

d

d

d=

−

−12

1

1 2

2



14.5 CONCLUDING REMARKS

In this chapter, a critical review of the strength of nanotubes, fabrication processes,and interfacial bonding properties of nanotube/polymer composites is given. It wasfound that there are still many uncertainties in measuring the mechanical propertiesof nanotubes due to the difficulty of conducting nanoscale property tests. MD andtheoretical analyses are mainly based on certain assumptions that may not be appli-cable to a real situation in nanotube/polymer composites. Dispersion properties andcontrol of alignment of nanotubes are also important issues that govern the globalproperties of the composites. Presently no one has reported a practical large-scalemanufacturing method to disperse and align nanotubes. This is the main factorlimiting enhancing the properties of nanotube/polymer composites. However, a goodinterfacial bonding between the nanotubes and matrix is also required. In the lastsection, we summarized some recent work on this particular issue. Two open prob-lems are (1) whether the chemical bonding between nanotubes and matrix exists ornot, and (2) whether nanotubes still maintain their extraordinary mechanical, elec-trical, and thermal properties in the presence of chemical bonding between thenanotube and matrix. Enhancement of the mechanical properties of advanced com-posite materials will require much further investigation, and this is definitely achallenging area for the composite community.

PROBLEMS

1. Which factors affect the mechanical, electrical, and thermal properties innanotube/polymer composites?

FIGURE 14.13 Plot of interfacial shear stress versus different types of nanotube.

0

50

100

150

200

250

0 40 80 120 160 200

z (nm)

She

ar s

tres

s (M

Pa)

Zigzag (5,0)

Chiral NT (5,3)

Armchair (5,5)



2. What are the differences between the mechanical properties of single-walled and multiwalled carbon nanotubes? Do all inner layers of thenanotubes contribute to the strength of the multiwalled nanotubes?

3. Fabrication processes of nanotube/polymer composites are a crucial factorthat controls the integrity of the composites. Which parameters have tobe controlled or studied in making high-strength composites?

4. Based on your own understanding, which type of nanotubes (MWNTs orSWNTs) is better to use to make nanotube/polymer composites?

ACKNOWLEDGMENTS

This work was supported by The Hong Polytechnic University Grant (G-T 861),Graduate Enhancement Fund from the University of New Orleans.

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