A Comprehensive Numerical Investigation on the Mechanical Properties of Hetero-Junction Carbon Nanotubes

Transcription

1 Commun. Theor. Phys. 64 (2015) Vol. 64, No. 2, August 1, 2015 A Comprehensive Numerical Investigation on the Mechanical Properties of Hetero-Junction Carbon Nanotubes Ali Ghavamian and Andreas Öchsner School of Engineering, Griffith University, Gold Coast Campus, Southport 4222, Australia (Received December 22, 2014; revised manuscript received March 13, 2015) Abstract A set of forty-three hetero-junction CNTs, made of forty-four homogeneous carbon nanotubes of different chiralities and configurations with all possible hetero-connection types, were numerically simulated, based on the finite element method in a commercial finite element software and their Young s and shear moduli, and critical buckling loads were obtained and evaluated under the tensile, torsional and buckling loads with an assumption of linear elastic deformation and also compared with each other. The comparison of the linear elastic behavior of hetero-junction CNTs and their corresponding fundamental tubes revealed that the size, type of the connection, and the bending angle in the structure of hetero-junction CNTs considerably influences the mechanical properties of these hetero-structures. It was also discovered that the Stone-Wales defect leads to lower elastic and torsional strength of hetero-junction CNTs when compared to homogeneous CNTs. However, the buckling strength of the hetero-junction CNTs was found to lie in the range of the buckling strength of their corresponding fundamental tubes. It was also determined that the shear modulus of hetero-junction carbon nanotubes generally tends to be closer to the shear modulus of their wider fundamental tubes while critical buckling loads of these heterostructures seem to be closer to critical buckling loads of their thinner fundamental tubes. The evaluation of the elastic properties of hetero-junction carbon nanotubes showed that among the hetero-junction models, those with armchair-armchair and zigzag-zigzag kinks have the highest elastic modulus while the models with armchair-zigzag connections show the lowest elastic stiffness. The results from torsion tests also revealed the fact that zigzag-zigzag and armchair-zigzag hetero-junction carbon nanotubes have the highest and the lowest shear modulus, respectively. Finally, it was observed that the highest critical buckling loads belong to armchair-armchair hetero-junction carbon nanotubes and the lowest buckling strength was found with the hetero-junction models with armchair-zigzag connection. PACS numbers: w, De, jd, U-, g Key words: finite element method, carbon nanotube, hetero-junction, pentagon-heptagon defect, Euler buckling, elasticity 1 Introduction Carbon nanotubes (CNTs), discovered by Iijima in 1991, are unique nanostructures which have drawn worldwide attention because of their exceptional mechanical, electrical and thermal properties. Among all the outstanding properties of these carbon-based nanomaterials, their substantial mechanical and physical properties such as strength and lightness have introduced wide application potentials in many nanoindustry fields as either standalone nanomaterials or composite materials reinforcement which have encouraged many scholars to investigate their mechanical properties. [1 2] Since the discovery of CNTs, these interesting nanostructures have been characterized by numerous approaches which are generally divided into two groups of experimental and computational approaches, based on which the Young s modulus of CNTs has been reported to be approximately equal to a surprisingly high value of 1 TPa. Among the computational techniques which are generally divided into the two groups of molecular dynamics (MD) and continuum mechanics (CM) techniques, the finite element method (FEM) has earned noticeable popularity among scholars. [3] 1.1 Literature Review: Mechanical Properties of Homogeneous CNTs In 1997, an experimental study was performed by Lu [4] for the evaluation of the tensile and shear moduli of CNTs in which the empirical force constant was considered. Based on his results, values of 0.97 TPa and 1 TPa were obtained as the Young s modulus for single-walled nanotubes (SWCNTs) and for multi-walled carbon nanotubes (MWCNTs), respectively. Furthermore, a general value of 0.5 TPa was reported for the shear modulus of CNTs. Also, the Young s modulus of CNTs was evaluated experimentally by Wu et al. by combining optical characterization and magnetic actuation techniques. According to their results, 0.97 TPa and 0.99 TPa were reported as the Young s modulus values for individual CNTs and for five equally weighted CNT bundles, respectively. [5] Li and Chou [6] employed a structural mechanics approach to investigate the elastic properties of CNTs. They Corresponding author, alighavamian@yahoo.com andreas.oechsner@gmail.com c 2015 Chinese Physical Society and IOP Publishing Ltd

2 216 Communications in Theoretical Physics Vol. 64 reported a Young s modulus in the range of 0.89 TPa and TPa and a shear modulus between 0.22 TPa and 0.48 TPa. To [7] obtained the Young s and shear moduli of CNTs to be TPa and 0.47 TPa, respectively by the finite element method based on the consideration of the effect of the Poisson s ratio on the CNT s elastic properties. Nahas and Abd-Rabou [8] also used the finite element method based on a structural mechanics approach for CNT modeling by which they calculated the elastic modulus that was equal to 1.03 TPa. Kalamkarov et al. [9] evaluated the mechanical properties of single-walled carbon nanotubes (SWCNTs) by two different approaches of analytical and finite element methods, based on continuum mechanics. According to their analytical calculations, the Young s and shear moduli of SWCNTs were reported to be 1.71 TPa and 0.32 TPa, respectively, while their finite element simulation yielded the elastic modulus to be between 0.9 TPa to 1.05 TPa for SWCNTs and 1.32 TPa to 1.58 TPa for double-walled carbon nanotubes (DWC- NTs). Furthermore, the shear modulus of 0.14 TPa to 0.47 TPa for single-walled and 0.44 TPa to 0.47 TPa for three- and four-walled carbon nanotubes was obtained. The elastic and torsional properties of SWCNTs were evaluated through analytical modeling by Natsuki et al. Finally, they reported the axial modulus of SWCNTs to be in the range of 0.73 TPa and 1.1 TPa and the shear modulus between 0.37 TPa and 0.32 TPa, [10] while these properties were investigated by Rahmandoust and Öchsner[11] by the finite element method, based on the substitution of covalent bonds between atoms by beam elements, to model MWCNTs which finally yielded a Young s modulus of MWCNTs in the range of 1.32 TPa and 1.58 TPa and the shear modulus to be between 0.37 TPa and 0.47 TPa. Meo and Rossi [12] also created an FE model of an SWCNT and eventually, reported a Young s modulus for SWCNTs of around 1 TPa. Ávila and Lacerda [13] employed the ANSYS commercial software for simulation of the three major SWCNT configurations, based on the finite element method. They finally obtained a Young s modulus between 0.97 TPa and 1.30 TPa for SWCNT models. Shokrieh and Rafiee [14] linked the lattice molecular structure and the equivalent discrete frame structure for evaluation of the elastic moduli of graphene sheets and CNTs. Classical mechanics yielded a Young s modulus of about 1.04 TPa for graphene sheets and in the range of TPa and TPa for CNTs. The elastic properties of CNTs were probed by Song et al. [15] in their perfect form and under the influence of Si-doping. According to their results, the Young s modulus of perfect SWCNTs varies in the range of ± TPa and decreases as a result of Si-doping. Jin and Yuan [16] predicted the elastic and torsional properties of SWCNTs by means of a numerical simulation using the energy and force approach. According to their study, the Young s modulus was reported to be between TPa and TPa and the shear modulus of SWCNTs was calculated in the range of TPa and TPa. Tserpes and Papanikos also developed a threedimensional finite element model, according to which the Young s modulus varied from 0.97 TPa to 1.03 TPa and the shear modulus was obtained in the range of TPa and TPa. [17] A molecular dynamics approach was employed to model SWCNTs under pure torsion by Yu et al. Their results showed that the shear modulus oscillates between 0.37 TPa and 0.5 TPa. [18] Liu and Chou studied the mechanical properties of MWCNTs in which MWCNTs were simulated as a frame-like structure, based on the finite element method. Accordingly, the value of the Young s modulus was predicted in the range of 1.05 ± 0.05 TPa and the shear modulus was found to be about 0.40 ± 0.05 TPa. [19] Fan et al. [20] also used the finite element method to investigate the mechanical properties of MWCNTs. Their results revealed that the Young s modulus of MWCNTs is about 1 TPa and their shear modulus varies between 0.35 TPa and 0.45 TPa. They also discovered that the critical buckling load of DWCNTs decreases by the increase in their aspect ratio. Lu et al. proposed an MD simulation to evaluate the buckling behavior of single and multi-walled CNTs, in terms of compressive strain. Their results demonstrated that smaller nanotubes have higher buckling stability. They also pointed out that the buckling behavior of MWCNTs is considerably dominated by the size of their outermost shell. [21] Liew et al. also investigated the buckling properties of SWCNTs and MWCNTs by MD models. They finally obtained an optimum diameter for buckling load peaks of SWCNTs. [22] A continuum mechanics analysis was performed by Ru to study the buckling properties of CNTs, using a simple shell model Their obtained results expressed the fact that the van der Waals (vdw) forces between the walls of CNTs do not have substantial influence on the critical strain for the infinitesimal buckling load of a DWCNT, [23] while Chang and Li s analytical study on the critical buckling strain of axially compressed chiral SWCNTs demonstrated that zigzag tubes have more buckling resistance than armchair ones with the same diameter. They also expressed that the effect of the van der Waals interaction between the layers of DWCNTs is rather negligible. [24] Yao et al. created finite element models of SWCNTs and MWCNTs to investigate their bending deformation and buckling behavior. Finally, they proposed an explicit relationship between the critical bending buckling curvature and the diameter, length and chirality of the CNTs. [25] Rahmandoust and Öchsner also investigated the buckling properties of SWCNTs by analytical and finite element approaches. Their results from analytical derivations noticeably complied with those from the finite element approach which demonstrates the fact that the classical Euler equation can successfully predict the buckling

3 No. 2 Communications in Theoretical Physics 217 behavior of SWCNTs, when they are assumed to be hollow cylinders. [26] A molecular dynamics approach was used by Xin et al. to evaluate the buckling behavior of perfect and defective SWCNTs under axial compression by MD simulation. Their research showed that the length, chirality, temperature, and the initial structural defects of the tubes have a considerable dominance on buckling and axially compressive properties of SWCNTs. They also reported that the classical Euler formula can be used to predict the axial critical buckling loads of SWCNTs with a large aspect ratio at normal temperatures. Finally, they mentioned that the buckling strength of CNTs with carbon vacancies is lower than that of perfect CNT structures. [27] Ghavamian et al. [28 29] also performed a numerical investigation on the elastic and torsional properties of MWCNTs in their perfect and atomically defective forms, based on a finite element simulation. Based on their study, the obtained Young s modulus of CNTs was about 1 TPa and their shear modulus varied between TPa and TPa. They also discovered that CNTs have anisotropic behavior and the existence of any type of atomic defect in the structure of CNTs results in lower elastic and torsional strength. They also discovered that the decrease in the value of the elastic modulus of defective CNTs follows a particular trend which could be expressed by mathematical relations in terms of the amount of atomic defects. Later on, Ghavamian and Öchsner[30] continued with the same approach and evaluated the buckling properties of MWCNTs. Their results revealed the fact that atomic defects in the structure of CNTs reduces their critical buckling load and for the first time they presented mathematical relations for the prediction of the buckling behavior of defective CNTs, in terms of the amount of atomic defects. 1.2 Literature Review: Mechanical Properties of Hetero-Junction CNTs Apart from the mechanical properties of individual CNTs, experimental observation shows that it is also possible that two CNTs link together by a heptagon-pentagon knee and construct hetero-junction (composite) CNTs. The importance of hetero-junction CNTs became highlighted when it was discovered that the junctions in their structure behave like nanoscale metal/semiconductor or semiconductor/semiconductor junctions and thus could be employed as the building blocks of nanoscale electronic devices made entirely of carbon. Moreover, these nanostructures which are widely employed as solar cells, offer high mobility, excellent air stability and high conductivity as well as exceptional mechanical characteristics i.e. lightness, and high stiffness and large aspect ratio which are inherited from their fundamental constructive tubes that lead to higher efficiency of these solar cells. [31 33] Therefore, several scholars devoted their research to the mechanical modeling and characterization of such composite CNTs. Jia et al. produced an efficient solar cell by combining hetero-junction carbon nanotubes and silicon, doped with diluted HNO 3 and achieved an efficiency of 13.8%. They realized that acid infiltration of nanotube networks enhances the efficiency of silicon-carbon nanotube heterojunction solar cells considerably by reducing their internal resistance. They also learned that the fabrication process is significantly simplified, compared with the conventional silicon cells. [33] Meunier et al. connected different CNTs by several pentagon-heptagon cell insertions between them. They discovered that the kink made of the pentagon-heptagon pair insertion causes a bending angle between two connected CNTs in the structure of hetero-junction CNTs. After creating hetero-junction CNTs, they calculated the energy of the pentagon-heptagon defect in the junction of the tubes of about 6 ev compared with each of the separate tubes. [34] Saito et al. [35] created a three-dimensional model of hetero-junction CNTs by pentagon and heptagon pairs through a projection method. Based on their modeling, the connecting kink introduced a three-dimensional dihedral angle. By considering the calculated tunneling conductance for a metal-metal CN junction, and a metalsemiconducting CN junction, they concluded that these junctions work as the smallest semiconductor devices. Xosrovashvili and Gorji [36] numerically simulated a hybrid heterojunction SWCNT and a GaAs solar cell by the AMPS-1D device simulation tool and analyzed their physics and performances by junction parameters. Finally, they reported that the electrical parameters of the system decrease as a result of an increase in the concentration of a discrete defect density in the absorber layer. Dunlap [37] employed two different approaches to connect different carbon tubules. First, a graphene ribbon with infinite length but finite width was assumed to be rolled into an infinite set of different tubules with different helicities. Then, connections between every two CNTs were constructed by locating different obtained configurations one after each other by translation and rotation. On the other hand, he added one pentagon and one heptagon cell to connect half of a perfect infinite tubule into a different tubule. His modeling illustrated the fact that the best way to connect sets of tubules is to use pentagonheptagon connections between them in a pair-wise way, which creates a bending angle in the kink location with an ideal bending angle of 30. Three-dimensional finite element models of five straight hetero-junction CNTs with armchair-armchair connections were created by Rajabpour et al. [38] and their elastic modulus was investigated with the consideration of two variables of length and chirality of the tubes. Their results demonstrated that with the constant length, the Young s modulus of hetero-junction CNTs increases with an increase in chirality of their fundamental tubes while with a constant chirality, by increasing the length, a slight decrease could be observed in the Young s modulus of hetero-junction CNTs. They finally emphasized

4 218 Communications in Theoretical Physics Vol. 64 that the finite element method is such an appropriate and valuable method for investigating the mechanical properties of hetero-junction CNTs. The same approach was also employed by Hemmatian et al. [39] for presenting the three-dimensional models of the defected, twisted, elliptic, bended and armchair-armchair hetero-junction CNTs and investigating their elastic properties, based on the finite element method. According to their results, the mechanical properties of twisted, elliptic and bended CNTs are significantly influenced by the torsional angle, the cross sectional aspect ratio and the bending angle of CNTs, respectively. They also pointed out that the mechanical properties of straight hetero-junction CNTs are considerably dominated by the choice of length and chirality of fundamental carbon nanotubes. They eventually reported the Young s modulus of different armchair-armchair heterojunction CNTs with the lengths between 40 Å and 80 Å to be in the range of 0.96 TPa and 1.27 TPa and obtained their shear modulus to be between 0.08 TPa to 0.53 TPa. They also concluded that the finite element method could be considered a valuable approach for the characterization of CNTs. Finally, Yengejeh et al. [40 41] studied buckling and torsional properties of nine straight hetero-junction CNT models by numerical simulation based on the finite element method. For such a research, they simulated nine straight hetero-junction CNTs with armchair-armchair and zigzag-zigzag connections and then evaluated their buckling and torsional properties under the cantilever and twisting boundary conditions, respectively. Their results showed that both the critical buckling load and twisting angle of hetero-junction CNTs are in the range of the values for their fundamental homogeneous CNTs when they are subjected to compressive load and twisting torque, respectively. They also reported the critical buckling load of straight hetero-junction CNTs to be in the range of about 0.2 nn and 13 nn. They finally concluded that the buckling and torsional strength of straight hetero-junction CNTs and their fundamental tubes increased by increasing the chiral number of both armchair and zigzag CNTs. Although many investigations have been done on the properties of CNTs, it seems that most researchers investigated the electrical properties of hetero-junction CNTs and that less attention has been paid to their mechanical properties. Literature on the mechanical properties of hetero-junction CNTs also confirms the fact that rarely has a comprehensive study on the mechanical and linear elastic behavior of hetero-junction CNTs been performed and presented. For example, most of the investigations have been restricted to insufficient numbers of heterojunction CNT models or their focus has only been on the hetero-junction CNTs with straight connections and the other possible connections (bent connections) have been neglected which does not allow reaching a fairly trustable conclusion about the mechanical properties of these nanomaterials. The actual research is devoted to evaluate the linear elastic behavior of a considerable number of heterojunction CNTs with all possible connection types (straight and bent connections), made of a variety of homogeneous CNTs of different configurations and chiralities, which provides a more realistic and trustable insight about the mechanical properties of these hetero-structures for their proper selection and applications in the nanoindustry. 2 Methodology 2.1 Atomic Geometry of Homogeneous CNTs Carbon nanotubes are often assumed to be hollow cylinders with the thickness of a single carbon atom (0.43 nm) [44] which are imagined to be created by rolling a graphene sheet, with diameters ranging from 1 nm to 50 nm and lengths over 10 µm. These nanomaterials are structured by hexagonal unit cells, made of carbon atoms, which are connected to each other by covalent carbon-carbon (C-C) bonds. These bonds are often modelled as 1D beam elements with a Young s modulus (E) of N/nm 2 and second moments of area (I xx = I yy ) of N/nm 2. [28] The CNTs atomic structure is defined by its tube chirality, or helicity which is described by the chiral vector C h and the chiral or twisting angle θ. The chiral vector is expressed by two unit vectors and a 1 and a 2 and two integers m and n (steps along the unit vectors) which is presented by Eqs. (1): [29] C h = n a 1 + m a 2, (1) Based on the chirality of the tube or the chiral angle by which the graphene sheet is rolled into a cylinder, three fundamental configurations (chiral, armchair and zigzag) are defined for CNTs. In terms of the chiral vector (m and n) or in terms of the chiral angle θ, a chiral structure is formed when (0 < θ < 30 ) or (m n 0). Likewise, if (m = n) or (θ = 30 ), an armchair CNT is formed and if (θ = 0 ) or (m = 0), a zigzag CNT is constructed which are illustrated in Figs. 1(a) 1(c). The radius of an SWCNT is also calculated by Eq. (2) where a 0 = 3b and and b = nm is the length of the C-C bond: [28,30] R CNT = a 0 m2 + mn + n 2 /2π. (2) 2.2 Atomic Geometry of Hetero-Junction CNTs From a local atomic view, hetero-junction CNTs are structured by carbon hexagonal unit cells, almost the same as their fundamental CNTs. However, from a global view, experimental observations revealed that heterojunction CNTs are constructed of two CNTs with different chiralities which connect together by a kink with pentagon-heptagon cell pairs (also known as Stone Wales defects) in the bent places of the connection [34,40] (see Figs. 1(d) 1(g)). For analytical calculations to determine the moduli and buckling loads of hetero-junction CNTs, the classical equations from continuum mechanics were employed to check if these simple relationships could be used to approximately describe the mechanical behavior

5 No. 2 Communications in Theoretical Physics 219 of the hetero-junction CNTs. Therefore, the lengths of hetero-junction CNTs are assumed to be calculated by summing up the length of the thinner and wider tubes and the kink part which is expressed by Eqs. (3) and (4) where d is the diameter of the fundamental tubes [40] and D d is the difference in the diameter of the wider and thinner fundamental tubes which is expressed by Eq. (5). Finally, hetero-junction CNTs were assumed to be homogeneous CNTs with a constant area and diameter which were addressed by A Hetero and D Hetero, respectively whose values were approximated by a general weight percentage relation, presented by Eqs. (6) (8). L Hetero = L thinner tube + L kink + L wider tube, (3) 3 L Kink = 2 π(d d), (4) D d = d wider tube d thinner tube, (5) A Hetero = L wider tube A wider tube + L kink A kink + L thinner tube A thinner tube L Hetero, (6) D Hetero = L wider tube D wider tube + L kink D kink + L thinner tube D thinner tube L Hetero, (7) A kink = A wider tube + A thinner tube 2. (8) Fig. 1 (a), (b) and (c) CNT configuration; (d) Stone-Wales defec; (e), (f) and (g) armchair-armchair, zigzagzigzag and armchair-zigzag kinks, respectively. According to previous investigations the anisotropic behavior of CNTs implies the fact that classical solid mechanics relations cannot be used for interrelating the elastic, shear and buckling behavior of CNTs. [29] Thus, these quantities must be examined and evaluated under pure tensile, torsion, and buckling loads. 2.3 Homogeneous and Hetero-Junction CNTs Finite Element Simulation and Connection Classifications In this research, fundamental homogeneous SWCNTs were simulated in a commercial finite element software, MSC. Marc in which each SWCNT was assumed to be

6 220 Communications in Theoretical Physics Vol. 64 formed of single rings that were copied along the principal axis of the SWCNT. Single rings themselves were constructed of hexagonal unit cells, each with six sides, representing a C-C bond as a 1D beam element arranged in a spatial network. Each of these beams has at its nodes with six degrees of freedom, i.e. three global displacements and three global rotations. For the simulation of heterojunction CNTs, first, the coordinates of carbon atoms (as model nodes) and their connections (as elements of the models, representing C-C bonds) were provided by CoNTub V1.0 (an algorithm for connecting two arbitrary carbon nanotubes) [45] which is specialized in CNT simulations. Then, this data was imported into MSC. Marc and modified to model hetero-junction CNTs. Finally, these models were subjected to tensile, torsion and buckling tests to evaluate their linear elastic behavior. On the other hand, it has been discovered that the existence and also the size of such kinks in the structure of CNTs seems to change their mechanical properties. These kinks also cause bending in the structure of heterojunction CNTs occasionally. Generally, the kink types are divided into two groups of straight and bent connections as illustrated in Figs. 1(e) 1(g). Straight connections happen when the orientations of the fundamental connecting tubes are parallel and they are of the same fundamental configurations. [35,42] Bent connections are also divided into two groups of large-angle (with bending angle about 36 ) [43] and small-angle connections (with bending angle about 12 ) [31,34] which are created when hetero-junction CNTs are made of the fundamental tubes with different orientations. Another classification of the kinks depends on their corresponding homogeneous tubes configurations which build the hetero-junction CNTs. Based on such a categorization, there are six possible hetero-junction CNTs, i.e. armchair-armchair, zigzag-zigzag, armchairzigzag, armchair-chiral, zigzag-chiral, and chiral-chiral hetero-junction CNTs. However, previous studies focused only on straight hetero-junction CNTs, which are constructed of two kink types of armchair-armchair and zigzag-zigzag connections. 3 Results and Discussion A collection of fortythree hetero-junction CNTs, made of fortyfour straight fundamental homogeneous CNTs of different chiralities and configurations with a length of about 15 nm, were simulated. All six possible hetero-connection types between homogeneous CNTs i.e. armchair-armchair, zigzag-zigzag, armchair-zigzag, armchair-chiral, zigzag-chiral and chiral-chiral were also included in the hetero-junction CNTs modeling. Finally, the mechanical properties of all the CNT models were evaluated with the assumption of linear elastic behavior, under tensile, torsion and buckling tests and compared as illustrated in Figs. 2(a) 2(c). Fig. 2 (a), (b) and (c) hetero-junction CNTs under tensile, torsion and buckling tests, respectively. 3.1 Elastic Modulus of Hetero-Junction CNTs and Their Fundamental Tubes After the simulation of different types of heterojunction CNTs and their fundamental tubes, they were elongated under the tensile test conditions by an arbitrary displacement and then the subsequent reaction forces of the models were obtained. Finally, the Young s moduli of the CNT models were calculated by Eqs. (9) (11). σ = stress = P A = ε = strain = L L reaction force cross-sectional area, (9) = axial displacement length of CNT, (10) E = Young s modulus = σ ε. (11)

7 No. 2 Communications in Theoretical Physics 221 According to the obtained results the Young s modulus of homogeneous CNTs was obtained about 1 TPa, which was in a considerable agreement with previous studies in the literature. For example, the Young s modulus of homogeneous armchair CNTs was about TPa. Zigzag CNTs elastic modulus varied between TPa and TPa and finally, the Young s modulus of chiral CNTs were obtained in the range of TPa and TPa. It was also observed that the Young s modulus of CNTs has a very slight increase by increasing their diameter in zigzag and chiral tubes while for armchair tubes the value of this quantity remained rather constant. The results from the tensile test on the hetero-junction CNTs demonstrate the fact that the Stone Wales defect in the structure of CNTs leads to lower tensile stiffness. This decrease in the Young s moduli of hetero-junction CNTs also depends on the type and size of the kink which is determined by the diameter of the wider and thinner fundamental tubes of the hetero-junction CNTs and the difference between their diameter (D d ) and also the magnitude of the bending angle of the kink. According to the results in Figs. 3 8, armchair-armchair and zigzag-zigzag heterojunction CNTs had the highest Young s moduli between 0.7 TPa and TPa which noticeably agree with the results from Hemmatian et al. [39] who reported this quantity in the range of 0.96 TPa and 1.27 TPa for straight hetero-junction CNTs while the models with armchairzigzag kinks showed the generally lowest elastic moduli in the range of 0.02 TPa and 0.21 TPa. The other heterojunction CNTs elastic modulus was also obtained in the range of about 0.05 TPa to 0.57 TPa. Fig. 3 Elastic modulus of armchair-armchair hetero-junction CNTs. Fig. 4 Elastic modulus of zigzag-zigzag hetero-junction CNTs.

8 222 Communications in Theoretical Physics Vol. 64 Fig. 5 Elastic modulus of armchair-zigzag hetero-junction CNTs. Fig. 6 Elastic modulus of armchair-chiral hetero-junction CNTs. Fig. 7 Elastic modulus of zigzag-chiral hetero-junction CNTs.

9 No. 2 Communications in Theoretical Physics 223 Fig. 8 Elastic modulus of chiral-chiral hetero-junction CNTs. 3.2 Shear Modulus of Hetero Junction CNTs and Their Fundamental Tubes Hetero-junction CNTs and their corresponding fundamental tubes were twisted for an arbitrary angle θ and then the reaction torque was obtained to calculate and compare their shear moduli for investigation of their torsional properties by Eq. (12). G = TL θj, (12) in which θ, T, L, and J represent the twisting angle, the torque, the length and the polar moment of inertia (calculated by Eq. (13)), respectively, whose values are calculated by the following equation: J = r 2 da = 2π rout r 3 dr = π(r4 out r4 in ) r in 2, (13) where r out and r in are the outer and inner radii of the CNTs, respectively which are equal to the radius of the CNT (r) plus and minus half of the wall thickness of a CNT. According to the results, reflected in Figs which confirm the results from Ghavamian et al. [29] and other researchers in the literature, the shear modulus of the homogeneous CNTs varied between 0.1 TPa and TPa and faced an overall gradual decrease by increasing the diameter of the CNTs. It was also determined that the shear modulus of hetero-junction CNTs generally decreased as a result of the Stone Wales defect in their structures, compared to the ones for their fundamental tubes. This quantity appeared to generally vary from about 0.1 TPa to 0.37 TPa which agrees with the figures, presented by Hemmatian et al. [39] who suggested the range of TPa and TPa for the shear modulus of straight hetero-junction CNTs. The results clearly postulated that the more D d is, the more difference will be observed between shear strength of hetero-junction CNTs and their fundamental tubes strength. In most cases, the shear modulus of hetero-junction CNTs seemed to be closer to the shear modulus of their corresponding wider tubes. Fig. 9 Shear modulus of armchair-armchair hetero-junction CNTs.

10 224 Communications in Theoretical Physics Vol. 64 Fig. 10 Shear modulus of zigzag-zigzag hetero-junction CNTs. Fig. 11 Shear modulus of armchair-zigzag hetero-junction CNTs. Fig. 12 Shear modulus of armchair-chiral hetero-junction CNTs.

11 No. 2 Communications in Theoretical Physics 225 Fig. 13 Shear modulus of zigzag-chiral hetero-junction CNTs. Fig. 14 Shear modulus of chiral-chiral hetero-junction CNTs. Finally, among all the hetero-connections, it seems that zigzag-zigzag hetero-junction CNTs have the highest (mostly about 0.3 TPa) and armchair-zigzag heterojunction CNTs have the lowest shear strength (mostly about 0.15 TPa). The shear modulus of other types of hetero-junction CNTs were generally obtained in the range of 0.15 TPa to 0.2 TPa. 3.3 Critical Buckling Loads of Hetero Junction CNTs and Their Fundamental Tubes Finally, buckling tests were performed on heterojunction CNTs and their fundamental tubes under cantilever boundary conditions. For the simulation of this test, first the critical buckling load of homogeneous CNTs was calculated analytically by the classical Euler formula, which is presented by Eq. (14) and then this quantity was obtained through buckling test simulation and compared with each other. In the following equation, P cr is the critical buckling load, n is the buckling mode, E is the elastic modulus of the model, I is the second moment of area, calculated by Eq. (15). K and L are the effective length constant and the length of the tube, respectively. Based on the type of boundary conditions and the number of mode, K is assumed to be equal to 2 and n = 1. P cr = n2 π 2 EI (KL) 2, (14) I = π[(d + t) 4 + (d t) 4 ]/64. (15) For the buckling test finite element simulation, an arbitrary buckling force was exerted on all the models and their critical buckling loads were obtained. Although in most cases, the chosen homogeneous and hetero-junction CNT models for this research are different from the models, studied by other researchers in the literature, for the common study cases, the results in this research are in a good agreement with the results, presented in the liter-

12 226 Communications in Theoretical Physics Vol. 64 ature. For example, the critical buckling loads of (9,9), (10,10), (11,11), (14,14), (0,9), (0,11), (0,12), and (0,16) homogeneous CNTs were obtained as 3.31 nn, 4.53 nn, 6.01 nn, nn, 0.62 nn, 1.15 nn, 1.49 nn, and 3.56 nn, respectively which confirms the results from research by Yengejeh et al. [40] who reported 3.15 nn, 4.31 nn, 5.68 nn, nn, 0.58 nn, 1.08 nn, 1.4 nn, and 3.34 nn for these models, respectively or the critical buckling loads of (5,5) (10,10) and (0,11) (0,12) straight hetero-junction CNTs were found to be about 1.6 nn and 1.1 nn which confirms the values, presented by Yengejeh et al. [40] for critical buckling load of these hetero-structures. The results in Tables 1, 2, and 3, demonstrate the fact that by increasing the diameter of the homogeneous CNTs, their critical buckling loads increase. The results from FEM revealed that this load varied between 0.58 nn and nn for armchair CNTs with diameters in the range of 0.68 nm and 2.58 nm, and 0.17 nn and 8.04 nn for zigzag CNTs with diameters between 0.47 nm and 1.64 nm. Finally, the values in the range of 0.23 nn and 9.51 nn were acquired for chiral CNTs with diameters varying from 0.51 nm to 1.74 nm. This fact needs to be noted that in most cases analytical calculation of the critical buckling loads of CNTs confirms the FEM results, but the figures listed in Tables 1, 2, and 3 and also the results reported by Yengejeh et al. [40] show that the analytical solution does not work in all cases, particularly for CNTs with small diameters which clearly demonstrates the necessity of a numerical approach for such an investigation. Table 1 Critical buckling load of armchair homogeneous SWCNTs. Tube chirality P cr (nn) P cr (nn) Relative Tube P cr (nn) P cr (nn) Relative (Analytical) (FEM) difference chirality (Analytical) (FEM) difference In % In % (5,5) (11,11) (6,6) (13,13) (7,7) (14,14) (8,8) (15,15) (9, 9) (18,18) (10, 10) (19,19) Table 2 Critical buckling load of zigzag homogeneous SWCNTs. Tube chirality P cr (nn) P cr (nn) Relative Tube P cr (nn) P cr (nn) Relative (Analytical) (FEM) difference chirality (Analytical) (FEM) difference In % In % (0,6) (0,14) (0,7) (0,15) (0,8) (0,16) (0,9) (0,17) (0,10) (0,18) (0,11) l.18 l (0, 19) (0, 12) l.50 l (0,20) (0,13) l.88 l.90 l.40 (0,21) Table 3 Critical buckling load of chiral homogeneous SWCNTs. Tube chirality P cr (nn) P cr (nn) Relative Tube P cr (nn) P cr (nn) Relative (Analytical) (FEM) difference chirality (Analytical) (FEM) difference In % In % (6,1) (13,4) (6,4) (14,2) (7,1) (15,1) (7,3) (15,3) (8,2) (15,4) (8,3) (16,1) (8,4) l.05 l.04 l.86 (19,3) (11,3) l.76 l (20,4)

13 No. 2 Communications in Theoretical Physics 227 Fig. 15 Critical buckling load of armchair-armchair hetero-junction CNTs. Fig. 16 Critical buckling load of zigzag-zigzag hetero-junction CNTs. Fig. 17 Critical buckling load of armchair-zigzag hetero-junction CNTs.

14 228 Communications in Theoretical Physics Vol. 64 Fig. 18 Critical buckling load of armchair-chiral hetero-junction CNTs. Fig. 19 Critical buckling load of zigzag-chiral hetero-junction CNTs. Fig. 20 Critical buckling load of chiral-chiral hetero-junction CNTs.

15 No. 2 Communications in Theoretical Physics 229 According to the results from buckling tests on heterojunction CNTs, illustrated in Figs , the critical buckling loads of hetero-junction CNTs depends on the critical buckling loads of their fundamental tubes and lies in their range. The obtained results show that the size of the kink, depending on D d and the diameters of the wider and thinner fundamental tubes and also the bending angle of the kink are determinant in the value of buckling strength of hetero-junction CNTs. It can be claimed that the lowern D d is, the closer is the critical buckling load of hetero-junction CNTs to the ones of their fundamental tubes. It was also observed that hetero-junction CNTs with armchair-armchair connections had the highest critical buckling loads (mostly varying about 5 nn to 10 nn) while those with armchair-zigzag connection had the lowest buckling loads (mostly varying about 1 nn). The critical buckling loads of the hetero-junction CNTs with the other types of connections were mostly seen to vary about the values of 1 nn and 2 nn however; critical buckling loads equal to considerably high vaule of about 15 nn and substantially low value of about 0.2 nn were uniquely observed too. Finally, it was also illustrated that the buckling strength of hetero-junction CNTs tends to be closer to their thinner fundamental tubes buckling strengths. 4 Conclusion In this study, different types of hetero-junction CNTs, made of homogeneous CNTs of different chiralities and configurations with all possible connections were created and their elastic, torsional and buckling properties with the assumption of linear elastic behavior were investigated numerically, based on the finite element method. The results showed that the size and the type of the connection and also the bending angle in the structure of hetero-junction CNTs which depend on the diameter and configuration of their fundamental CNTs are substantially determinant for the mechanical properties of these heterostructures. It was also learned that generally, the Stone Wales defect leads to lower elastic and torsional strength of hetero-junction CNTs. However, the buckling strength of these hetero-junction CNTs lies in the range of the buckling strength of their fundamental CNTs. The shear modulus of hetero-junction CNTs generally tended to be closer to the one of their wider fundamental CNTs while the critical buckling loads of these hetero-structures seemed to be closer to their thinner fundamental tubes. The evaluation of the elastic properties of hetero-junction CNTs and their fundamental tubes yielded that the Young s modulus of homogeneous CNTs is about 1 TPa. Among the heterojunction CNTs, those with armchair-armchair and zigzagzigzag connections had the highest elastic modulus (from 0.7 TPa to TPa) and the models with armchairzigzag kinks showed the lowest elastic moduli (between 0.02 TPa and 0.21 TPa). Shear moduli of the homogeneous CNTs oscillated between 0.1 TPa and TPa. However, the shear modulus of hetero-junction CNTs decreased as a result of the kink in their structures. The highest shear modulus (mostly about 0.3 TPa) was obtained for zigzag-zigzag hetero-junction CNTs and the lowest one (mostly about 0.15 TPa) was acquired for models with armchair-zigzag connections. Eventually, the results from buckling tests demonstrated that critical buckling loads of homogeneous CNTs which correlates to the diameter of the tubes, varies in the range of 0.17 nn and nn for the tubes with diameters between 0.47 nm and 2.58 nm. 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