Elsevier

Computational Materials Science

Volume 62, September 2012, Pages 110-116
Computational Materials Science

A numerical evaluation of the influence of defects on the elastic modulus of single and multi-walled carbon nanotubes

https://doi.org/10.1016/j.commatsci.2012.05.003Get rights and content

Abstract

Finite element models of single-walled and multi-walled carbon nanotubes in their perfect and fundamental forms (zigzag and armchair) were constructed. Then, after obtaining the mechanical properties of perfect carbon nanotubes, three types of imperfections, i.e., doping with Si atoms, carbon vacancy and perturbation of the ideal location of the carbon atom were introduced in different amounts to the perfect models to make them imperfect. Finally, the mechanical properties of the imperfect carbon nanotubes were numerically simulated and compared with those of perfect ones. Simple relations which predict the change of Young’s modulus as a function of the imperfection percentage were derived. The results show that the existence of any kind of imperfection in the perfect models leads to lower stiffness values. This study allows to realistically judge any simulation based on perfect structures and gives for the first time a good estimate to which extend the values based on perfect structures must be lowered in order to account for common imperfections to predict the mechanical properties of carbon nanotubes which are nowadays used in the production of advanced nanocomposites and reinforced materials.

Highlights

► Single and multi-walled carbon nanotubes are numerically simulated. ► Three likely defects are considered: Si-doping, carbon vacancy and perturbation. ► Influence of imperfections on mechanical properties is quantified. ► Simple relations for the prediction of the influence of imperfections are proposed.

Introduction

Carbon nanotubes (CNTs) are unique nanostructures because of their outstanding mechanical and electrical properties and their potential in the production of advanced nanocomposites and reinforced materials which have attracted worldwide attention and encouraged many researchers to work on the properties of these structures. Since the discovery of CNTs, numerous methods have been applied to study the mechanical properties of these interesting nanostructures [1], [2]. The methods are generally divided into the two groups of experimental and computational approaches. Among the computational methods, molecular dynamics (MD) and continuum mechanics (CM) techniques such as the finite element (FE) method are the two most commonly used approaches to study the behavior of CNTs. Based on these investigations, Young’s modulus of CNTs has been reported to be approximately equal to 1 TPa which is an interestingly high value [3]. Noticing the fact that nothing is perfect in nature and that only a few investigations have been performed on imperfect CNT structures, it is necessary to study imperfect CNT structures to realistically evaluate their mechanical properties. In the following, the results of several investigations on the evaluation of CNTs mechanical properties are presented.

Wu et al. [4] performed an experimental study in which optical characterization and magnetic actuation techniques are combined for evaluating the CNTs elastic stiffness, i.e., Young’s modulus. They obtained a Young’s modulus of 0.97 TPa for individual CNTs and an average value of 0.99 TPa for five equally weighted CNT bundles. A practical method was used by Lu [5] to apply the empirical force constant model. In this study, Young’s modulus was calculated to be about 0.97 TPa for single-walled (SW) and 1 TPa for multi-walled carbon nanotubes (MWCNTs).

A structural mechanics approach was used by Li and Chou [6] that had acquired Young’s modulus in the range of 0.89–1.033 TPa. To [7] used the finite element method in which the effect of Poisson’s ratio was included in the determination of Young’s modulus which was obtained as 1.024 TPa. Chang and Gao [8] used an analytical method based on a molecular mechanics approach to relate the elastic properties of a single-walled carbon nanotube (SWCNT) to its atomic structure and finally obtained the value of 1.024 TPa for the elastic modulus. Nahas and Abd-Rabou [9] applied a structural mechanics approach for CNT modeling in which they used the finite element method. Consequently, they calculated the elastic modulus equal to 1.03 TPa.

An analytical method for modeling the elastic properties of SWCNTs was applied by Natsuki et al. [10], who obtained the axial modulus in the range of 1.1–0.73 TPa. Jin and Yuan [11] evaluated the elastic modulus of SWCNTs by numerical simulation and predicted the elastic modulus about 1.350 and 1.238 TPa, using energy and force approaches, respectively. Ávila and Lacerda [12] simulated the three major SWCNT configurations by the finite element method, using the ANSYS V.10 software, and evaluated the Young’s modulus of CNTs to be between 0.97 and 1.30 TPa.

Kalamkarov et al. [13] used two different continuum-based approaches to model the behavior of CNTs and investigated their mechanical properties. They used first an analytical approach which gave Young’s modulus equal to 1.71 TPa for SWCNTs. The second approach which was based on the FE method predicted the elastic modulus between 0.9 to 1.05 TPa for SW and 1.32 to 1.58 TPa for double-walled carbon nanotubes (DWCNTs). Meo and Rossi [14] also proposed a FE model of a SWCNT and he finally obtained the value of Young’s modulus around 1 TPa.

Tserpes and Papanikos [15] defined a three-dimensional FE model. According to their results, the Young’s modulus varied from 0.97 to 1.03 TPa. Fan et al. [16] also used a FE simulation technique to evaluate the mechanical properties of MWCNTs. They calculated Young’s modulus about 1.0 TPa for both SW and MWCNTs.

Rahmandoust and Öchsner [17] also applied the FE method, based on the substitution of covalent bonds between atoms by beam elements, to model MWCNTs. They finally reported the value of Young’s modulus of MWCNTs in the range of 1.32 and 1.58 TPa. Brcic et al. [18] used the same method for evaluation of the Young’s modulus of MWCNTs. According to their results, Young’s modulus of DWCNTs was obtained about 1.04 TPa. Li and Chou [19] used the FE method to simulate MWCNTs as frame-like structure to evaluate the mechanical properties. They reported the value of Young’s modulus in the ranges of 1.05 ± 0.05 TPa.

An analytical formulation was developed by Shokrieh and Rafiee [20] for evaluation of the elastic moduli of graphene sheets and CNTs by linking the lattice molecular structure and the equivalent discrete frame structure. By classical mechanics formulae, they finally obtained Young’s modulus for graphene sheets about 1.04 TPa and for CNTs in the range of 1.033–1.042 TPa, which was in a good agreement with previous investigations.

A comprehensive study on the influence of three most likely types of structural imperfections on the mechanical properties of SWCNTs was performed for the first time by Rahmandoust and Öchsner [21] by numerical simulation. According to their results, application of any type of imperfections on the structure of SWCNTs led to lower stiffness. The actual research continues this work and extends the prediction to MWCNTs based on the same approach. Song et al. [22] also investigated the elastic properties of CNTs in their perfect form and under the influence of Si-doping. Their results show that the Young’s modulus of perfect SWCNTs varies in the range of 1.099 ± 0.005 TPa and decreases as a result of Si-doping.

Section snippets

Geometric definition

Carbon nanotubes are similar to hollow cylinders with diameters ranging from 1 to 50 nm and lengths over 10 μm. They are structured only by carbon atoms with hexagonal unit cells [24]. A SWCNT is assumed to be formed by rolling a graphene sheet, with a thickness in the nanometre range. The geometry of the atomic structure of CNTs is described in terms of the tube chirality, or helicity, which is defined by the chiral vector (roll-up vector) Ch and the chiral angle θ. The following relationship

Results and discussion

In this study, CNTs from SW to 5-walled were simulated in the FE software MSC.Marc. These CNTs and their characteristics are presented in Table 2. For all models, the thickness was assumed to be equal to 0.34 nm [21].

To evaluate the elastic modulus of CNTs from SW to 5-walled, the models were loaded by an arbitrary displacement boundary condition. Since the investigations are performed in the linear-elastic range, the absolute value of the displacement condition is not important. By evaluating

Conclusion

According to the presented results, by increasing the number of CNT walls, Young’s modulus increases with a converging trend. This means that after reaching a particular number of CNT walls, adding more walls does not increase Young’s modulus noticeably and Young’s modulus can be assumed to remain unchanged.

From the results of imperfect CNTs, it could be extracted that applying any kind of imperfection on the nanotubes leads to a lower Young’s modulus. By increasing the percentage of these

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