On the estimation of mechanical properties of single-walled carbon nanotubes by using a molecular-mechanics based FE approach

Composites Science and Technology 69 (2009) 1394–1398

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On the estimation of mechanical properties of single-walled carbon nanotubes by using a molecular-mechanics based FE approach Marco Rossi, Michele Meo * Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK

a r t i c l e

i n f o

Article history: Received 4 December 2007 Received in revised form 13 May 2008 Accepted 7 September 2008 Available online 15 September 2008 Keywords: B. Fracture Carbon nanotubes C. Finite element analysis Molecular mechanics

a b s t r a c t Carbon nanotubes (CNTs) have attracted considerable attention in scientific communities due to their remarkable mechanical, thermal and electrical properties (high stiffness, high strength, resilience, etc.). In particular, mechanical properties of single wall nanotubes (SWNTs) have a Young’s modulus of about 1 TPa if normalized to their diameter showing why they are widely considered as reinforcing elements in advanced low weight composite structures. The determinations of mechanical properties of SWNT are currently investigated both experimentally and theoretically. However, to determine CNTs mechanical properties in a direct experimental way is a challenging and not economical task because of the technical difficulties and the costs involved in the manipulation of nanoscale objects. Due to the handling difficulty, estimation of mechanical properties using computer simulations are being performed by several author with different approaches. In this work a Finite Element Model of SWNTs based on molecular mechanics theory is proposed to evaluate mechanical properties as Young’s modulus, ultimate strength and strain. The novelty of the model lies on the use of non-linear and torsional spring elements, to evaluate SWNTs mechanical properties and tensile failure. With this approach, it was possible to model the bond interaction without making any assumption on non-physical variable, i.e. area, inertia of atom interaction when using beam approach. With the proposed model it was able to understand the evolution of the tensile failure of nanotubes. Moreover, it is important to point out that while most of the fracture evolution studies use molecular dynamics theory and technique, the proposed approach leaded to a minor computational time with the possibility to simulate large atoms system. The calculated mechanical properties show good agreement with existing other work and experimental results. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Since their discovery carbon nanotubes [1–2] have attracted considerable attention in scientific communities. This is partly due to their remarkable mechanical, electrical and thermal properties. A detailed summary of CNTs mechanical properties can be found in [3]. To determine CNTs mechanical properties in a direct experimental way is a challenging and not economical task because of the technical difficulties and the costs involved in the manipulation of nanoscale objects. Based on the above mentioned difficulties, atomistic simulations (AS) have being used successfully to study and predict mechanical properties, and failure of CNT [4]. Several carbon nanotubes were simulated and fracture behaviour investigated in [5–7] using both molecular mechanics and dynamics methods. Fracture dependence chirality, dissociation energy, inflection point etc. were studied. Furthermore, ultimate strength and strain were evaluated and evolution of the crack was studied * Corresponding author. Tel.: +44 1225384224; fax: +44 1225386928. E-mail address: [email protected] (M. Meo). 0266-3538/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2008.09.010

[5]. An interesting molecular structural mechanics model for the mechanical properties of defect free carbon nanotubes, was proposed by Xiao et al. [8]. The calculated Young’s modulus tended to approach graphite Young’s modulus (1.13 TPa). A tensile strength for armchair (126.2 GPa) and zig-zag (94.5 GPa) structures were predicted respectively around 23.1% and 15.6–17.5% of strain. Li and Chou built a single carbon nanotube in order to evaluate the Young’s modulus and shear modulus [9]. This approach was based on carbon bonds beam-like representation and a linkage between the beam sectional stiffness parameters and constants of force field. The method was validated on a graphene model, and a Young’s modulus between 0.995 and 1.033 TPa was found, showing good agreement with known graphite Young’s modulus. It was recognized a not rigorously definition of the bond angles bending and torsion. A finite element approach based on molecular mechanics was proposed by Sun [10]. The chemical bond was simulated with a two node elastic rod element with an elastic joint at each end. A 12*12 stiffness matrix was derived to describe the element behaviour and only stretching and bending potential energy were considered. A tensile stiffness equal to

M. Rossi, M. Meo / Composites Science and Technology 69 (2009) 1394–1398

0.4 TPa was found and a tensile strength from 77 GPa up to 101 GPa was calculated. A 4 node finite element for computation of nano-structured materials was developed in [11] and a (17,0) CNT was stretched up to a strain of 100% [11]. Finally, they showed that using beam elements to model the angle bending leads to wrong results. Some authors studied the effects of healed vacancy defect [12] and of a pre-existing Stone-Wales defect [4] on the mechanical properties. Specifically, Li and Bhattacharya [4], studied the fracture evolution in a (6,6) CNT nucleated from a StoneWales defect by a molecular dynamics simulation. Most of these works show that the Atomistic Simulation [11] has been proved to be an attractive tool due to its accuracy, economy, time and versatility. In this work a carbon nanotube finite element approach based on molecular mechanics are proposed to estimate their mechanical properties. In particular, in the next section a description of the molecular mechanics theory is reported. Then a description of the FE model of carbon nanotubes is presented. Finally results are discussed and compared with data available in literature. 2. Molecular mechanics theory In order to define the SWNT mechanical properties, the molecular mechanics theory can be used. For a random assembly of molecules, mechanical properties cannot be expressed analytically and, to evaluate mechanical properties of CNTs, the motion of the whole system under an assumed potential function can be simulated according to the classical or the quantum mechanical equation of motion. In its general formula, the potential energy (Fig. 1) is described as:

V ¼ RV r þ RV h þ RV u þ RV x þ RV vdW þ RV el

ð1Þ

where Vr is the bond stretching, Vh bond angle bending, Vu, dihedral angle torsion, Vx, inversion terms, VvdW, van der Walls interaction, Vel, electrostatic interaction. Various functional forms may be used for these potential energy terms depending on the particular material and loading conditions considered. In some paper regarding CNTs, the effects of Vu, Vx, VvdW, are neglected under the uniaxial loading and small strain hypothesis [7–8,13–14]. Regarding the bond stretching, an attempt to represent the experimentally determined bond energy curves of diatomic molecules by simple analytical functions is the Morse function [15]:

V r ¼ Deij ½eð2aij Drij Þ  2eðaij Drij Þ 

ð2Þ

where Deij represent the energy required to stretch the bond rij from its equilibrium distance to infinity, Drij is the bond length variation

r0

and aij is equal to (ke =Deij )1/2 where ke is the force constant at the minimum of the well. Another expression for the Morse potential with parameters for hybridised sp2 bonds is given

V r ¼ De f½1  ebðrr0 Þ 2  1g

Vh ¼

1 kh ðDhÞ2 ½1 þ ksextic ðDhÞ4  2

FðDrÞ ¼ 2bDe ð1  ebDr ÞebDr

Torsion

ð5Þ

An elastic non-linear spring element was chosen to simulate the carbon-carbon atom stretching behaviour described from Eq. (5). This was useful to link together non-linear spring element with the torsional spring, as will be later explained. Since the chemical bond always remains straight regardless to the applied loads, the spring bending was neglected. The next step was to simulate the bending angle energy variation. Deriving (4) it was possible to find the relationship between momentum and CACAC angle variation. The result is reported below:

MðDhÞ ¼ kh Dh½1 þ 3ksextic ðDhÞ4 

ð6Þ

To simulate the bending behaviour a linear elastic torsional spring was used at atoms location. The proper bending stiffness was set according to Belytschko [5]. A comparison of the analytical solution and the simulated FE behaviour of CAC bonds, were performed. Using the force-displacement curve of Fig. 2, the breaking of the bond can be described. At displacement equal to zero the spring representing the CC-bond is in equilibrium, then as the displacement increase the spring reacts with a force up to critical distance. When the spring reach the critical distance no force is generated and the spring, and hence the bond, can be imagined as broken.

Van der Walls

Bending Δϕ

ð4Þ

with: kh = 0.9*1018 Nm/rad2, Dh = hh0 with h0 = 2.094 rad, ksextic = 0.754 rad 4 [5]. Based on Molecular mechanics, in order to test the idea of simulating the structural behavior of SWCNT using a combination of linear and torsional spring finite element, a carbon-carbon bond was analyzed under tensile loading and the results were compared with the analytical solution. For the axial stretching the Morse potential with the parameters set described by Belytschko et al. [5] was used. Deriving (11) it is possible to have the relation Force/Bond length variation:

Δθ

Stretching

ð3Þ

where r0 is the bond equilibrium length, De is the dissociation energy and b describes the distance dependence of atoms interactions. The following parameters were used both in [5] and in the present work: r0 = 0.139 nm, De = 6.03105*1010 N nm, b = 2.625*1010 m1 = 26.25 nm1. For the CACAC bond angle energy this expression is given

θ0

Δr

1395

Δω

Inversion or Out of plane

Fig. 1. Interatomic interaction in molecular mechanics theory. Ball and stick representation.

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a

b

CC bond stretching 9.00E-09 8.00E-09

6.00E-10

7.00E-09

N*nm

5.00E-09

Analitycal solution

5.00E-10

Analytical Simulated Results

6.00E-09

Force

Momentum vs Angle Variation 7.00E-10

4.00E-09 3.00E-09

Simulation

4.00E-10 3.00E-10 2.00E-10

2.00E-09 1.00E-09

1.00E-10

0.00E+00

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Displacement [nm]

0.00E+00 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Rad

Fig. 2. (a) Force vs. displacement curve (b) momentum vs. rotation curve.

3. SWCNT mechanical properties evaluation The next step of this work was the construction of a FE model of carbon nanotubes. To calculate the Young’s modulus, one nanotube extremity was totally restrained and on the opposite extremity a displacement was imposed. The nanotube Young’s modulus was evaluated with the equation:

EYoung ¼ r=e ¼ ðF tot =A0 Þ=ðDL=L0 Þ

ð7Þ

where Ftot is the sum of the reaction force generated after the displacement imposed, A0 = p Dn t (with Dn = nanotube diameter, thickness t = 0.34 nm is the interlayer graphite distance), L0 is the nanotube length and DL is the displacement imposed. Three carbon nanotubes structure under uniaxial load were simulated. In agreement with [11], a length to radius ratio smaller than 10 could affect the results, thus, tests were performed with a ratio greater, to limit the edge effects. The results are summarized in Table 1 are in very good agreement with the experimental results of 1.002 TPa reported in [12]. Finally, the ultimate strength of several carbon nanotubes was calculated. Looking at the CC-bond stretching curve (fig. 2), the strain at which the tensile force is at its maximum will be labelled as inflection point [5]. The strain at which the tensile force vanishes will be called as bond-breaking strain. Moreover, as stated in [5], the failure process can be considered completed when all bonds around the nanotube circumference are broken. But, on the other hand, once a single bond failed, no equilibrium solution can be easily found. Further, it was found, that the shape of the CC-bond stretching curve after the inflection point does not affect

the fracture behaviour. Based on these observations, in the present work, it was assumed that the nanotube fracture starts when one or more bonds overcome the inflection point. In fact, looking at the stretching curve, if a force greater than the maximum is applied, no equilibrium force reaction can be generated from the bond. To calculate the ultimate tensile strength, one nanotube extremity was totally restrained and on the opposite extremity, a displacement was imposed. The reaction forces were calculated. The stress value, for each displacement, was calculated by the cross sectional area described before. An Armchair carbon nanotubes (12,12) was tested, and the results compared with the work of Belytschko [5] and Liew [17]. The stress-strain diagram is reported in Fig. 3, and both the ultimate strength and curve shape are in good agreement with 5 and [17]. However, the predicted strengths and failure strains were significantly higher than the experimental values of Yu et al. [18] (11–63 GPa for strengths and 10–13% for failure strains). This disagreement is explained by the presence of defects as reported in [12]. This work showed that the defects, especially vacancies, deteriorate axial mechanical properties of nanotubes. In particular, it was found that the tensile strength and critical strain of single-walled nanotubes could decrease by nearly a factor of 2 in the case of unreconstructed vacancy showing that the strength of nanotubes is inherently governed by the statistical distribution of defects. Then other nanotubes configurations were tested. The results are summarized in Tables 2 and 3. The results showed no tensile strength dependence on SWNT radius. On the other hand, the zig-zag configurations lead to a smaller tensile strength value than the Armchair configuration. These results are in agreement with 10. Further, in Fig. 4, the stress-strain diagram of two nanotubes with different chirality and similar radius are reported and compared. The results carried

1.40E+11 1.20E+11 1.00E+11

Stress [Pa]

With the proper set of constant, it is possible to simulate the other bond deformations, i.e. torsion and out-of-plane angle variations, but in agreement with [6–8,16], these effects are negligible in small strain conditions. Moreover, in [6–8,16], these effects were neglected also for carbon nanotube ultimate strength evaluation. The torsional potential, the out-of-plane potential, the van der Walls forces and electrostatic forces were not considered. During the present work this assumption was confirmed by some numerical tests performed to evaluate the effective negligible influence of these parameters on the simulation results.

8.00E+10 6.00E+10 4.00E+10

Table 1 Nanotube Young’s modulus evaluation

2.00E+10

Kind of nanotubes

Young modulus (TPa)

Armchair Zig-zag Chiral

Average = 0.920 +/ 0.005 Average = 0.912 +/ 0.009 Average = 0.915 +/ 0.009

0.00E+00 0.00

5.00

10.00

15.00

20.00

Strain % Fig. 3. Nanotube Armchair (12,12) under uniaxial load.

25.00

M. Rossi, M. Meo / Composites Science and Technology 69 (2009) 1394–1398 Table 2 Armchair Nanotube ultimate properties Chiral index (n,m)

Diameter (nm)

Length (nm)

L/d

Ultimate strength (GPa)

Strain (%)

(5,5) (6,6) (10,10) (12,12)

0.678 0.814 1.360 1.628

14.644 17.106 13.475 7.261

21.6 21.0 9.9 4.5

117.3 117.1 117.2 117.9

19.95 19.60 20.00 20.00

Table 3 Zig-zag nanotube ultimate properties Chiral index (n,m)

Diameter (nm)

Length (nm)

L/d

Ultimate strength (GPa)

Strain (%)

(5,0) (7,0) (8,0) (9,0) (10,0) (15,0) (20,0)

0.391 0.548 0.626 0.705 0.783 1.178 1.566

8.313 11.723 13.286 14.707 16.697 12.576 12.434

21.0 21.4 21.2 20.8 21.3 10.8 8.0

93.8 94.4 94.7 94.3 94.7 94.5 94.5

17.20 18.00 18.40 18.40 18.40 18.00 18.00

Stress / Strain Curve Amchair (5,5) - ZigZag (9,0) comparison

1.40E+11 1.20E+11

Nanotube Zig Zag 9,0 Nanotube Armchair 5,5

Stress

1.00E+11 8.00E+10 6.00E+10 4.00E+10 2.00E+10 0.00E+00 0.00

21,6% 5.00

10.00

15.00

20.00

25.00

30.00

Strain % Fig. 4. Stress/Strain curve for Armchair (5,5) and zig-zag (9,0).

out by the simulation performed, are in good agreement with the results of other authors and a summary is reported in Table 4. In order to compare the effect of chirality on the force/displacement curve and therefore the amount of energy that can be absorbed by SWCNT, nanotubes with the same diameter i.e. a (5,5) and a (9,0) nanotube were studied. The nanotube fracture

Table 4 Nanotube ultimate properties results Reference number

Method

SWNT chiral index

Ultimate strength (GPa)

Ultimate strain

Present Present Present Present [8] [8] [16] [16] [23] [23] [17] [17]

MM MM MM MM MM MM MM MM DFT DFT MD MD

5,5 10,10 10,0 20,0 12,12 20,0 10,10 17,0 5,5 10,0 10,10 12,12

117.3 117.2 94.7 94.5 126,2 93,5 88 70 110 105 140.4 110

0.199 0.200 0.184 0.180 0.231 0.156 0.15 0.11 0.3 0.20 0.280 0.20

work work Work Work

MD = Molecular Dynamics; MM = Molecular Mechanics; DFT = Density Functional Theory.

1397

was considered complete when all the bonds along the circumference are broken. A bond was considered broken when the distance between atoms is greater than a critical distance labeled ‘‘rf” [4]. In this study a critical distance of 0.3 nm was used. This distance was chosen because, referring to Fig. 2, almost zero reaction force is generated by the bond at that distance. In [4] a critical distance of 0.177 nm was used, but as it is possible to observe even for this distance the bond force is negligible. In order to promote the beginning of the fracture in the middle of the nanotube rather than the extremity, a defect was introduced by weakening a bond in the middle of the tube as in [5]. At atomic level, displaying force rather then stress is more meaningful, however stress–strain curves were used as in [5]. A length/diameter ratio round 10 was used for both the nanotubes. The results of the simulations performed on the nanotubes (5,5) and (9,0) are reported in terms of of stress/strain. For the Armchair (Fig. 5), a maximum stress equal to round 123 GPa was found at around 21.6% of strain. These results are in good agreement with the work of several authors [5]. Focusing the attention on the (5,5) Armchair nanotube and enlarging the stress/strain curve in the fracture area it was possible to visualize the relationship between the points on the curve and the crack evolution in the nanotube. The evolution of the nanotube fracture process is shown in Fig. 5a. The fracture starts from the weakened bond (shown in Fig. 5 with a dashed oval). The weak bond started to fail at a strain of 21.6%. When the weak bond reached the critical distance it was not able to carry any more loads. As the strain increases, more defects were generated along the nanotube circumference, following an ideal path that was skewed with respect to the nanotube’s axes as shown in Fig. 5a. At 22.4% strain, more bonds were broken leading to a big hole, as shown in Fig. 5a. Upon reaching the maximum strain of 23.6%, the CNT experienced complete rupture. This fracture evolution was in good agreement with the cracking mechanism showed in 5. A similar kind of study was performed on the (9,0) zig-zag nanotube (Fig. 5b). For the zig-zag, a maximum stress of 94 GPa was found at 16% of strain. In the zig-zag nanotube the fracture evolved in a very small strain interval (less than 1%). Further, even if the model is based on a discrete domain it is possible to imagine a crack tip moving across the nanotube section up to separate it in two different parts. The mechanism of fracture reported for the zig-zag nanotube is almost the same reported for the Armchair. Specifically, starting from the weakened bond, the other bonds parallel and close to the weakened one, start to enlarge up to reach the critical length. The main difference lies in the fact that crack evolution follows a path along a circumference orthogonal to nanotube longitudinal axis. A bending deformation was experienced by this CNT configuration while a shear fracture deformation was observed for the considered Armchair configuration. The reliability of the model was provided both by the simulated nanotubes ultimate properties and fracture model evolution proposed by other authors. In fact, the values of ultimate strength and strain reported in Fig. 4 are in agreement with the work of [8–10,16]. Further, the evolution of the cracking process is in agreement with the simulation work carried out by [4] for the similar type of defects introduced.

4. Conclusions The results of this work are in good agreement with numerical and experimental works performed by various researchers. The graphene Young’s modulus calculated with the model used, is in good agreement with commonly accepted value of 1.025 TPa, and with the results of several authors [7–9,24]. Regarding the ultimate properties, the calculated tensile strength and strain of several nanotubes was in good agreement with previous works. In partic-

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Fig. 5. (a) Fracture evolution in an Armchair (5,5) and (b) a zig-zag (9,0) nanotube.

ular, a tensile strength of 94 GPa at strain 16.40% was found for a (9,0) zig-zag nanotube and a tensile strength of 123 GPa at strain 21.60% for (5,5) Armchair. Finally, the results show strong dependence of the ultimate stress on the chirality of the nanotubes. References [1] Iijima S. Helical microtubes of graphitics carbon. Nature 1991;354:56–8. [2] Robertson J. Realistic application of CNTs. Mater Today 2004;7(10):46–52. [3] Salvetat JD, Rubio A. Mechanical properties of carbon nanotubes: a fiber digest for beginners. Carbon 2002;40:1729–34. [4] Lu Q, Bhattacharya B. The role of atomistic simulations in probing the smallscale aspect of fracture – a case study on a single walled carbon nanotube. Eng Fracture Mech 2005;72:2031–71. [5] Belytschko T. Atomistic simulations of nanotube fracture. Phys Rev B 2002;65:235430. [6] Natsuki T, Tantrakan K, Endo M. Effects of carbon nanotube structures on mechanical properties. Appl Phys A 2004;79:117–24. [7] Chang T, Gao H. Size dependent elastic properties of a single carbon nanotube via molecular mechanics. J Mech Phys Solids 2003;51:1059–74. [8] Xiao JR, Gama BA, Gillespie Jr JW. An analytical molecular structural mechanics model for the mechanical properties of CNT. Int J Solids Struct 2005;42:3075–92. [9] Li C, Chou T. A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 2003;40:2487–99. [10] Sun X, Zhao W. Prediction of stiffness and strength of single-walled carbon nanotubes bymolecular-mechanics based finite element approach. Mater Sci Eng A 2005;390:366–71.

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