Mechanical property evaluation of single-walled carbon nanotubes by finite element modeling

https://doi.org/10.1016/j.compositesb.2012.02.002Get rights and content

Abstract

Computational simulation for predicting mechanical properties of carbon nanotubes (CNTs) has been adopted as a powerful tool relative to the experimental difficulty. Based on molecular mechanics, an improved 3D finite element (FE) model for armchair, zigzag and chiral single-walled carbon nanotubes (SWNTs) has been developed. The bending stiffness of the graphene layer has been considered. The potentials associated with the atomic interactions within a SWNT were evaluated by the strain energies of beam elements which serve as structural substitutions of covalent bonds. The out-of-plane deformation of the bonds was distinguished from the in-plane deformation by considering an elliptical cross-section for the beam elements. The elastic stiffness of graphene has been studied and the rolling energy per atom has been calculated through the analysis of rolling a graphene sheet into a SWNT to validate the proposed FE model. The effects of diameters and helicity on Young’s modulus and the shear modulus of SWNTs were investigated. The simulation results from this work are comparable to both experimental tests and theoretical studies from the literatures.

Introduction

Carbon nanotubes (CNTs), since their discovery in 1991 [1], have attracted much interest due to their extraordinary physical properties (mechanical, thermal and electrical properties) [2], [3], [4], [5], [6], [7], [8]. From structural application perspectives, CNTs’ high stiffness, coupled with their low density, implies that nanotubes might be useful as nanoscale fibers in strong, lightweight nano-composite materials. These potential usages have originated the need to explore their mechanical properties, study their deformation under loading and identify their failure mechanism that may appear.

A great number of experimental studies have been undertaken to estimate the mechanical properties of CNTs. Experimental methods for measuring the mechanical properties of CNTs are mainly based on transmission electron microscopy (TEM) and atomic force microscopy (AFM). Due to complexity of characterization on nano-size, there is a large variability among these experiment results. In 1996, Treacy et al. [9] have been the first to perform an experimental measurement of Young’s modulus in single-walled carbon nanotubes (SWNTs) and estimated a value of 0.4–4.15 (average 1.8) TPa by measuring thermal vibrations using TEM. A cantilevered beam model has been used by Wong et al. [10] in which individual multi-walled carbon nanotubes (MWNTs) were bent using an AFM tip, and a lower value range of 1.28 ± 0.59 TPa has been obtained. Krishnan et al. [11] have concluded an estimated mean value of Young’s modulus of 1.3–0.4/+0.6 TPa and a weighted average value of 1.25 TPa for SWNTs having a diameter range of 1.0–1.5 nm using TEM. Lourie et al. [12], [13] used a bar model in their experiment, in which the compressive response was measured using micro-Raman spectroscopy. They reported Young’s modulus of 2.8–3.6 TPa for SWNT and 1.7–2.4 TPa for MWNT. Yu et al. have used direct tensile loading tests for SWNT [14] and MWNT [15], and obtained Young’s modulus ranged from 0.32 to 1.47 TPa (mean 1.00 TPa) for SWNT and from 0.27 to 0.95 TPa for MWNT. The simple-supported beam model in AFM was used by Salvetat et al. [16], [17] for individual MWNTs and of different-sized SWNTs, Young’s modulus of ∼1 TPa for MWNTs grown by arc discharge was reported, whereas CNTs grown by the catalytic decomposition of hydrocarbons had a modulus of 1–2 order of magnitude smaller. Tombler et al. reported the Young’s modulus of 1.2 TPa for SWNT by 3-point bending using AFM [18]. The shear modulus of SWNTs was measured to be of the order of only 1 GPa due to the unexpectedly low intertube shear stiffness dominating the flexural behavior of the SWNT ropes [17]. Wei et al. [19] have led simultaneously Young’s modulus and shear modules of SWNTs through fitting the experimental fL (natural frequencies–length) data with the Timoshenko beam model, combining the electric-field-induced resonance method and the “nanoknife” technique. The shear modulus is found to be of the order of several hundred MPa, which is about two to three orders of magnitude smaller than Young’s modulus. Guhados et al. [20] have developed a technique to directly determine effective values of both Young’s and shear moduli of MWNTs using AFM, and measured a Young’s modulus of 350 ± 110 GPa, and also reported a direct measurement of the shear modulus (G = 1.4 ± 0.3 GPa) of MWNTs. These experiments have confirmed that CNTs have exceptional mechanical properties. However, the experimental error bars are too large to state the exact characteristics of CNTs of different sizes and structures.

Computational simulation for predicting mechanical properties of CNTs has been regarded as a powerful tool relative to the experimental difficulty. The computational approaches can be classified into two categories: namely the ’bottom up’ approach, based on quantum/molecular mechanics including the classical molecular dynamics (MD) and ab initio methods, and the ’top down’ approach, based on continuum mechanics. ’Bottom up’ methods have been extensively applied in the characterization of material properties of CNTs [21], [22], [23], [24], [25], [26], [27]. Generally, ab initio methods give more accurate results than MD, but it is computationally expensive and only effective for small systems containing a few hundreds of carbon atoms. MD could be used in a larger systems, but it is still limited to simulating on up to millions of atoms on a too-short time scale (less than 10−6–10−9 s) even if the new TeraGrid hardware and software are used [28], [29], since the frequency of molecular thermal vibration is so high. The simulation of a larger system and longer time must currently be left to continuum methods (’top down’ approach).

Equivalent-continuum modeling (ECM) approach is one of the major developments of continuum method. It has been regarded as a very efficient method, especially for nano-structures with large scale. Molecular mechanics (MM) combined with finite element method (FEM) is a main ECM approach. Over the past years, many ECM models were presented in literature. The ECM approaches mainly involve continuum shell modeling, continuum truss modeling, and continuum beam modeling. Yakobson et al. [23] and Ru [30], [31] proposed a study of continuum shell modeling. Odegard et al. [32] developed a continuum truss model to obtain a relation between effective bending rigidity and molecular properties of a graphene sheet by equating the molecular potential energy of nanotubes to the mechanical strain energy of a representative continuum truss model. Meo and Rossi [33], [34], using the same method, gave a step forward by combining the use of non-linear elastic and torsional elastic spring. Mechanical properties, such as Young’s modulus, ultimate strength and strain for several CNTs were calculated. Furthermore, a stress–strain curve for large deformation (up to 70%) is reported for a nanotube zigzag (9,0). In particular, the stress–strain curve of a nanotube with one-vacancy defect was evaluated and compared with the curve of a pristine one, showing a reduction of the ultimate strength and strain for the defected nanotube.

In a paper of Li and Chou [35], a finite element (FE) model was built with beam elements substituting C–C bond. The elastic moduli of beam elements were determined by using a linkage between molecular and continuum mechanics. The cross section of C–C bond’s equivalent beam is assumed to be circular. Young’s modulus of armchair and zigzag CNTs between 0.995 and 1.033 TPa was found, showing good agreement with known graphene Young’s modulus. The same authors studied a MWNT [36]. The same relationship between cross-sectional properties of beam and molecular force field constants was proposed, and a non-linear truss rod model was used to simulate van der Waals interaction among the layers. Young’s modulus of the MWNTs was calculated to be 1.05 ± 0.05 TPa and the shear modulus calculated was 0.40 ± 0.05 TPa. Using the same method, Xiao et al. [37] developed a FE beam model with a modified Morse potential incorporated. Young’s modulus, Poisson’s ratio and stress–strain relationships were evaluated for SWNTs under axial and torsion load. Other research in this approach could be referred [38], [39], [40], [41], [42].

ECM approach can be used to calculate both static and dynamic properties of CNTs. However, as it simulates CNTs with shells, it neglects their atomic characteristics, eliminating in that way any possible effect that may have on the mechanical behavior of CNTs. Another disadvantage of continuum shell modeling is that it does not consider the interatomic forces. For the truss model, it fails to take the inversion and bending of C–C bond into account. The latest developed finite element modeling based on multi-scale mechanics is a very cost-effective method. It is simple and computational efficient for analyzing the mechanical behaviors of SWNTs. However, the bending stiffness of the graphite layer in a CNT (which is related to the bond inversion: the out-of-plane deformation) was not taken into account in both the truss model and the beam model. In order to study problems where the effect of local bending is significant, a modified model is needed.

In this paper, an improved beam element which includes the bond inversion energy is proposed to evaluate mechanical properties of graphene and SWNTs based on molecular mechanics. The physical parameters of C–C bond beam element are derived analytically by using linkage between molecular and continuum mechanics. Young’s modulus and shear modulus of SWNTs and graphene sheet are simulated, through the use of FE model, as well as a parametric study of wall thickness, diameter and chirality effects on stiffness and comparison of rolling energy per atom with literatures.

Section snippets

Atomic structure of SWNTs

A single-wall carbon nanotube (SWNT) is best described as a rolled-up tubular shell of graphene sheet which is made of benzene-type hexagonal rings of carbon atoms [43]. The body of the tubular shell is thus mainly made of hexagonal rings (in a sheet) of carbon atoms, whereas the ends are capped by half-dome-shaped half-fullerene molecules. The hexagonal pattern is repeated periodically leading to binding of each carbon atom to three neighboring atoms with covalent bonds. The covalent bond is

Molecular mechanics of SWNTs

In SWNTs, carbon atoms are bonded together with covalent bonds. The displacement of individual carbon atoms under external forces is constrained by the bonds. The total force on each atomic nuclei or nucleus is the vectorial sum of the force generated by the electrons and electrostatic force between the positively charged nuclei themselves. The molecular force field is expressed in the form of steric potential energy, which depends solely on the relative positions of the nuclei constituting the

Young’s modulus evaluation of graphene

The proposed FE formulations can be efficiently implemented by self-compiled or commercial FEM programs. In this work, a commercial software ANSYS was used to create the FE model. The C–C bonds are simulated by beam element with six degrees-of-freedom at each node: translational and rotational displacements along axes x, y, and z. In order to validate the proposed C–C bond, a graphene model was developed [64]. The structural responses were tested under uniaxial load shown in Fig. 8. One side

Conclusion

A 3-D FE model has been generated by using a linkage between molecular and continuum mechanics. A chemical bond beam element has been developed with an elliptical cross-section, taking into account the bending stiffness of the graphene layer. The elastic stiffness of graphene has been studied and the rolling energy per atom has been calculated through the analysis of rolling a graphene sheet into a SWNT to validate the proposed FE model. The computational results agreement well with the

Acknowledgment

This work was supported by NSF/SD EPSCoR Funds #0554609, the State of South Dakota, and South Dakota State University RSF Fund. Computational facility support from the College of Engineering and the Department of Mechanical Engineering at SDSU are gratefully acknowledged.

References (72)

  • K.I. Tserpes et al.

    Finite element modelling of single-walled carbon nanotubes

    Compos: Part B Eng

    (2005)
  • M.S. Dresselhaus et al.

    Physics of carbon nanotubes

    Carbon

    (1995)
  • T.C. Chang et al.

    Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model

    J Mech Phys Solids

    (2003)
  • G.X. Jia et al.

    Electronic structures and hydrogenation of a chiral single-wall (6,4) carbon nanotubes: a density functional theory study

    Chem Phys Lett

    (2006)
  • M. Meo et al.

    Prediction of Young’s modulus of single wall carbon nanotubes by molecular-mechanics-based finite element modelling

    Compos Sci Technol

    (2006)
  • T. Natsuki et al.

    Stress simulation of carbon nanotubes in tension and compression

    Carbon

    (2004)
  • A. Pantano et al.

    Mechanics of deformation of single-and multi-wall carbon nanotubes

    J Mech Phys Solids

    (2004)
  • X.L. Gao et al.

    Finite deformation continuum model for single-walled carbon nanotubes

    Int J Solid Struct

    (2003)
  • S. Iijima

    Helical microtubules of graphitic carbon

    Nature

    (1991)
  • K.T. Lau et al.

    The revolutionary creation of new advanced materials-carbon nanotube

    Compos: Part B

    (2002)
  • K.T. Lau et al.

    Carbon nanotubes for space and bio-engineering applications

    J Comput Theor Nanosci

    (2008)
  • D. Qian et al.

    Mechancis of carbon nanotubes

    Appl Mech Rev

    (2002)
  • Y.W. Zhu et al.

    Carbon-based supercapacitors produced by activation of graphene

    Science

    (2011)
  • M.D. Stoller et al.

    Methods and best practices for determining an electrode material’s performance for ultracapacitors

    Energy Environ

    (2010)
  • M.M.J. Treacy et al.

    Exceptionally high Young’s modulus observed for individual carbon nanotubes

    Nature

    (1996)
  • E.W. Wong et al.

    Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes

    Science

    (1997)
  • A. Krishnan et al.

    Young’s modulus of single-walled nanotubes

    Phys Rev B

    (1998)
  • O. Lourie et al.

    Evaluation of Young’s modulus of carbon nanotubes by micro-Raman spectroscopy

    J Mater Res

    (1998)
  • O. Lourie et al.

    Buckling and collapse of embedded carbon nanotubes

    Phys Rev Lett

    (1998)
  • M.F. Yu et al.

    Tensile loading of ropes of single wallcarbon nanotubes and their mechanical properties

    Phys Rev Lett

    (2000)
  • M.F. Yu et al.

    Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load

    Science

    (2000)
  • J.P. Salvetat et al.

    Elastic modulus of ordered and disordered multiwalled carbon nanotubes

    Adv Mater

    (1999)
  • J.P. Salvetat et al.

    Elastic and shear moduli of single-walled carbon nanotube ropes

    Phys Rev Lett

    (1999)
  • T.W. Tombler et al.

    Reversible electromechanical characteristics of carbon nanotubes under local-probe manipulation

    Nature

    (2000)
  • X.L. Wei et al.

    The very low shear modulus of multi-walled carbon nanotubes determined simultaneously with the axial Young’s modulus via in situ experiments

    Adv Funct Mater

    (2008)
  • G. Guhados et al.

    Simultaneous measurement of Young’s and shear moduli of multiwalled carbon nanotubes using atomic force microscopy

    J Appl Phys

    (2007)
  • Cited by (138)

    • Molecular structure based study on the elastic properties of carbon nanotubes in a thermal environment

      2022, Journal of Molecular Structure
      Citation Excerpt :

      The coefficient of thermal expansion is 7.1 × 10−6 K −1, which is also adopted in this paper. Cornwell and Wille [11] and Lu and Hu [12] applied the different methods, molecular dynamics and finite element analysis to study the changes in strain energy of carbon nanotubes with strain and radius. Moreover, the temperature related strain energy and thermodynamic properties are also interested by many researchers [13–15].

    View all citing articles on Scopus
    1

    Present address: Department of Materials Science and Engineering, Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles, CA 90095-1595, USA.

    View full text