Estimating the Effect of Chirality and Size on the Mechanical Properties of Carbon Nanotubes Through Finite Element Modelling

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Estimating the Effect of Chirality and Size on the Mechanical Properties of Carbon Nanotubes Through Finite Element Modelling

ZUBERI, Muhammad Jibran Shahzad, ESAT, Volkan

Abstract

Carbon nanotubes (CNTs) are considered to be one of the contemporary materials exhibiting superior mechanical, thermal and electrical properties. A new generation state-of-the-art composite material, carbon nanotube reinforced polymer (CNTRP), utilizes carbon nanotubes as the reinforcing fibre element. CNTRPs are highly promising composite materials possessing the potential to be used in various areas such as automotive, aerospace, defence, and energy sectors. The CNTRP composite owes its frontline mechanical material properties mainly to the improvement provided by the CNT filler. There are challenging issues regarding CNTRPs such as determination of material properties, and effect of chirality and size on the mechanical material properties of carbon nanotube fibres, which warrant development of computational models. Along with the difficulties associated with experimentation on CNTs, there is paucity in the literature on the effects of chirality and size on the mechanical properties of CNTs. Insight into the aforementioned issues may be brought through computational modelling time- and cost-effectively when compared to [...]

ZUBERI, Muhammad Jibran Shahzad, ESAT, Volkan. Estimating the Effect of Chirality and Size on the Mechanical Properties of Carbon Nanotubes Through Finite Element Modelling. In: ASME 2014 12th Biennial Conference on Engineering Systems Design and Analysis;

Volume 1: Applied Mechanics; Automotive Systems; Biomedical Biotechnology Engineering; Computational Mechanics; Design; Digital Manufacturing; Education;

Marine and Aerospace Applications . ASME, 2014. p. 7

DOI : 10.1115/ESDA2014-20156

Available at:

http://archive-ouverte.unige.ch/unige:45698

ESTIMATING THE EFFECT OF CHIRALITY AND SIZE ON THE MECHANICAL PROPERTIES OF CARBON NANOTUBES THROUGH FINITE ELEMENT

MODELLING

Muhammad Jibran Shahzad Zuberi Sustainable Environment & Energy Systems

Middle East Technical University – Northern Cyprus Campus

Kalkanli, Guzelyurt, Mersin 10, Turkey

Volkan Esat

Mechanical Engineering Program Middle East Technical University - Northern

Cyprus Campus

Kalkanli, Guzelyurt, Mersin 10, Turkey

ABSTRACT

Carbon nanotubes (CNTs) are considered to be one of the contemporary materials exhibiting superior mechanical, thermal and electrical properties. A new generation state-of-the- art composite material, carbon nanotube reinforced polymer (CNTRP), utilizes carbon nanotubes as the reinforcing fibre element. CNTRPs are highly promising composite materials possessing the potential to be used in various areas such as automotive, aerospace, defence, and energy sectors.

The CNTRP composite owes its frontline mechanical material properties mainly to the improvement provided by the CNT filler. There are challenging issues regarding CNTRPs such as determination of material properties, and effect of chirality and size on the mechanical material properties of carbon nanotube fibres, which warrant development of computational models. Along with the difficulties associated with experimentation on CNTs, there is paucity in the literature on the effects of chirality and size on the mechanical properties of CNTs. Insight into the aforementioned issues may be brought through computational modelling time- and cost-effectively when compared to experimentation.

This study aims to investigate the effect of chirality and size of single-walled carbon nanotubes (SWNTs) on its mechanical material properties so that their contribution to the mechanical properties of CNTRP composite may be understood more clearly. Nonlinear finite element models based on molecular mechanics using various element types substituting C-C bond are generated to develop zigzag, armchair and chiral SWNTs over a range of diameters. The predictions collected from simulations are compared to the experimental and computational studies available in the literature.

Keywords: Carbon nanotubes, finite element modelling, chirality, mechanical properties.

INTRODUCTION

Material science is presently undergoing a shift from developing traditional materials to developing nanostructured materials as they are functionalised, self-assisting and sometimes self-healing. Field of conventional composite materials can be redefined by the nanocomposites both in terms of performance and applications. Polymer nanocomposites show the potential to replace current composites and create new markets offering superior properties. Carbon nanotubes (CNTs) are unique nanostructured materials which possess extraordinary physical and mechanical properties. It is their remarkable properties that bring interest in using CNTs as filler in polymeric matrix to get ultra-light high strength structural materials. The excellent mechanical properties of CNTs encourage their use as reinforced fibres in high-toughness nanocomposites such as carbon nanotube reinforced polymers (CNTRPs), where strength, stiffness and low weight characteristics are required [1]. But CNTRPs’ feasibility is still questionable to figure out whether these materials can meet aforementioned requirements or not. Although they are utilised in various important and appealing applications, CNTRPs still need major breakthroughs. As the structure of CNTs determines the performance of these CNTRPs in their specific applications, it is therefore important to study the structural behaviour of CNTs first. In this study, it is aimed to investigate the effects of chirality and size on the stiffness, namely on the Young’s modulus (E), of CNTs.

A robust growth in computer technology has enabled researchers to characterize and envisage the properties of these CNTs via modelling and simulations. Computational modelling for the prediction of mechanical properties of CNTs is considered a powerful tool when compared to the experimental handicaps. These computational approaches can be categorized as molecular mechanics approach including ab initio and Proceedings of the ASME 2014 12th Biennial Conference on Engineering Systems Design and Analysis ESDA2014 June 25-27, 2014, Copenhagen, Denmark

ESDA2014-20156

classical molecular dynamics (MD) methods, and continuum mechanics approach. Although ab initio methods give results more accurate than MD methods, it is computationally expensive and effective only for small systems having a few hundreds of carbon atoms. MD methods have been applied extensively in material characterization of CNTs and can be used in large systems but still are limited to simulating up to millions of atoms on a very short time scale. Continuum methods are capable of simulating larger systems with longer time scale [2]. One of the key developments of continuum methods is the ‘Equivalent-continuum modelling (ECM)’

approach. ECM methodology combines molecular mechanics (MM) and finite element method (FEM). For large scale nanostructured materials, it has been regarded as an efficient technique. ECM approach primarily involves continuum shell, continuum truss, and continuum beam modelling.

Yakobson et al. studied CNT behaviour beyond Hooke’s law by using a continuum shell model and calculated the Young’s modulus of nanotube as 5.5 TPa [3]. Pantano et al.

also proposed a study for CNTs based on continuum shell modelling reporting Young’s modulus to be 4.84 TPa [4].

Odegard et al. developed a relationship between effective bending rigidity and molecular properties of a graphene sheet by comparing the molecular potential energy of CNTs with the mechanical strain energy by continuum truss modelling [5].

Using the same method, Meo and Rossi predicted the ultimate strength and strain of SWNTs as well as the effects of chirality and deflections on it. They found the Young’s modulus as 0.920 TPa for armchair structures and 0.912 TPa for zigzag structures [6]. Li and Chou developed a continuum beam model substituting C-C bond with beam elements. The elastic moduli of these beam elements were found by linking molecular and continuum mechanics together. The cross section of C-C bond’s equivalent beam was assumed as circular. Their results demonstrated the Young’s modulus for armchair and zigzag CNTs to be in the range of 0.995 TPa and 1.033 TPa [7]. Xiao et al. also developed a finite element beam model by incorporating modified Morse potential. They determined the value as 1 to 1.2 TPa for the Young’s modulus of SWNTs under tension and torsion, respectively [8]. Tserpes and Papanikos evaluated the Young’s modulus of SWNT using the similar approach developed by Li and Chou and investigated the influence of chirality on the Young’s modulus. They reported a Young’s modulus of 2.377 TPa for chirality (8,8) [9].

Jalalahmadi and Naghdabadi determined the Young’s modulus utilizing FEM and Morse potential to find the mechanical properties of beam elements. Their results demonstrate the modulus to be in the range of 3.296 to 3.514 TPa [10].

Recently, Lu and Hu simulated C-C bond considering its cross section area to be elliptical. They predicted the mechanical properties of CNTs using FEM and obtained Young’s modulus of 0.989 TPa to 1.058 TPa for a range of CNT diameters ranging from 0.375 nm to 1.8 nm [2].

In this article, a novel 3D beam element model is proposed to evaluate the Young’s modulus of graphene sheet and a number of SWNTs based on molecular mechanics. The

physical parameters of C-C bond beam element providing linkage between molecular and continuum mechanics are chosen carefully from the literature. The finite element model is constructed using MSC Marc 2010 which is a multi-physics simulation software particularly for nonlinear finite element analysis of static and dynamic problems. The FE model is used to investigate the effects of chirality and diameter on the stiffness, i.e. Young’s modulus, of the SWNTs. Current study presents the factors influencing the Young’s modulus of SWNTs. The results obtained are in good agreement with the published data in literature based on other models developed with other commercial FE software.

ATOMIC STRUCTURE OF SWNTs

A single walled carbon nanotube (SWNT) is a hexagonal network of carbon atoms rolled into hollow, seamless cylinder capped with half of a fullerene molecule at each of its ends. The hexagonal arrangement is repeated periodically as each carbon atom binds to three adjacent atoms with strong covalent bonds.

These bonds play significant role to the striking mechanical properties of graphene and CNTs. The atomic structure of a SWNT depends upon its chirality represented by a chiral vector.

Chiral vector (n, m) and angle of the SWNTs are usually defined by the packing of carbon hexagons in the graphene sheets. The indices of the chiral vector demonstrate the morphology of the SWNT. Change in nanotube morphology, alters nanotube properties. The geometry representatives are summarized in Figure 1. The chiral vector is defined as follows [2]:

𝐶_ℎ= 𝑛𝑎₁+ 𝑚𝑎₂ (1)

where a1 and a2 are the unit cell base vectors of the graphene sheet. ‘a’ is the length of unit vector defined as:

𝑎 = √3 𝐿 (2)

where the value of bond length L is taken as 0.142 nm. The circumference (C), the diameter (d) and the chiral angle (θ) of the SWNT is defined as:

𝐶 = 𝑎√𝑛²+ 𝑛𝑚 + 𝑚² (3)

𝑑 = 𝐶 𝜋

(4)

Ɵ = 𝑠𝑖𝑛⁻¹ √3𝑚 2√𝑛²+ 𝑛𝑚 + 𝑚²

(5)

There are three classifications of nanotube structures;

1. armchair (n, n), 2. zigzag (n, 0), and 3. chiral (n, m)

where n and m are not equal to each other [11]. The chiral angles of the zigzag and armchair SWNTs are 0^o and 30^o, respectively, while the chiral angle is between 0^o and 30^o for a chiral SWNT.

Figure 1: Chiral vector and angle representation FINITE ELEMENT MODELLING OF SWNTS

Carbon atoms of a SWNT are bonded together by covalent bonds. When external forces are applied, the displacement of each carbon atom is restrained by these bonds. The total force on each atomic nucleus is obtained by the sum of electrostatic forces between the positively charged nuclei and the force caused by the electrons [2]. In molecular mechanics, CNTs or specifically SWNTs can be observed as large molecules consisting of carbon atoms. The total steric potential energy (Vt) between the C-C bonds of the SWNTs under minor linear elastic deformations, neglecting the electrostatic interactions, can be expressed as the sum of the following energies [6], [7], [9]:

𝑉𝑡 = 𝑉𝑟 + 𝑉Ɵ + 𝑉𝜏+ 𝑉𝑤 (6) where Vr is the energy due to bond stretching, Vθ is the energy due to bond angle variation or bending, Vτ is the combined energy due to dihedral angle (φ) and out-of-plane torsion (ω), and Vw is the energy due to non-bonded van der Waals interactions. A schematic representation of the interatomic interactions in molecular mechanics is shown in Figure 2. The major contributors towards the total steric energy for covalent systems are the first four terms of Equation 6 where the effects of van der Waals interaction energies are often neglected. The harmonic representation of the single energies is given as [2], [7], [9], [12]:

𝑉𝑟=𝑘𝑟 (𝛥𝑟)² 2

(7)

𝑉Ɵ=𝑘Ɵ (𝛥Ɵ)² 2

(8)

𝑉𝜏=𝑘𝜏 (𝛥𝜑)² 2

(9)

Figure 2: Interatomic interactions in molecular mechanics where kr is the bond stretching force constant, kθ is the bond angle variation force constant and kτ is the torsional stiffness force constant. Δr, Δθ and Δφ indicates bond stretching variation, in-plane, and twisting angle increments respectively.

The values of kr , kθ and kτ are taken as 652 nN/nm, 0.876 (nN/nm)/rad² and 0.278 (nN/nm)/rad², respectively, as adopted from Li and Chou [7]. If C-C bonds act as uniform three dimensional beams capable of stretching, bending and torsion, the strain energies associated with pure axial and torsion loading can be expressed as [2], [7], [9], [12]:

𝑉𝑎𝑥𝑖𝑎𝑙=𝐸𝐴(𝛥𝐿)² 2𝐿

(10)

𝑉𝑏𝑒𝑛𝑑𝑖𝑛𝑔=𝐸𝐼(2𝛼)² 2𝐿

(11)

𝑉𝑡𝑜𝑟𝑠𝑖𝑜𝑛=𝐺𝐽(𝛥𝛽)² 2𝐿

(12)

where E, I, G, J, A, and L represents the equivalent Young’s modulus, moment of inertia, shear modulus, polar moment of inertia, cross-sectional area, and length of the beam, respectively. ΔL is the axial deformation, 2α is the change in rotational angle, and Δβ is end beam rotation. Equation 8 corresponds to a slender uniform beam under pure bending.

For thick beams, shear deformation of the cross sectional area under bending should be taken into account in order not to overrate the beam deflection. To calculate the equivalent continuum diameter of the C-C bond, shear deformation and the effect of Poisson’s ratio of the equivalent continuum material for the covalent bond should be considered [12].

Assuming the beam cross section as circular and isotropic with diameter d, the geometric parameters are to be expressed as [2], [9], [12]:

𝐴 =𝜋𝑑² 4

(13)

𝐼 =𝜋𝑑⁴ 64

(14)

𝐽 =𝜋𝑑⁴ 32

(15)

Scarpa and Adhikari obtained the following relations for Young’s modulus (E) and shear modulus (G) from the equivalence between the steric and mechanical strain energies by employing the equivalence between Δr and ΔL, also between Δβ and Δθ [12]:

𝐸 = 4 𝑘𝑟 𝐿 𝜋𝑑²

(16)

𝐺 = 32 𝑘Ɵ 𝐿 𝜋𝑑⁴

(17)

Scarpa and Adhikari also developed an implicit relation between the bond diameter (d) and the Poisson’s ratio (). They imposed the isotropic relationship G = E/2(1+) for the equivalent C-C bond medium which lead them to sectional properties of the beam element shown in Table 1.

Table 1: Sectional properties of the 3D beam element

Diameter (d) 0.0844 nm

Young’s Modulus (E) 16.71 TPa Shear Modulus (G)

Poisson’s Ratio ()

8.08 TPa 0.0344

SWNTs are simulated at nanoscale in MSC Marc 2010 using the aforementioned section properties of the beam element. A 3D solid section beam element (Type 98) is used for modelling SWNTs. This element has six degrees of freedom per node which are translational, and rotational in and about x, y, and z axes, respectively. It is a straight beam in space which includes transverse shear effects and can be used to model linear or nonlinear elastic response by entering the cross- section properties directly. This element is capable for modelling inelastic and nonlinear elastic material response when employing numerical integration over the cross section.

Figure 3 shows the isometric view of an FE mesh developed with these elements. Simulated SWNTs are treated as isotropic materials.

Figure 3: Isometric view of the FE mesh of the armchair (10,10) SWNT

FINITE ELEMENT MODELLING OF SWNTS Young’s Modulus of Graphene Sheet

A graphene sheet model is developed and tested under uniaxial load as shown in Figure 4. The graphene sheet is constrained from one end and displaced from the opposite end to observe the structural responses. In this work, a wall thickness t is taken as the diameter of the carbon atom, i.e 0.140 nm, in line with literature [13]. Also, a similar value for thickness is calculated by Tserpes & Papanikos [9]. Young’s modulus Es can be evaluated via the following equation:

𝐸𝑠= 𝐹 𝐿𝑜 𝑊 𝑡 𝛥𝐿

(18)

where F, Lo, W, t, and ΔL are the total applied force, initial length, width, thickness and change in length of the sheet respectively.

Figure 4: Graphene sheet model with boundary conditions It is observed from the same model that length of the sheet does not cause any significant effect on the Young’s modulus of the graphene sheet. Therefore, graphene sheets with fixed length of 10.082 nm and different widths are investigated through this model and results are demonstrated in Table 2.

Figure 5 shows the variation of Young’s modulus with width to length ratio. As the ratio is increased from 0.07 to 2.34, the Young’s modulus reaches a plateau approximately at 2.13 TPa.

The value is comparable to the values found in literature [2], [14], [15]. No rotation of the bond or out of plane displacement is observed from the simulation results.

Figure 5: Young's modulus of the graphene sheet versus width to length ratio

1.85 1.9 1.95 2 2.05 2.1 2.15

0 0.5 1 1.5 2 2.5

Young's Modulus, E(TPa)

Width to Length Ratio Fixed

displacements

Stretched end of the sheet

Table 2: Characteristics of graphene sheet FE models

Width, W (nm) Length, L (nm) W / L ΔL (nm) F (nN) E (TPa)

0.738 10.082 0.073 0.10 1.910 1.864

1.476 10.082 0.146 0.10 4.094 1.997

2.952 10.082 0.293 0.10 8.476 2.068

5.904 10.082 0.586 0.10 17.240 2.103

11.808 10.082 1.171 0.10 34.950 2.132

23.616 10.082 2.342 0.10 70.120 2.138

Young’s Moduli of SWNTs Finite element models of SWNTs

The mechanical properties of CNTs depend on their chirality and size. Effects of both are analysed for the evaluations of Young’s moduli of SWNTs. The FE model is employed to assess the effect of diameter and chirality on the Young’s moduli of SWNTs E given by the following equation:

𝐸 = 𝐹 𝐿𝑜 𝜋 𝑑𝑐𝑛𝑡 𝑡 𝛥𝐿

(19) where dcnt is referred to as the diameter of SWNTs.

These SWNTs are restrained completely from one extremity and a displacement of 0.1 nm is applied at the other extremity. Figure 6 shows the model mesh for zigzag (8,0), chiral (8,4), and armchair (8,8) SWNTs with the boundary conditions imposed. Table 3 provides the characteristics of all SWNTs investigated in this study.

Table 3: Characteristics of SWNT FE models

SWNTs Diameter (nm) Chirality (θ^o) Length (nm) ΔL (nm) Zigzag

(6,0) 0.470 0 12.354 0.1

(8,0) 0.626 0 12.638 0.1

(10,0) 0.783 0 12.060 0.1

(13,0) 1.018 0 12.070 0.1

(20,0) 1.566 0 12.336 0.1

Chiral

(4,2) 0.414 19.1 12.398 0.1

(6,2) 0.565 13.9 12.288 0.1

(8,4) 0.829 19.1 12.398 0.1

(10,6) 1.096 21.8 11.928 0.1

(12,6) 1.243 19.1 12.398 0.1

Armchair

(3,3) 0.406 30 12.296 0.1

(5,5) 0.678 30 12.052 0.1

(6,6) 0.814 30 12.052 0.1

(8,8) 1.085 30 12.544 0.1

(10,10) 1.356 30 12.052 0.1

Figure 6: Imposed boundary conditions on the zigzag (8,0), chiral (8,4) and armchair (8,8) SWNTs.

Effect of diameter and chirality on Young’s moduli of SWNTs

In majority of the works reported, only zigzag and armchair tubes have been included. In current work all the three types of SWNTs are brought under examination. Figure 7 shows the variation of Young’s modulus of zigzag, chiral and armchair SWNTs with the tube diameter. It can be seen that there is a considerable effect of diameter on the moduli of zigzag and chiral SWNTs while the effect is not significant for armchair type. As the diameter of zigzag and chiral SWNTs increases, Young’s modulus also increases. Li and Chou [16]

pointed out that this increase is due to the influence of nanotube curvature. Curvature increases with decreasing nanotube diameter resulting in large distortion of the C-C bonds and therefore, in large displacement of the nanotube. Young’s modulus of chiral SWNTs is concluded to be the greatest followed by armchair SWNTs, and then zigzag SWNTs.

Figure 7: Variation of Young's modulus with SWNT diameter for zigzag, chiral, and armchair configurations (data fitted by

polynomial regression).

Three SWNTs have also been selected in order to study the effect of chirality. Zigzag (13,0), chiral (10,6) and armchair

(8,8) are chosen to possess approximate diameters of 1.018 nm, 1.096 nm, and 1.085 nm; and angles of 0^o, 21.8^o and 30^o respectively. It can also be observed from Figure 7 that on an average armchair attains relatively greater Young’s modulus when compared to chiral whereas zigzag gets the lowest. This can be interpreted as the greater the chiral angle, larger the Young’s modulus for similar nanotube diameters for smaller diameters. When the SWNT diameter reaches about 1.3, all configurations yield a similar Young’s modulus.

Similar trends are observed by Meo and Rossi [6] and Yakobson et al. [3]. Also, the values for Young’s modulus found in this research approaches to that of the graphene model developed for the study validating the proposed C-C bond. As prior research suggests that SWNTs have Young’s moduli ranging between 1 to 5 TPa approximately [1], [17], the Young’s moduli computed in this study are in good agreement with those in literature [2]–[4], [7], [9], [18]–[23].

CONCLUSION

FE models for zigzag, chiral and armchair SWNTs are generated in this study. In order to use harmonic potential for equivalent mechanical properties of C-C bond appropriately, shear deformation and equivalent Poisson’s ratio of the equivalent bond material should be taken into consideration whilst introducing the values for Young’s and shear moduli of the beam elements.

The models are used to investigate the effects of diameter and chirality on the Young’s moduli of SWNTs. Young’s moduli values agree well with the corresponding theoretical and computational results. The FE model results suggest that the Young’s modulus depends on the chirality and size of nanotubes particularly for smaller diameters. Increasing the tube diameter and the chiral angle increases the Young’s moduli of SWNTs. The Young’s modulus of armchair SWNTs is found to be greater than that of chiral and zigzag SWNTs. The proposed approach may provide a valuable tool for determining the mechanical properties of CNTs and related nanocomposites.

1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25

0.35 0.55 0.75 0.95 1.15

Young's Modulus, Y (TPa)

SWNT Diamater (nm) Armchair Zigzag Chiral

Fixed displacements Stretched end of the tube

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