On the tensile behavior of hetero-junction carbon nanotubes

Accepted Manuscript On the tensile behavior of hetero-junction carbon nanotubes Sadegh Imani Yengejeh, Mojtaba Akbar Zadeh, Andreas Öchsner PII:

S1359-8368(15)00068-2

DOI:

10.1016/j.compositesb.2015.02.001

Reference:

JCOMB 3400

To appear in:

Composites Part B

Received Date: 12 June 2014 Revised Date:

29 January 2015

Accepted Date: 1 February 2015

Please cite this article as: Yengejeh SI, Zadeh MA, Öchsner A, On the tensile behavior of heterojunction carbon nanotubes, Composites Part B (2015), doi: 10.1016/j.compositesb.2015.02.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

On the tensile behavior of hetero-junction carbon nanotubes

Sadegh Imani Yengejeh1,2,a,* , Mojtaba Akbar Zadeh1,b, Andreas Öchsner 3,4,c

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Department of Mechanical Engineering, The University of Birjand, Birjand, Iran School of Engineering, Griffith University, Gold Coast Campus, Southport 4222, Australia

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Department of Solid Mechanics and Design, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia - UTM, 81310 UTM skudai, Johor, Malaysia

School of Engineering, The University of Newcastle, Callaghan New South Wales 2308, Australia

*

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Corresponding author. Tel.: +989359459747

E-mail address: [email protected] (Sadegh Imani) b

[email protected] (Andreas Öchsner)

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c

[email protected] (Mojtaba Akbarzade)

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ACCEPTED MANUSCRIPT Abstract Several straight hetero-junctions carbon nanotubes (CNTs) are constructed and their tensile behavior are investigated. It is pointed that the Young’s modulus of hetero-junctions is lower

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than the values of their fundamental homogeneous CNTs due to the existence of pentagonheptagon pair defects. In addition, it is revealed that the Young’s modulus of homogeneous zigzag CNTs increases by increasing the chiral number of these nano-structures. Finally, it is concluded that as the connecting length of the hetero-junctions increases the Young’s

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modulus of these particular CNTs decreases. Therefore, the tensile strength of hetero-

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junctions depends on the presence of pentagon-heptagon pair defects.

Keywords: A. Young’s modulus; B. Mechanical properties; C. Finite Element Analysis

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(FEA).

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ACCEPTED MANUSCRIPT 1. Introduction Since the discovery of carbon nanotubes (CNTs) by Iijima (1991), these nanostructures have been widely studied in basic and applied experiments due to their remarkable mechanical properties, as well as their specific electrical and thermal properties (Falvo et al.,1997;

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Sanvito et al., 2000). During this period until the present time, numerous approaches have been conducted to study the mechanical properties of these nanostructures (Dresselhaus et al., 1995; Yang and Wang, 2007). The methods are basically divided into the two groups of

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computational and experimental approaches. Among the computational approaches, continuum mechanics techniques such as the finite element method (FEM) is one of the most

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popular and common approaches to study the behavior of the CNTs. Based on these investigations, Young’s modulus of homogenous CNTs has been reported to be approximately equal to 1 TPa which is an interestingly high value (Wu et al., 2006). Noticing the fact that there are other configurations of CNTs, e.g. hetero-junction CNTs, it is necessary

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to evaluate the mechanical behavior of these specific CNTs. In the following, the results of various studies on the evaluation of homogenous CNTs mechanical properties are presented.

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Meo and Rossi (2006) proposed a finite element (FE) model, based on the molecular dynamic theory, in order to investigate fracture progress in armchair and zigzag CNTs with defects

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under uniaxial tensile stress. They found the value of Young’s modulus to be around 1 TPa. They obtained the complete load-displacement relationship of the force/displacement curve for a (5,5) and a (9,0) CNT up to the complete fracture. Their results indicated that the effect of chirality on the mechanical properties and failure mode of CNTs was quite significant. Afterwards, Tserpes and Papanikos (2007) studied the effect of Stone-Wales defects on the elastic behavior and fracture of armchair, chiral and zigzag single-walled carbon nanotubes (SWCNTs), applying an atomistic-based progressive fracture model. Their model used the FEM for analyzing the structure of SWCNTs and the modified Morse interatomic potential 3

ACCEPTED MANUSCRIPT for describing the nonlinear force-field of the C-C bonds. They predicted a significant reduction in failure stress and failure strain in armchair SWCNTs, contrary to zigzag ones, ranging from 18% to 25% and from 30% to 41%, respectively. The value of Young’s modulus in their investigations was in the range of 0.97 to 1.03 TPa. Xiao et al. (2008)

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conducted tensile tests on SWCNTs. They applied a self-developed nano-mechanical testing device. Using a statistical approach, they studied the tensile strength of SWCNT bundles, as well as reported data of multi-walled carbon nanotubes (MWCNTs) and SWCNT ropes. They

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found that the tensile distribution of CNTs can be adequately described by a two-parameter Weiball model. They finally proposed an approach for the tensile strength distribution of

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individual CNTs. Then, Kuang and He (2009) investigated the effect of chemical functionalization on the axial Young’s moduli of SWCNTs based on molecular mechanics simulation. They obtained the axial Young’s moduli of both functionalized and nonfunctionalized SWCNTs. Their finding revealed the fact that the Young’s moduli strongly

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depend on the chirality of the CNTs. In addition, molecular mechanics showed that the functionalization of SWCNTs results in a decreases of Young’s moduli of the corresponding SWCNT composites. After that, Talukdar, and Mitra (2010) investigated the influence of

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Stone-Thrower-Wales defects on the mechanical behavior of a zigzag (5,0) single-walled

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CNT, applying molecular dynamics simulation. In their study, the nanotube was subjected to axial stretch and the potential energy was computed for gradually increasing values of strain. Using both the potentials, firstly with a perfect lattice and then by introducing and increasing number of Stone-Thrower-Wales defects, they obtained the mechanical characteristics such as Young’s modulus from the energy strain curve. Their results revealed significant reduction in the values of the mechanical properties with changes in the plastic deformation pattern. An FE model based on the molecular mechanics theory was developed by Mohammadpour and Awang (2011) in order to evaluate tensile properties of SWCNTs. They used the commercial

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ACCEPTED MANUSCRIPT finite element software Ansys in order to study the deformation and fracture of CNTs under tensile strain conditions. In their model, they simulated an individual CNT as a frame-like structure, and the primary bonds between two nearest-neighboring atoms were treated as beam elements. The novelty of their model lies in the use of nonlinear beam elements to

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evaluate SWCNTs tensile failure. They obtained the value of Young’s modulus in the range of 0.88 to 2.04 TPa. Later, Lu and Hu (2012) conducted a computational simulation in order to investigate the mechanical properties of CNTs. They developed an improved 3D finite

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element model for armchair, chiral and zigzag SWCNTs. They evaluated the potentials associated with the atomic interactions within a SWCNT. Then, the elastic stiffness of

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graphene has been investigated. Finally, they examined the effects of diameters and helicity on Young’s modulus of SWCNTs. According to their investigations, the average value of the Young’s modulus was around 1.12 TPa. Afterwards, Ghavamian et al. (2012) constructed several finite element configurations of SWCNTS and MWCNTs in their perfect form. Then,

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after obtaining the mechanical properties such as the elastic modulus of perfect CNTs, they introduced some types of imperfections in different amount to the perfect models in order to make them imperfect. Finally the elastic moduli of the imperfect models were obtained and

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compared with those of perfect ones. Recently, Kinoshita et al. (2013) investigated the

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mechanical properties of SWCNTs with one-dimensional intramolecular junctions (IMJs), applying first-principles density functional theory calculations. The average value of the Young’s modulus obtained in this study was around 0.9 TPa. Despite the fact that the previous investigations have been comprehensive and substantial progress in the study of the mechanical properties of CNTs has been obtained, the concentration of hetero-junction CNTs is only rarely addressed. Hetero-junction CNTs are connections of two dissimilar tubes where pentagon-heptagon pair defects play a significant role in the transition region. The basic configuration of hetero-junction CNTs is formed when 5

ACCEPTED MANUSCRIPT a pair of heptagon and pentagon is introduced to the perfect hexagonal graphite lattice after connecting two nanotubes with different chiralities. These particular CNTs are remarkably important based on their significant characteristics. Although extensive studies have been done to investigate the atomic structural as well as electrical properties of hetero-junctions

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and great achievements have been accomplished, mechanical responses of hetero-junctions are still not fully explored (Kang et al., 2010; Lambin et al., 2006). The main purpose of this study is to continue and broaden the previous investigations, and to study and obtain the

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elastic modulus of straight hetero-junction CNTs. In addition, comparison between the

is taken under special consideration.

2.1 Geometric definition

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2. Material and methods

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mechanical properties of straight hetero-junctions and their fundamental homogeneous CNTs

Carbon nanotube configurations are assumed to be similar to hollow cylinder shaped

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structures based on the major similarity between a CNT and graphene atomic structure with diameters ranging from 1 to 50 nm and length over 10 µm. A single-walled carbon nanotube

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(SWCNT) can be imagined by rolling a graphene sheet into a cylinder. The atomic geometry of CNTs is defined by the chiral or roll-up vector
 and the chiral angle θ . The chiral vector is introduced by two unit vectors,  and  , and two integers, m and n (steps along the unit vectors), as it is presented by the following equation (Dresselhaus et al., 1995):
 =  + ,

(1)

The basic formation of CNTs is defined based on the chiral vector or angle by which the sheet is rolled into a cylinder, and categorized as zigzag, armchair, and chiral. An armchair

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ACCEPTED MANUSCRIPT CNT is built if, in terms of chiral vector ( = ) or in terms of chiral angle ( = 30 ). In the case of ( = 0 ) or ( = 0) the zigzag CNT is constructed and finally a chiral CNT is shaped if (0 < < 30 ) or ( ≠ ≠ 0 ).

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Based on the following equation, the diameter of the CNT can be calculated:  =  √ +  +  /,

(2)

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where  = √3 and  = 0.142 nm is the length of the C-C bond (Dresselhaus et al., 1995). A homogeneous CNT can be viewed as a rolled graphene sheet with specified width, as

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shown in Fig. 1. A hetero-junction CNT, treated as a wrapped graphene sheet with specific geometry, is built by connecting two CNTs through the introduction of Stone-Wales defects (pentagon and heptagon defects) into the connected region, as illustrated in Fig. 2. --- FIGURE 1 ---

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Hetero-junction CNTs are categorized based on the configurations of their two constituent CNTs. For example, a hetero-junction CNT constructed by connecting a (12,0) zigzag CNT

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and a (16,0) zigzag CNT is referred to as a (12,0)-(16,0) hetero-junction CNT (Imani Yengejeh et al., 2014). When investigating the elastic behavior of straight hetero-junction

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CNTs, we also compare them with those of their fundamental CNTs. The geometry of a hetero-junction CNT is more sophisticated as compared with that of a CNT owing to the irregular structure of the connecting region. The axis of the wider CNT is defined as the reference axis of the junction due to the fact that the two constituent CNTs of a heterojunction are able to be aligned on a planar surface. --- FIGURE 2 --The overall length of a hetero-junction is defined as:

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ACCEPTED MANUSCRIPT  =  !"# + $%##&$ "#' + (")&

(3)

where  !"# , $%##&$ "#', and (")& are the length of the thinner tube, the connecting region and the wider tube, respectively (see Fig. 3). By the geometries of the connected CNTs, the length

$%##&$ "#' =

√* ((")& 

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of the connecting region can be approximated as (Qin et al., 2008): −  !"# )

(4)

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where (")& and  !"# are the diameters of the wider and thinner tubes, respectively (see Fig. 3). As the junction consists of two homogeneous CNTs with different diameters, the diameter of

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junction as the average diameter of both CNTs is defined as:  ̅ =  ((")& +  !"# )

(5)

The aspect ratio of the junction is defined as (Li et al., 2013): 0

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/ = 12

(6)

Another technique to characterize a hetero-junction is based on the difference in carbon

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atoms per cross section. This difference is calculated by subtracting the number of carbon

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atoms of the thinner tube’s cross section from those of the wider tube’s cross section (Imani Yengejeh et al., 2014).

Our modeling method follows the idea first suggested by Melchor and Dobado (2004) where the theory of classical structural mechanics was extended into the modeling of carbon nanotubes: In a carbon nanotube, carbon atoms are bonded together by covalent bonds which have their characteristic lengths and angles in a three-dimensional space. It was then assumed that carbon nanotubes, when subjected to loading, behave like space-frame structures. Therefore, the bonds between carbon atoms are considered as connecting load-carrying 8

ACCEPTED MANUSCRIPT generalized beam members, while the carbon atoms act as joints of the members. This idea is illustrated in Fig. 2. Further details on this finite element modeling approach can be found in (Tsai et al., 2010; Rahmandoust and Öchsner, 2011 ; Lu and Hu, 2012; Zuberi and Esat,

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2015). In this paper, the models of hetero-junction CNTs are constructed by the CoNTub software (Melchor and Dobado, 2004; Melchor et al., 2011), a computer program for determining the coordinates of hetero-junctions between two arbitrary carbon nanotubes. Defining the

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chirality and the length of the tubes, the spatial coordinates of the C-atoms and the

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corresponding connectivities (i.e. the primary bonds between two nearest-neighboring atoms) are calculated. Afterwards, the gathered data is transferred to a commercial finite element package, where C–C bonds are modeled as circular beam elements (Kang et al., 2010). These elements (straight two-node beams with Hermitian polynomials, i.e. cubic shape functions) are based on Timoshenko’s beam theory with shear deformation effects included and the

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ability of torsional deformations around the longitudinal axis. Various types of straight hetero-junctions with different chirality are examined as illustrated in Fig. 3.

2.2

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--- FIGURE 3 ---

Material parameters and boundary conditions

Linear-elastic properties of hetero-junctions and their fundamental CNTs under cantilevered boundary conditions are investigated in which one end is fully fixed and the other end is completely free and exposed to an arbitrary small tensile axial load. Since the assigned material properties of the beam elements are linear-elastic and we are evaluating the macroscopic linear-elastic properties, any small applied end load allows the evaluation of the

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ACCEPTED MANUSCRIPT simulated tensile test. Figure 4 shows a (5,5)-(10,10) hetero-junction and its constituent CNTs under two different types of cantilevered boundary conditions. For one case, the wider tube is completed fixed at one side and for the other case, the thinner tube is completely

results. --- FIGURE 4 ---

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fixed. Considering these two different cases, we are able to obtain a wider spectrum of

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For macroscopic space frame structures in the scope of classical structural mechanics, the material properties (e.g. Young’s and shear moduli) and element sectional parameters (e.g.

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second moment of area) can be easily obtained from basic material tests and calculated based on the given dimensions of the cross-sectional area. Nevertheless, for the C-C bonds of CNTs, no such classical tests or geometric derivations are available to define the properties of the equivalent beam elements. Thus, we assume the same values for the equivalent beam

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elements as in the method proposed in (Li and Chou, 2003; To, 2006; Kalamkarov et al., 2006). These effective material (linear-elastic) and geometrical properties were obtained in the mentioned references according to a molecular mechanics method where CNTs were

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regarded as a large molecule consisting of carbon atoms with atomic nuclei treated as material points. Their motions are regulated by a force field, which is generated by electron–

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nucleus interactions and nucleus-nucleus interactions, and usually expressed in the form of steric potential energy. This steric potential energy is in general the sum of contributions from bond stretch interaction, bond angle bending, dihedral angle torsion, and improper (out of plane) torsion, see Fig. 5. Li and Chou (2003) used the harmonic approximation and applied further simplification to derive the energy expressions. These remaining energy expressions were equated with the corresponding energy expressions for a beam obtained from classical structural mechanics, see Fig. 5. Finally, the stiffness parameters of the beam

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ACCEPTED MANUSCRIPT for elongation, bending and torsion were obtained as function of three force field constants (Li and Chou, 2003) for stretching, bond angle bending and torsional resistance. --- FIGURE 5 ---

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--- TABLE 1 ---

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These introduced constants and the element properties are listed in Table 1.

Results and discussion

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In this study, several straight hetero-junctions and their fundamental homogeneous CNTs are simulated by the commercial finite element software MSC.Marc. For all configurations, the thickness is assumed to be equal to 0.34 nm (Kalamkarov et al., 2006). To evaluate the elastic modulus of hetero-junctions and their constituent homogeneous CNTs, an arbitrary

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displacement is applied to one of the CNT’s ends. Thus, the computations yields the reaction force and consequently the elastic behavior of hetero-junctions and homogeneous CNTs under different boundary conditions. Applying one-dimensional Hooke’s law, the Young’s

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modulus of homogeneous CNTs is calculated by the following equations: reaction force (7) cross − sectional area

C = strain = ΔE⁄E =

displacement B. C. (8) length of CNT

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3 = stress = 8A9 =

Q = Young’s modulus = 3⁄C (9) The homogeneous CNTs and their characteristics are presented in Table 2. --- TABLE 2 ---

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ACCEPTED MANUSCRIPT In order to obtain the cross-sectional value A for the case of hetero-junctions, the assumption of series springs is applied. Therefore, the A value for hetero-junctions is defined as: 9 = 29 9 ⁄(9 + 9 )

(10)

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Where 9 and 9 are the cross-sectional areas of the wider and thinner tube, respectively. After simulating the hetero-junction CNTs, their elastic response is investigated and then their finite element results are compared with those of fundamental homogeneous CNTs. It is

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expected in this study that the Young’s modulus of the hetero-junction CNTs should be lower than the values of Young’s modulus of their constituent homogeneous CNTs due to the

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imperfection of the hetero-junctions, i.e. the pentagon-heptagon pair defects which make these specific configurations non-symmetric and restricted their tensile strength. The elastic behavior of hetero-junctions is evaluated for two types of boundary conditions. For the first case, the wider tube is fixed, and for the second case the thinner tube is fixed (see Fig. 4). The

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investigated hetero-junction and their characteristics are listed in Table 3. --- TABLE 3 ---

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The computational results of zigzag homogenous CNTs reveal the fact that as the chiral number of CNTs increases, the Young’s modulus of these homogenous configuration

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increases, as shown in Fig. 6.

--- FIGURE 6 ---

Based on the results, it should be noted that by increasing the connecting length of the heterojunctions, which is listed in Table 3, the Young’s modulus of these specific configurations decreases, as illustrated in Fig. 7. --- FIGURE 7 ---

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ACCEPTED MANUSCRIPT Having a closer look on the computational results, it is noticeable that the Young’s modulus of hetero-junctions with similar connecting length increases by increasing the chiral number of their fundamental CNTs. Figure 8 supports this claim for hetero-junction CNTs.

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--- FIGURE 8 --Finally, as it was already assumed, it can be observed that the Young’s modulus of heterojunctions is lower than the values of their fundamental constituent CNTs for both cases of

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boundary conditions, as illustrated in Fig. 9.

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--- FIGURE 9(a) and 9(b) ---

Conclusions

In this study, several configurations of hetero-junctions and their fundamental homogenous

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CNTs are constructed by an FE approach, and their tensile behavior is investigated through performing computational simulations with different cases of boundary conditions. The Young’s modulus of several hetero-junctions and their constituent homogeneous CNTs are

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obtained and compared with each other. Based on our findings, it is shown that the Young’s

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modulus of zigzag homogeneous CNTs increases as the chirality increases. In addition, it is pointed out that the Young’s modulus of hetero-junctions with similar connecting length increases by increasing the chiral number of their constituent CNTs. One of the most important conclusions of this study reveals that by increasing the connecting length of the hetero-junctions, the Young’s modulus of these specific configurations decreases. Therefore, the tensile strength of hetero-junctions depends directly on the position of pentagon-heptagon pair defects. As it was already assumed, the Young’s modulus of hetero-junctions with pentagon-heptagon pair defects is lower than the values of their fundamental homogeneous 13

ACCEPTED MANUSCRIPT CNTs. Investigation of the behavior of hetero-junction CNTs will be helpful in order to have substantial and comprehensive progress in the study of the mechanical properties of these specific CNTs. Since the actual work is focused on zigzag and armchair configurations in order to reduce the presented cases, we are going to investigate chiral CTNs in our future

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research work.

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References:

[1] Iijima S. Helical microtubules of graphitic carbon. Nature 1991; 354: 56–58. [2] Falvo M.R, Clary G.J, Taylor RM., Chi V, Brooks Jr FP, Washburn S, Superfine R. Bending and buckling of carbon nanotubes under large strain. Nature 1997; 389: 582– 584. [3] Sanvito S, Kwon YK, Toma´nek D, Lambert CJ. Fractional quantum conductance in

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carbon nanotubes. Phys Rev Lett 2000; 84: 1974–1977.

[4] Dresselhaus MS, Dresselhaus G, Saito R. Physics of carbon nanotubes. Carbon 1995; 33: 883–891.

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[5] Yang HK, Wang X.T. Torsional buckling of multi-wall carbon nanotubes embedded in an elastic medium. Compos Struct 2007; 77: 182–192.

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[6] Wu Y, Zhang X, Leung AYT, Zhong W. An energy-equivalent model on studying the mechanical properties of single-walled carbon nanotubes. Thin Wall Struct 2006; 44: 667–676.

[7] Meo M, Rossi M. Tensile failure prediction of single wall carbon nanotube. Eng Fract Mech 2006; 73: 2589–2599. [8] Tserpes KI, Papanikos P. The effect of Stone–Wales defect on the tensile behavior and fracture of single-walled carbon nanotubes. Compos Struct 2007; 79: 581–589.

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ACCEPTED MANUSCRIPT [9] Xiao T, Re Y, Liao K, Wu P, Li F, Cheng HM. Determination of tensile strength distribution of nanotubes from testing of nanotube bundles. Compos Sci Technol 2008; 68: 2937–2942. [10] Kuang YD, He XQ. Young’s moduli of functionalized single-wall carbon nanotubes

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under tensile loading. Compos Sci Technol 2009; 69: 169–175. [11] Talukdar K, Mitra AK. Comparative MD simulation study on the mechanical properties of a zigzag single-walled carbon nanotube in the presence of Stone- Thrower-Wales defects. Compos Struct 2010; 92: 1701–1705.

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[12] Mohammadpour E, Awang M. Predicting the nonlinear tensile behavior of carbon nanotubes using finite element simulation. Appl Phys A 2011; 104: 609–614.

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[13] Lu X, Hu Z. Mechanical property evaluation of single-walled carbon nanotubes by finite element modeling. Compos Part B 2012; 43: 1902–1913.

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[15] Kinoshita Y, Murashima M, Kawachi M, Ohno N. First-principles study of mechanical properties of one-dimensional carbon nanotube intramolecular junctions. Comput Mater Sci 2013; 70: 1–7.

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[16] Kang Z, Li M, Tang Q. Buckling behavior of carbon nanotube-based intramolecular junctions under compression: Molecular dynamics simulation and finite element analysis.

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Comp Mater Sci 2010; 50:253-259. [17] Lambin PH, Triozon F, Meunier V. Electronic transport in nanotubes and through junction of nanotubes, editor. V.N. Popov, P. Lambin (Springer, Netherlands), 2006. P. 123142.

[18] Imani Yengejeh S, Akbar Zadeh M, Öchsner A. On the buckling behavior of connected carbon nanotubes with parallel longitudinal axes. Appl Phys A 2014; 115: 1335-1344. [19] Qin Z, Qin Q, Feng X. Mechanical property of carbon nanotubes with intramolecular junctions: Molecular dynamics simulations. Phys Lett A 2008; 372: 6661– 6666.

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ACCEPTED MANUSCRIPT [20] Li M, Kang Z, Yang P, Meng X, Lu Y. Molecular dynamics study on buckling of singlewall carbon nanotube-based intramolecular junctions and influence factors. Comp Mater Sci 2013; 67: 390-396.

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[21] Tsai J-L, Tzeng S-H, Chiu Y-T. Characterizing elastic properties of carbon nanotubes/polyimide nanocomposites using multi-scale simulation. Compos Part B-Eng 2010; 41: 106-115. [22] Rahmandoust M, Öchsner A. Buckling behavior and natural frequency of zigzag and armchair single-walled carbon nanotubes. J Nano Res 2011; 16: 153-160.

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[23] Lu X, Hu Z. Mechanical property evaluation of single-walled carbon nanotubes by finite element modeling. Compos Part B-Eng 2012; 43: 1902-1913.

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[24] Zuberi MJS, Esat V. Investigating the mechanical properties of single walled carbon nanotube reinforced epoxy composite through finite element modelling. Compos Part B-Eng 2015; 71: 1-9. [25] Melchor S, Dobado JA. CoNTub: An algorithm for connecting two arbitrary carbonnanotubes. Chem J Inf Comput Sci 2004 44: 1639-1646.

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[26] Melchor S, Martin-Martinez FJ, Dobado JA. CoNTub v2. 0-algorithms for constructing c 3-symmetric models of three-nanotube junctions. J Chem Inf Model 2011; 51: 1492-1505. [27] Li C, Chou TW. A structural mechanics approach for the analysis of carbon nanotubes.

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Int J Solids Struct 2003; 40: 2487-2499.

[28] To CWS. Bending and shear moduli of single-walled carbon nanotubes. Finite Elem

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Anal Des 2006; 42: 404-413.

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ACCEPTED MANUSCRIPT FIGURE and TABLE captions Fig. 1. Front view and single ring of the FE models of (a) (20,0), (b) (10,10) homogeneous CNTs. Fig. 2. Simulation of a (12,0)–(16,0) hetero-junction carbon nanotube as a space-frame

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structure and its Pentagon–heptagon pair defects, adapted from (Imani Yengejeh et al., 2014). Fig. 3. Different types of straight hetero-junctions: (a) (3,3)–(5,5); (b) (7,7)-(9,9); (c) (12,0)–

(i) (14,14)– (16,16).

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(16,0); (d) (5,5)–(7,7); (e) (9,0)–(12,0); (f) (11,0)–(12,0); (g) (5,5)–(10,0); (h) (9,9)–(11,11);

Fig. 4. Homogeneous (left- and right-hand side) and straight hetero-junction (middle) CNTs

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under tensile load.

Fig. 5. Equivalence of molecular mechanics and structural mechanics for covalent and noncovalent interactions between carbon atoms. Molecular mechanics model (left) and structural mechanics model (right), adapted from (Imani Yengejeh et al., 2014). Fig. 6. Change in Young’s modulus for zigzag homogeneous CNTs.

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Fig. 7. Change in Young’s modulus for hetero-junction CNTs with different connecting length.

Fig. 8. Change in Young’s modulus for armchair hetero-junction CNTs with similar

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connecting length for wider tube fixed and thinner tube fixed. Fig. 9. Computational results of longitudinal displacement of hetero-junctions and their

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fundamental CNTs with (a) wider tube fixed, and (b) thinner tube fixed.

Table 1. Material and geometric properties of C–C covalent bonds (Lu, 1997). Table 2. Characteristic of simulated homogeneous CNTs. Table 3. Investigated straight hetero-junctions and their characteristics.

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ACCEPTED MANUSCRIPT Table 1 Material and geometric properties of C-C covalent bonds (Lu, J.P., 1997)
 = 651.97 nN/nm

Corresponding force field constants


 = 0.8758 nN nm/rad

 = Young  s modulus =

!

"

#

$% &

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 = 0.2780 nN nm/rad

5.484 × 10)* N/nm

.

2.159 × 10)* N/nm

4# = bond radius = 27

0.0733 nm

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+ = shear modulus = (012) !

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&

%>? @ $

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899 = 8:: = second moments of area =

2.2661 × 10)A nm$

ACCEPTED MANUSCRIPT CNT Type Chirality (n,m)

Length (nm) Diameter (nm) Young’s modulus (TPa)

(3,3)

15

0.407

1.039

Armchair

(5,5)

15

0.678

1.039

Armchair

(6,6)

15

0.814

1.039

Armchair

(7,7)

15

0.949

1.039

Armchair

(9,9)

15

1.22

Armchair

(10,10)

15

1.356

Armchair

(11,11)

15

1.492

Armchair

(14,14)

15

1.898

Armchair

(16,16)

15

2.171

Zigzag

(8,0)

15.05

0.626

Zigzag

(9,0)

15.05

Zigzag

(10,0)

15.05

Zigzag

(11,0)

15.05

Zigzag

(12,0)

15.05

Zigzag

(16,0)

Zigzag

(20,0)

1.039

1.039

1.039

M AN U

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1.039 1.039

1.021 1.025

0.783

1.028

0.861

1.031

0.939

1.032

15.05

1.253

1.037

15.05

1.566

1.039

TE D

0.705

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Armchair

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Chirality (n,m) (n , m ) Difference in c-atoms per cross section (3,3)-(5,5) 4

Diameter of the junction ̅ (nm) 0.539

(7,7)-(9,9)

4

1.707

7.131 7.131

0.738

0.974

0.981

(c)

(12,0)-(16,0)

4

1.088

7.074 7.074

0.852

0.962

0.962

(d)

(5,5)-(7,7)

4

0.808

7.131 7.131

0.738

0.952

0.953

(e)

(9,0)-(12,0)

3

0.816

7.131 7.131

0.639

0.961

0.960

(f)

(11,0)-(12,0)

1

0.894

7.394 7.394

0.213

1.023

1.023

(g)

(5,5)-(10,10)

10

1.01

6.578 6.578

1.845

0.824

0.865

(h)

(9,9)-(11,11)

4

1.346

7.131 7.131

0.738

0.993

0.992

(i)

(14,14)-(16,16)

4

2.02

7.131 7.131

0.738

1.001

1.001

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EP

TE D

M AN U

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(b)

Length (nm) Young’s modulus (TPa) Lthin Lwide Lconnecting First case Second case 7.131 7.131 0.738 0.888 0.889

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Hetero-junction cf. Fig. 3 (a)

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