Finite Element Modeling of Multi-Walled Carbon Nanotubesmaterials, carbon nanotubes, CNT,...

International Journal of Engineering & Technology IJET-IJENS Vol:10 No:04 63

102804-5959 IJET-IJENS © August 2010 IJENS

I J E N S

Finite Element Modeling of Multi-Walled Carbon

Nanotubes

Prof. Mahmoud Nadim Nahas* and Prof. Mahmoud Abd-Rabou [email protected] and [email protected]

*Mechanical Engineering Department, King Abdulaziz University,

P.O.Box 80204, Jeddah, 21589, Saudi Arabia

Abstract-- Finite element modeling is used to simulate

carbon nanotubes, CNTs. A three-dimensional finite element

(FE) model for Armchair, Chiral and Zigzag single-walled

carbon nanotube (SWCNT), double-walled carbon nanotube

(DWCNT) and multi-walled carbon nanotube (MWCNT) is

presented here. The model development is based on the

assumption that carbon nanotubes, when subjected to loading,

behave like space-frame structures. The bonds between the

carbon atoms are considered as connecting load-carrying

members, while the carbon atoms themselves as joints of the

members. The nodes are placed at the locations of the carbon

atoms and the bonds between them are modeled using three-

dimensional elastic beam elements. Young modulus is finally

calculated. The Young's modulus of Chiral configuration is

found to be the largest among the three configurations, while

that of the Zigzag is the lowest.

Index Term-- finite element method, nanostructured

materials, carbon nanotubes, CNT, multi -walled carbon

nanotubes, MWCNT.

I. INTRODUCTION

The potential use of carbon nanotubes (CNTs) as reinforcing

materials in nano-composites has originated the need to

explore their exact mechanical properties. The

characterization of CNTs is more complex than that of

conventional materials due to the dependence of their

mechanical properties on size and nano-structure.

Computational approach can play a significant role in the

development of the CNT-based composites by providing

simulation results to help on the understanding, analysis and

design of such nanocomposites.

At the nanoscale, analytical models are difficult to establish

or too complicated to solve, and tests are extremely difficult

and expensive to conduct due to the very small size of CNTs.

Modeling and simulations of nanocomposites, on the other

hand, can be achieved readily and cost effectively on even a

desktop computer. Characterizing the mechanical properties

of CNT-based composites is just one of the many important

and urgent tasks that simulations can accomplish.

This paper is a continuation of previous work conducted by

the authors where they used finite element to characterize

grapheme sheet [1] and single-walled carbon nanotubes [2].

In the present project, they modelled the multi-walled carbon

nanotube composites using also finite element modeling.

Work of other authors on MWNT composites includes

Pantano et al [3] who studied the nature of the carbon

nanotube/polymer bonding and the curvature of the carbon

nanotubes within the polymer. The effects of carbon

nanotube curvature and interface interaction with the matrix

on the composite stiffness were investigated using

micromechanical analysis. In particular, the effects of poor

bonding and thus poor shear lag load transfer to the carbon

nanotubes were studied. In the case of poor bonding, carbon

nanotubes waviness was shown to enhance the composite

stiffness.

Lau et al [4] presented a critical review on the validity of

different experimental and theoretical approaches to the

mechanical properties of carbon nanotubes for advanced

composite structures. They stated that due to the use of

different fundamental assumptions and boundary

conditions, inconsistent results were reported. MD

simulation is a well-known technique that simulates

accurately the chemical and physical properties of structures

at atomic-scale level. However, it is limited by the time step.

The use of finite element modeling combined with MD

simulation can further decrease the processing time for

calculating the mechanical properties of nanotubes. Since

the aspect ratio of nanotubes is very large, the elastic rod or

beam models can be adequately used to simulate their overall

mechanical deformation. Although many theoretical studies

reported that the tensile modulus of multi-walled nanotubes

may reach 1 TPa, this value, however, cannot be directly

used to estimate the mechanical properties of multi-walled

nanotube/polymer composites. Hence there is a need for

prediction of properties.

Adding a small amount of carbon nanotubes into polymers

enhances the mechanical properties of carbon

nanotube/polymer composites. Qian et al [5-6] reported that

the addition of 1% (by weight) MWNTs into polystyrene

resulted in 36–42% increase in tensile modulus and 25%

increase in tensile strength. They also observed via

transmission electron microscopy (TEM) micrographs that

the nanotubes were able to bridge the crack surface of the

composite once a crack was initiated. The crack was

nucleated at the low nanotube density area and propagated

along the weak nanotube/polymer interfaces or regions with

the relatively low nanotube density regions. The pullout of

the nanotubes was observed when the crack opening

displacement reached ≈800 nm. Therefore, the function of

carbon nanotubes was to bridge up the crack in the

nanocomposites.

Lau et al [7] found that the hardness of carbon

nanotube/polymer composite increased with increasing

nanotube weight fraction. They also found that the hardness

was dropped at the low nanotube weight fraction samples

because of the weak bonding interface between the

nanotube and polymer matrix. Increasing the nanotube

weight fraction would result in forming a mesh-like

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I J E N S

networking structure by high aspect ratio nanotubes, which

could enhance the hardness of the composites.

Lau et al [8-9] studied the flexural strength of carbon

nanotube/epoxy beams under different ambient temperature

environments. They found that the flexural properties of the

nanotube beams decreases compared with a beam without

the nanotubes. The cause of the strength reduction was due

to structural nonhomogeneity and/or the existence of a

weak-bonding interface between the nanotubes and the

surrounding matrix. Although the mesh-like structure was

formed inside the beams, it did not improve the flexural

strength because this is mainly determined by the

nanotube/matrix bonding. The fracture surfaces of carbon

nanotube/epoxy composites after flexural strength tests have

shown different failure mechanisms for composites

pretreated at different temperatures. It was found that the

nanotubes within the composites after being treated at warm

and cryogenic temperatures were aligned perpendicular and

parallel to the fracture surfaces, respectively.

The potential applications of using carbon nanotubes as

nanoreinforcements and nanoconductors in polymer or

metallic-based composite structures are significant. Besides

the structural applications, the electrical and electronic

applications of using carbon nanotube/polymer composites

are exclusive. Much work in this area has recently been

conducted by many research groups around the world. The

major focus has paid much attention on the determination of

the resistance, which in turn could be converted to strain or

other chemical quantities, of different types of carbon

nanotube/polymer thin films [10]. The films could be used as

sensors for any tiny instruments and coatings for

electrostatic discharge protection for high-speed vehicle

applications. The investigation on the durability and

reliability of carbon nanotube/polymer composites subjected

to different mechanical and thermal loading cycles is a key

issue. In addition, the incorporation of carbon nanotubes

into polymeric materials for wear-sensitive components is

possible to decrease the generation of the wear debris.

Wagner et al [11] found that the stress transfer efficiency in

multi-walled nanotubes/polymers is at least one order of

magnitude larger than that of conventional fiber-based

composites. Zeng et al [12] reported a 50% increase in

Young’s modulus in CNTs/PMMA composites when

5.0 wt% carbon nanofibers were introduced into the

composites. Allaoui et al [13] found twice and triple

improvement on Young’s modulus and yield strength

respectively, when 1.0 wt% MWNTs was added to epoxy

matrix. Other researches on carbon nanofiller/polymer

composites [14-16] reported increases in mechanical

properties with increasing amount of CNTs. Tai et al [16]

showed a double improvement in tensile strength of the

MWNTs/phenolic composites when 3.0 wt% carbon

nanotube network was introduced into the phenolic matrix.

Ryan et al [17] addressed the issue of formation of

crystalline polymer coatings around the nanotubes in

solution by doping a semi-crystalline polymer, poly (vinyl

alcohol), with multiwalled carbon nanotubes. Dynamic

mechanical analyzer (DMA) measurements of thin films

identified a three- to five-fold increase in the Young's

modulus of the polymer depending on nanotube type.

Dynamic differential scanning calorimetry (DSC) of thin films

shows that the increase in modulus is accompanied by an

increase in polymer crystallinity. In addition, the results

verify that multiwalled carbon nanotubes nucleate

crystallization of the polymer and a link between polymer

crystallinity and composite reinforcement is established.

Furthermore, transmission electron microscopy (TEM)

images confirm an excellent dispersion and wetting of the

nanotubes in the polymer solution providing visual evidence

of matrix reinforcement.

Bakshi et al [18] In the present work, multiwalled carbon

nanotube (MWNT) reinforced UHMWPE composite films

were prepared by electrostatic spraying followed by

consolidation. X-ray diffraction and differential scanning

calorimetry studies showed a decrease in the crystallinity of

UHMWPE due to the nature of the fabrication process as

well as addition of MWNT. Tensile test showed an 82%

increase in the Young’s modulus, decrease in stress to

failure from 14.3 to 12.4 MPa and strain to failure from 3.9%

to 1.4% due to 5% addition of MWNT. Raman spectra

showed the presence of compressive stresses in the

nanotubes. Fracture surface showed presence of pullout like

phenomena in the MWNT reinforced film.

Yu et al [19] prepared multiwall carbon

nanotube/polystyrene (MWCNT/PS) composites based on

latex technology. MWCNTs were first dispersed in aqueous

solution of sodium dodecyl sulfate (SDS) driven by

sonication and then mixed with different amounts of PS latex.

From these mixtures MWCNT/PS composites were prepared

by freeze-drying and compression molding. The dispersion

of MWCNTs in aqueous SDS solution and in the PS matrix is

monitored by UV–vis, transmission electron microscopy,

electron tomography and scanning electron microscopy.

When applying adequate preparation conditions, MWCNTs

are well dispersed and homogeneously incorporated in the

PS matrix. The percolation threshold for conduction is about

1.5 wt% of MWCNTs in the composites, and a maximum

conductivity of about 1 S m−1

can be achieved. The approach

presented can be adapted to other MWCNT/polymer latex

systems.

More recent work by Chen et al [20] reported that the

Young’s modulus of the armchair type is slightly larger than

the zigzag type because of the difference in intrinsic atomic

structure, while Jing et al [21] reported that there presents a

strong size dependence in the material properties of

nanomaterials.

Cheng et al [22] used atomistic-continuum modeling (ACM)

approach. The modeling approach incorporates atomistic

modeling, by virtue of molecular dynamics (MD) simulation,

for simulating the initial unstrained equilibrium state, and

equivalent-continuum modeling (ECM), by way of finite

element approximations (FEA), for modeling the subsequent

static/dynamic behaviors.

Moreover, several numerical models, e.g., Odegard et al [23],

Fisher et al [24], Bradshaw et al [25] and Shi [26] have been

developed in attempts to improve the understanding of the

stiffening effects of MWCNTs in a polymer matrix. These

studies are based on micromechanical models, since



I J E N S

atomistic/molecular models are too computationally

demanding for direct application to MWCNT composites. All

of these continuum micromechanical models either explicitly

or effectively adopt an assumption of an ideally bonded

interface between the compliant polymer matrix and the stiff

MWCNTs, and often lead to predictions of overall

composite stiffness that are quite optimistic compared to

experimental results.

II. MATERIALS AND METHODS

The finite element package used is ABAQUS. To give

complete picture of the problem the authors have developed

finite element models for single wall carbon nanotubes

(SWCNT), double walls carbon nanotubes (DWCNT) multi

walls carbon nanotube (MWCNT) with triple walls. All

models are for the three different configurations of

nanotubes, i.e. Armchair, Fig. 1, Chiral, Fig. 2, and Zigzag,

Fig. 3.

Fig. 1. Armchair Configuration

Fig. 2. Chiral Configuration

Fig. 3. Zigzag Configuration

Finite Element Modelling and Idealization of MWCNT

CNTs carbon atoms are bonded together with chemical

bonds forming hexagonal lattice. These bonds have a

characteristic bond length aC–C. By considering the bonds as

connecting load-carrying elements, and the atoms as joints

of the connecting elements, CNTs is simulated as space-

frame structures, Fig. 4. In this way the mechanical behavior

can be analyzed. In this investigation, the carbon nanotube

structure is modeled using a 3D beam element. Each element

consists of 2 nodes. Each node has 6 degrees of freedom

(DOF), where the first three degrees of freedom are

translation in x,y and z directions, and the rest are the

degrees of freedom for rotation around x,y and z directions.

This means that each element has 12 degrees of freedom.

Fig. 4. Simulating of CNT as a space-frame structure

This leads to a good idealization of the whole structure in

space under different loading conditions. To find the

equivalent properties of the MWCNT of different

configuration, i.e. Armchair, Chiral and Zigzag, boundary

conditions simulating the actual loading case are shown in

figure 5. In this figure, lower nodes are totally fixed, while the

upper nodes are to be extende one unit length in the vertical

direction. Also, Fig. 6 shows the main dimensions,

parameters and mechanical properties of unit carbon

nanotube cell used in the next different models.

Fig. 5. Main boundary conditions of the developed model

Upper nodes to be

extended one unit

length

Lower nodes to

be totally fixed



I J E N S

Fig. 6. Main dimensions and mechanical properties of developed

carbon nanotube cell

Table I presents the total number of nodes, total number of

elements and total number of active degrees of freedom

(DOF) for each of the three Armchair configuration models.

Fig. 7 shows the developed armchair model of single wall

carbon nanotube (SWCNT). Fig. 8 shows the developed

armchair model of double walls carbon nanotube (DWCNT).

Moreover, Fig. 9 shows the developed armchair model of

triple (multi) walls carbon nanotube (MWCNT).

T ABLE I

ARMCHAIR MODELS

No. of

nodes

No. of

elements

No. of

degrees of

freedom

(DOF)

Single wall

tube

2416 2624 14384

Double wall

tube

5134 5576 30566

Triple wall

tube

8179 8906 48696

Fig. 7. The developed FE mesh for Armchair DWCNT

Fig. 8. The developed FE mesh for Armchair DWCNT

Fig. 9. The developed FE mesh for Armchair MWCNT

Table II presents the total number of nodes, total number of

elements and total number of active degrees of freedom for

each of the three Chiral configuration models. Fig. 10 shows

the developed Chiral model of single wall carbon nanotube

(SWCNT). Fig. 11 shows the developed Chiral model of

double walls carbon nanotube (DWCNT). Moreover, Fig. 12

ac-c(nm) 0.1421

d (nm) 0.147

Modulus of

elasticity

E (TPa) 5.49

Modulus of

rigidity

G(TPa) 0.871

a (nm) 0.246

b (nm) 0.284



I J E N S

shows the developed Chiral model of triple (multi) walls

carbon nanotube (MWCNT).

T ABLE II

CHIRAL MODELS

No. of

nodes

No. of

elements

No. of

degrees of

freedom

Single wall

tube

2548 2768 15176

Double wall

tube

5464 5936 32543

Triple wall

tube

8726 9480 51975

Fig. 10. The developed FE mesh for Chiral SWCNT

Fig. 11. The developed FE mesh for Chiral DWCNT

Fig. 12. The developed FE mesh for Chiral MWCNT

Table III presents the total number of nodes, total number of

elements and total number of active degrees of freedom for

each of the three Zigzag configuration models. Fig. 13

shows the developed Zigzag model of single wall carbon

nanotube (SWCNT). Fig. 14 shows the developed Zigzag

model of Double walls carbon nanotube (DWCNT).

Moreover, Fig. 15 shows the developed Zigzag model of

triple (multi) walls carbon nanotube (MWCNT).

T ABLE III

ZIGZAG MODELS

No. of

nodes

No. of

elements

No. of

degrees of

freedom

(DOF)

Single wall

tube

2576 2800 15358

Double wall

tube

5336 5800 31813

Triple wall

tube

8280 9000 49365

Fig. 13. The developed FE mesh for Zigzag SWCNT



I J E N S

Fig. 14. The developed FE mesh for Zigzag DWCNT

Fig. 15. The developed FE mesh for Zigzag MWCNT

Loading the Armchair onfiguration

Fig. 16 shows SWCNT Armchair model with its 4992

elements, 6239 nodes and 6 DOF. Fig. 17 shows the same

model with the upper nodes having a 1e-3

nm displacement

while the lower nodes are totally fixed, while Fig. 18 shows

the Von Mises stress for the model.

Similar figures exist for the DWCNT and MWCNT Armchair

models but were omitted here for brevity.

Fig. 16. SWCNT Armchair (4992 elements, 6239 nodes, 6 DOF)

Fig. 17. SWCNT Armchair (upper nodes with a 1e

-3 nm displacement,

lower nodes are totally fixed)

Fig. 18. SWCNT Armchair Von Mises stress

Loading the Chiral Configuration

Fig. 19 shows DWCNT Chiral model with its 13168 elements,

16458 nodes and 6 DOF. Fig. 20 shows the same model with

the upper nodes having a 1e-3

nm displacement while the

lower nodes are totally fixed, while Fig. 21 shows the Von

Mises stress for the model. Here the figures for SWCNT and

MWCNT were omitted for brevity.

Fig. 19. DWCNT Chiral (13168 elements, 16458 nodes, 6 DOF)



I J E N S

Fig. 20. DWCNT Chiral (upper nodes with a 1e

-3 nm displacement,


Fig. 21. DWCNT Chiral Von Mises stress

Loading the Zigzag Configuration

Here the SWCNT and DWCNT Zigzag models were omitted.

Fig. 22 shows MWCNT Zigzag model with its 17280

elements, 21597 nodes and 6 DOF. Fig. 23 shows the same

model with the upper nodes having a 1e-3

nm displacement

while the lower nodes are totally fixed, while Fig. 24 shows

the Von Mises stress for the model.

Fig. 22. MWCNT Zigzag (17280 elements, 21597 nodes, 6 DOF)

Fig. 23. MWCNT Zigzag (upper nodes with a 1e

-3 nm displacement,




I J E N S

Fig. 24. MWCNT Zigzag Von Mises stress

DISCUSSION

1) Carbon Nanotube Stiffness

The carbon nanotube (CNT) longitudinal stiffness for the

three configurations (as obtained from the finite element

models) is shown in Fig. 25 as function to the number of

walls. Obviously the stiffness increases with the number of

CNT walls. The stiffness of the Chiral model is the highest,

while fro the Zigzag model is the lowest. The Armchair model

stiffness comes in between.

Fig. 25. CNT stiffness with respect to No. of walls

2) Young's Modulus

The Young's modulus, E, for the CNT in the axial direction is

related to the longitudinal stiffness, K, by the usual relation:

L

AEK (1)

where A is the cross sectional area of the CNT, and

L is the length of the CNT

The area, A, equals the length of the perimeter of the CNT

times the thickness of one wall, t, times the number of walls,

n, i.e.:

DtnA (2)

where D is the diameter of the CNT.

Consequently, Young's modulus is:

Dtn

KLE

(3)

The ratio K/D is the aspect ratio of the CNT.

The CNT wall thickness is equal to the interlayer spacing of

graphite (0.346 nm)

Table IV summarizes the calculation of E for the different

configurations.

T ABLE IV

MODULUS OF ELASTICITY

Modulus of Elasticity (MPa)

SWCNT DWCNT MWCNT

Armchair 908 965 1020

Chiral 961 1016 1071

Zigzag 876 907 937

The value of Young's modulus computed by the present

finite element model agrees very well with those results

reported in [27]. This reference also concludes that in order

to enable comparison with other theoretical work, the most

widely used value of 0.346 nm is adopted for the wall

thickness. This is what is done in this project.

CONCLUSIONS

The finite element (FE) model developed in this work

performs very well and gives good results.

The model has been used to investigate the properties of

single-walled, double-walled and multi-walled carbon

nanotubes (SWCNT, DWCNT and MWCNT) with different

configurations, viz. Armchair, Chiral and Zigzag.

The model development is based on the assumption that

CNTs, when subjected to loading, behave like space-frame

structures. As the FE model comprises small number of

elements, it performs under minimal computational time. This

advantage, in combination with the modeling abilities of the

FE method, extends the model applicability to CNTs with

large number of atoms.



I J E N S

For the values of wall thickness used in the literature, the

obtained values of Young’s modulus agree very well with

the corresponding theoretical results.

The Young’s modulus of Chiral configuration is found to be

larger than that of Armchair and Zigzag. The results

demonstrate that the proposed FE model may provide a

valuable tool for studying the mechanical behavior of CNTs

and nanocomposites.

The obtained values of Young's modulus agree very well

with the corresponding theoretical results and experimental

measurements that are available in the literatue.

The results demonstrate that the proposed finite element

model may provide a valuable tool for studying the

mechanical behavior of carbon nanotubes and nano-

composites based on them.

ACKNOWLEDGEMENTS

This project is funded by the Deanship of Scientific

Research at King Abdulaziz University (Project No. 4-

012/429). The authors would like to express their sincere

gratitude and appreciation to both the Deanship of Scientific

Research and King Abdulaziz University for the help and

support offered during the whole period of conducting this

work.

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