Energy consistent modified molecular structural mechanics model for the determination of the elastic properties of single wall carbon nanotubes

Accepted Manuscript Energy consistent modified molecular structural mechanics model for the determination of the elastic properties of single wall carbon nanotubes Oliver Eberhardt, Thomas Wallmersperger PII:

S0008-6223(15)30118-4

DOI:

10.1016/j.carbon.2015.07.092

Reference:

CARBON 10161

To appear in:

Carbon

Received Date: 20 May 2015 Revised Date:

28 July 2015

Accepted Date: 31 July 2015

Please cite this article as: O. Eberhardt, T. Wallmersperger, Energy consistent modified molecular structural mechanics model for the determination of the elastic properties of single wall carbon nanotubes, Carbon (2015), doi: 10.1016/j.carbon.2015.07.092. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Energy consistent modified molecular structural mechanics model for the determination of the elastic properties of single wall carbon nanotubes Oliver Eberhardt∗ and Thomas Wallmersperger†

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July 28, 2015

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Institute of Solid Mechanics, Technische Universit¨at Dresden, 01069 Dresden, Germany

Abstract

A new, modified molecular structural mechanics model for the determination of the elastic properties of carbon nanotubes is presented. It is designed specifically to overcome drawbacks in existing molecular structural mechanics models, which are not consistent with

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their underlying chemical force fields in terms of energy. As a result, modifications are motivated, developed and implemented in order to create a new, energy consistent molecular structural mechanics model. Hence, the new model leads to a better prediction of the material parameters for single wall carbon nanotubes, while the simple applicability of the approach is maintained. The results calculated for the elastic constants (Young’s modu-

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lus, Poisson ratio) of armchair and zig-zag CNTs are given and discussed. Both elastic constants were found to be dependent on the chirality as well as on the carbon nanotube diameter. An asymptotic value of approximately 800 GPa was obtained for the Young’s

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modulus and a value of approximately 0.28 for the Poisson ratio.

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Introduction

The extensively reported outstanding material properties of carbon nanotubes (CNTs) regarding their mechanical [1], electrical [2] and thermal [3] behavior make them promising candidates for a wide range of applications. However, the quantitative knowledge of these properties is vital before the application of the CNTs can be considered. Since the experimental determination of material properties is in many cases at least challenging, sometimes ∗ Corresponding author. † Corresponding author.

Tel. +49 351 463 334 01. E-mail: [email protected] Tel. +49 351 463 370 13. E-mail: [email protected]

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ACCEPTED MANUSCRIPT even close to impossible, the desire of determining the CNT material properties with the help of simulation methods was raised. As a result of this, several models have been developed in order to calculate the CNTs mechanical parameters like for instance the Young’s modulus and the Poisson ratio. In general, the models developed to calculate the material parameters of CNTs can be divided into two groups. The first group is based on atomistic ab-initio approaches, which are

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derived from the solution of the Schr¨odinger equation. As it is yet not possible to solve this equation for a complex structure like a carbon nanotube, it is necessary to introduce sev-

eral simplifications in the original ab-initio theory. Methods involving simplifications are

for instance the density functional theory (DFT) and tight-binding-methods (TBM) which

have both been applied to calculate the mechanical properties of carbon nanotubes, see [4],

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[5], [6], [7] and [8].

Besides the methods stemming from the solution of the Schr¨odinger equation, the second group of approaches consists of models based on classical mechanics. In the present paper

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we distinguish the classical mechanics approaches further into (i) methods developing an equivalent continuum structure of the CNTs and then applying continuum mechanics, see e.g. [9], [10], [11], [12] and [13], and (ii) methods using discrete continuum elements like beams or springs to represent the covalent bonds in the CNTs. The approach developed in the present paper belongs to this group of methods. One of the first approaches of this kind was presented in 2003 by Li and Chou [14]. They used truss-beam elements in order to represent the covalent bonds in CNTs. The basic idea is, that the potentials describing the

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bond deformations in a harmonic chemical force field description can be equated with the mechanical potentials (elastic strain energy) describing the corresponding deformations of a truss beam element. Tserpes and Papanikos [15] made a major extension of the model with respect to the applicability of the model when using commercial finite element pack-

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ages. The model aims at the calculation of mechanical parameters like the Young’s and shear modulus of armchair and zig-zag single wall carbon nanotubes (SWCNTs). Eberhardt and Wallmersperger [16] used the original model of Li and Chou for the investigation

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of the deformation behavior of armchair and zig-zag SWCNTs in more detail. Based on the original model of Li and Chou [14], several modifications have been proposed, e.g. by Chen et al. [17] and by Lu and Hu [18]. In contrast, Nasdala and Ernst [19] developed a special 4-node element designed for the application in mechanical models describing CNTs. Furthermore, several useful remarks have been made by Nasdala et al. [20] regarding the application of Finite Element methods for the simulation of molecular systems. Some of these annotations have been considered for the present research. A comprehensive review on the modeling of carbon nanotubes including a multitude of results regarding the mechanical behavior of CNTs was published recently by Rafiee and Moghadam [21]; a review on nanocarbon actuators, including their modeling and simulation, has been given by Kosidlo et al. [22]. However, studies shown in the present work reveal, that the model developed by Li and

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ACCEPTED MANUSCRIPT Chou as well as the modifications of the original model mentioned above are not consistent to the chemical force field descriptions they are based on in terms of energy. The present paper identifies this drawback as well as the reasons for it. As a result of this, a new, modified model is developed in order to create a model which is consistent to the chemical force field it is based on. The new model is then used for the calculation of the elastic properties of the CNTs (Young’s modulus, Poisson ratio).

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The following section deals with all the aspects necessary for modeling the elastic behavior of CNTs. At first the original approach developed by Li and Chou [14] is briefly introduced

and the results obtained for the elastic constants are given. In a next step the drawbacks

of the model are identified and discussed. This motivates the development of a new, modi-

fied molecular structural mechanics model. This new model is subsequently developed. In

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section 3 the results for the elastic properties obtained with the new model are given and discussed. The present paper is then closed by summarizing the findings and improvements

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in section 4.

Modeling and methods

2.1 Original MSM approach

In this section a short overview of the original molecular structural mechanics approach by to their paper.

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Li and Chou [14] is given. For a more detailed description, the interested reader is referred The modeling of CNTs can be divided into two basic parts: The first one is the consideration of the CNT geometry. In Li and Chou’s model this is done with the roll up model. This model is based on the idea, that a cut out in a graphene sheet can be defined which is then

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rolled up in order to create a specific CNT. The second part is the proper representation of the covalent bond. Li and Chou have chosen truss-beam elements to be the mechanically applicable representation of the covalent bonds. As a result of this, it is necessary to cal-

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culate the properties mandatory to define the beam geometry and the beam material. This is done with the help of a chemical description of the CNTs based on a chemical force field description. In the following, the necessary steps for the calculation of the mechanical parameters of the beam elements according to Li and Chou [14] with extensions given by Tserpes and Papanikos [15], are explained. During deformations of SWCNTs, e.g. due to the application of loads, the individual covalent bonds deform in specific manners. The possible bond deformations which are considered to contribute significantly to the realization of the total CNT deformation are depicted in Figure 1 (top). They include bond stretching, bond angle bending and bond torsion. Each of these deformations can be described by a potential stemming from a chemical force field representation. Li and Chou [14] applied harmonic potentials for their approach. We refer to them as “chemical potentials“. The potentials are given in Figure 1 and include the so

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bond stretching

bond angle bending

bond torsion

Dj

Dq

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Uθ = 12 kθ ∆θ 2

Ur = 21 kr ∆r2

UN =

Eb A l

kθ =

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kr =

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Dr

chemical potential

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1 Eb A 2 2 l ∆l

UM =

1 Eb I 2 l

Uτ = 12 kτ ∆ϕ 2

Eb I l

kτ =

(2 ∆α )2

UT =

Gb J l

1 Gb J 2 l

∆β 2

N

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mechanical potential M

M

N

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Dl

beam stretching

Db

T Da

Da

beam bending

T

beam torsion

Figure 1: Schematic description of the molecular structural mechanics model according to Li and Chou [14].

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ACCEPTED MANUSCRIPT called force constants kr for bond stretching, kθ for bond angle bending, and kτ for bond torsion. The next step is to observe the corresponding deformations of the truss-beam elements which are the beam stretching, bending and torsion of the truss-beam element, see Figure 1 (bottom). The energy involved during the deformation of the beam is described by the elastic strain energies which can be addressed as “mechanical potentials“, see Figure 1. The mechanical potentials are formulated (i) by using the corresponding stiffness values

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Eb A, Eb I and Gb J being the axial, flexural and torsional rigidity and (ii) by using the deformation variables given in Figure 1 which are the beam stretching ∆l, the beam bending

angle ∆α and the torsion angle ∆β . The beam material is assumed to behave linear elastic.

By comparing the corresponding deformations in bonds and beams respectively, one can recognize, that the mechanical potential and the chemical potential can be correlated, since

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they describe the same deformation (stretching, bending and torsion, respectively). This correlation results in a set of three equations, where each equation connects the properties of the beam geometry and the beam material with the force constants of the chemical force

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field

kr =

kθ

=

kτ

=

Eb A l Eb I l Gb J . l

(1) (2) (3)

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The equations are also displayed in the middle row of Figure 1. It has to be noted, that the approach explained so far strictly distinguishes the different deformations stretching, bending and torsion. The mechanical as well as the chemical potentials include only stretching, bending or torsion, respectively. No coupling between the different deformation modes is present in the model. This provides, that the bond and beam deformations can be correlated

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directly, i.e. bond stretching to beam stretching, bond angle bending to beam bending and bond torsion to beam torsion.

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The geometrical beam parameters which are used here are the undeformed length of the beam l, the cross sectional area A, the area moment of inertia I and the torsional moment of inertia J. The model assumes a circular cross section of the beam with diameter db . Hence, the cross sectional properties read

π 2 d 4 b π 4 d 64 b π 4 d . 32 b

A = I = J =

(4) (5) (6)

The length l of the beam can be easily identified as the length of a single covalent bond ˚ Hence, the only geometrical parameter which needs to be determined is aC−C = 1.421 A. the diameter. The mandatory parameters of the beams material are the Young’s modulus

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ACCEPTED MANUSCRIPT Eb and the shear modulus Gb . All of these parameters are defined locally for the bond and should not be mistaken with the global diameter, Young’s modulus and shear modulus of the carbon nanotube. Solving the set of the three equations (1)-(3) given above enables the calculation of the three identified beam parameters. After the consideration of the cross sectional properties (4)-(6) the results are in accordance with Tserpes and Papanikos [15] kr2 l 4 π kθ kτ l kr2 = 8 π kθ2 r kθ = 4 . kr

Eb =

(8) (9)

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db

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Gb

(7)

With these parameters and the geometrical CNT model (roll up model) it is possible to assign the properties to the truss-beam elements and create the CNT model. This enables

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the calculation of the global mechanical properties like the Young’s modulus or the Poisson ratio for the CNTs. The elastic constants are determined during a virtual tensile test. In this tensile test, one end of the investigated carbon nanotube is fixed, while the other end is subjected to a tensile force. The total force applied on the nanotube is distributed on the last atoms at one nanotube end. The boundary conditions creating the fixed support are given on the opposite end of the nanotube.

The Young’s modulus is then calculated as it can be done for a real specimen in a macro-

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scopic tensile test. For this, the nanotube displacement, the applied force and the original length as well as a definition of a cross-sectional area for the nanotube are used. The crosssection used by Li and Chou [14] is shaped like an annulus with a wall thickness t of 0.34

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nm. The area of the CNT cross-section is then calculated as A = π d t,

(10)

where d is the CNT diameter.

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The results for the Young’s modulus obtained with the original approach by Li and Chou are given in Figure 2.

The Poisson ratio is here defined as the negative ratio of the strains in radial directions to the ones in lateral direction. The results for the Poisson ratio obtained with the model developed by Li and Chou are given in Figure 3. The results obtained with this model state that the Young’s modulus for armchair tubes varies very little and can be found at approximately 1040 GPa, see Figure 2. For the zigzag tubes an increasing Young’s modulus for increasing diameters can be observed. The Young’s modulus for zig-zag tubes is lower than for armchair tubes. Figure 3 depicts the results for the Poisson ratio. The armchair tubes show an almost constant value of approximately ν = 0.06. The zig-zag tubes again show a different behavior compared to the armchair tubes. Here, a decrease of the Poisson ratio with increasing tube

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1200

(2,2)

(12,12) (20,0)

(3,0)

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800

600

400

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Young‘s modulus [GPa]

1000

0

2

4

6

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200

8

10

12

armchair (n,n) zig-zag (n,0)

14

16

18

CNT diameter [Å]

Figure 2: Young’s modulus for armchair and zig-zag tubes obtained by applying the original model of Li and Chou [14].

0.25

(3,0)

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0.2

0.15

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Poisson ratio [-]

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0.3

0.1

(20,0)

(2,2)

(12,12)

0.05

0

armchair(n,n) zig-zag(n,0) 2

4

6

8

10

12

14

16

18

CNT diameter [A]

Figure 3: Poisson ratio for armchair and zig-zag tubes obtained by applying the original model of Li and Chou [14].

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ACCEPTED MANUSCRIPT diameters is observed. For larger tube diameters the Poisson ratio seems to converge to the value of the armchair tubes which is ν = 0.06. It can be observed that the value for the Young’s modulus is comparable to experimental results, e.g. see Wu et al. [23]. However, the obtained values for the Poisson ratio seem disputable since they are very small. Unfortunately, from the authors’ point of view and to our best knowledge, no resilient experimental data for the Poisson ratio is available so

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far. Most of the experimental data available for the Poisson ratio we found are based on the measurement of the Young’s modulus and the shear modulus. With these two parameters

the Poisson ratio was in many cases calculated by the well known relationship between the Young’s modulus, the shear modulus and the Poisson ratio valid for a linear elastic,

isotropic material. Besides the circumstance, that already the definition of the Young’s and

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the shear modulus for CNTs is subject to debatable assumptions, the application of this relationship seems questionable.

Due to the unusual results for the Poisson ratio, it seems reasonable to investigate the model

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of Li and Chou [14] in more detail in order to get ideas on how the model can be modified in order to enhance it. This is the topic of the next section of the present paper.

2.2 Motivation for a new, modified molecular structural mechanics model

As already mentioned, the modeling of CNTs can be divided into two parts: (i) a geo-

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metrical part and (ii) a part where the modeling of the covalent bond is considered. The first modification proposed in this work concerns the geometric model of the CNT. In the original model the roll up model is applied in order to represent the nanotubes geometry. However, this simple model has a major drawback. During the transformation from a plane

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into a tubular structure (rolling), the nanotube geometry is not considered properly which results in unequal lengths of bonds which should have the same length. As a result of this, Cox and Hill developed another geometrical representation of the carbon nanotubes

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[24] which is able to properly consider the CNT structure without the drawback mentioned above. The principle of this approach is the placement of predefined tetrahedrons on helical trajectories. As a result of this, the present work applies the exact polyhedral model of Cox and Hill [24] as geometrical representation of the CNT. The consideration of this advanced geometrical model instead of the roll up model, combined with the original bond representation, leads to the results for the Young’s modulus and the Poisson ratio given in Figures 4 and 5. All the results given in the present work from now on will be obtained using the exact polyhedral model for the calculation of the nanotube geometry. The Young’s modulus obtained with the exact polyhedral model (see Figure 4) shows some differences compared to the results calculated with the roll up model depicted in Figure 2. At first one can notice that the Young’s modulus for armchair nanotubes shows a different behavior. Using the roll up model, the Young’s modulus of the armchair nanotubes is al-

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1200

(12,12) (20,0)

(2,2)

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800

(3,0)

600

400

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Young‘s modulus [GPa]

1000

0

2

4

6

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200

8

10

12

armchair (n,n) zig-zag (n,0)

14

16

18

CNT diameter [Å]

Figure 4: Young’s modulus for armchair and zig-zag tubes obtained by applying the original MSM approach when using the exact polyhedral geometrical model.

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0.3

0.25

EP (3,0)

0.15

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Poisson ratio [-]

0.2

0.1

(20,0)

(2,2)

(12,12)

0.05

0

armchair (n,n) zig-zag (n,0) 2

4

6

8

10

12

14

16

18

CNT diameter [Å]

Figure 5: Poisson ratio for armchair and zig-zag tubes obtained by applying the original MSM approach when using the exact polyhedral geometrical model.

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ACCEPTED MANUSCRIPT most constant. With the exact polyhedral model an increase of the Young’s modulus from smaller to larger tubes can be observed. Furthermore, the results for small diameter zig-zag tubes are much smaller when applying the exact polyhedral model compared to the results obtained with the roll up model. On the other hand, the results for the Young’s modulus of nanotubes with larger diameters are nearly the same. This shows, that the consideration of a proper geometry is very important when investigating smaller nanotubes. For larger

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nanotubes the drawbacks of the roll up model decrease with increasing nanotube diameters.

Regarding the Poisson ratio, only small differences can be found when the results calcu-

lated with the exact polyhedral model (Figure 5) are compared to the ones derived with the roll up model (Figure 3). This means, that the choice of the geometrical nanotube repre-

sentation has no major impact on the results obtained for the Poisson ratio. Since the small

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value of the Poisson ratio is one of our main motivations for the identification of possible

improvements, it is necessary to consider modifications in the representation of the chemical bonds.

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Regarding possible modifications concerning the representation of the covalent bond, a virtual tensile test of several armchair and zig-zag tubes was conducted and analyzed in detail. In particular, the deformation work occurring in the CNT during the virtual tensile test is calculated. The deformation work is the work necessary to elongate the carbon nanotube from its relaxed state due to the tensile force. This deformation work can be calculated by using two different methods which can be applied independently of each other. The first method is based on the application of the chemical potential from the chemical

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force field description. In order to obtain the overall deformation work, in a first step it is necessary to calculate the local deformations in the CNT occurring during the virtual tensile test. In order to do this, the results obtained from the finite element solution, which are displacements and rotations of the atomic locations (nodes), have to be transformed into the local bond deformations. These deformations include the bond stretching ∆r, the bond an-

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gle change ∆θ and the bond torsion ∆ϕ . With these deformations and their corresponding potentials it is now possible to calculate the deformation works of the individual deforma-

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total , the individual contributions need tions. In order to obtain the total deformation work Wchem

to be summed up

Nθ Nτ Nr 1 1 1 total = ∑ kr ∆r2 + ∑ kθ ∆θ 2 + ∑ kτ ∆ϕ 2 . Wchem i=1 2 i=1 2 i=1 2

(11)

Nr , Nθ and Nτ are here the generalized descriptions for the corresponding number of stretched, angle bended and twisted bonds occurring in the CNT during the virtual tensile test. The second method for the calculation of the overall deformation work utilizes the mechanical potentials, which represent the elastic strain energy in the truss-beam elements. In analogy to the first method, the contributions of all the elements have to be summed up in order to obtain the overall deformation work

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Ne

Ne 1 Eb A 2 Ne 1 Eb I 1 Gb J 2 ∆l + ∑ (2∆α )2 + ∑ ∆β . 2 l 2 l i=1 i=1 2 l i=1

total Wmech =∑

(12)

Ne is the total number of truss-beam elements in the CNT model. It corresponds to the number of bonds in the investigated CNT. Since the CNT model was implemented with the help of the commercial finite element tool ABAQUS, it was possible to calculate the elastic

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strain energy in the beams by using an already implemented function.

total is comAs a next step, the deformation work calculated with the chemical approach Wchem

total . In order to do this, an pared to the one obtained by the mechanical calculations Wmech

energy deviation ∆W is calculated as:

(13)

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total total W −W ∆W = mech total chem · 100% . Wchem

In theory, the values obtained by equations (11) and (12) should be equal, since they

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both describe the same deformation work of the CNTs occurring in the virtual tensile test. Hence, the energy deviation calculated by equation (13) should be zero. However, having a closer look at Figure 6 - where the energy deviation ∆W is plotted versus the nanotube diameter - one can recognize, that a substantial energy deviation can be detected. 30

(2,2) 25

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20

15

(12,12) (20,0)

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energy deviation DW [%]

(3,0)

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10

5

0

2

4

armchair(n,n) zig-zag(n,0) 6

8

10 CNT diameter [Å]

12

14

16

18

Figure 6: Energy deviation ∆W when comparing the chemically calculated deformation work with the mechanically calculated one. For nanotubes with very small diameters the energy deviation reaches values of up to 26 %, while a reduction of the energy deviation is observed when the nanotube diameter is increased. For larger tube diameters, ∆W converges to an value of approximately 17.5 %. As

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ACCEPTED MANUSCRIPT already mentioned, in theory this deviation should not exist and indicates some drawbacks in the original model which need to be identified and, if possible, need to be corrected. One may assume, that the energy deviation could be caused by the application of the exact polyhedral geometrical model, but the comparison of the results obtained with the roll up model also show energy deviations in the same range. Hence, the geometrical representation of the CNT is not responsible for the observed energy deviation.

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For the purpose of verification of the observed phenomenon, a plane graphene sheet was investigated in the same manner as the SWCNTs. A zig-zag graphene sheet with the dimen-

sions 3.14 nm by 37.51 nm was subjected to a tensile test, where the smaller boundaries of

the sheet were used to apply the tensile force and the fixation. Here, an energy deviation of 17.23 % was observed. As a result of the investigation of the deformation work of the vir-

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tual tensile test, the next step will be the identification of possible reasons of this deviation.

In order to answer this question, the internal forces and moments occurring during the tensile test have been investigated. Figure 7 depicts the internal bending moments. To be

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more specific, the Figure only shows the component of the internal bending moments in the bond angle bending plane. In armchair nanotubes, this component is the only internal bending moment present in the CNT, i.e. the internal bending moment in the bond angle plane is congruously the resulting internal bending moment. In zig-zag CNTs this behavior is different. For zig-zag CNTs an additional internal bending moment component perpendicular to the bond angle bending plane can be observed. In the present work this bending moment component is neglected due to the following reasons: First, the neglected bending

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moment component is small compared to the internal bending moment in the bond angle bending plane, and second, the neglected bending moment decreases fast with increasing CNT diameter. Hence, effects caused by this bending moment should only - if at all - be noticeable in small diameter CNTs. The bending moments are projected onto the unit cells

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of a representative armchair (6,6) and zig-zag (10,0) tube, however, the results given in the Figure are transferable to any other CNT of the type armchair or zig-zag. As a result of this, it has to be noted, that the results given in Figure 7 are not to be scaled, but only give a

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qualitative representation of the internal moments. In the pictures, one can easily recognize the linear progression of the internal bending moments on the beam elements. The absolute values of the bending moment are the same at both ends of the beam element. This investigation of the internal moments revealed a possible starting point for modifications of the original model. This is because the examination of the bond angle bending term of the original model given in Figure 1 (middle column) shows, that it only accounts for constant bending moments which leads in our opinion at least partly to the energy deviation shown in Figure 6. Hence, the present work aims at the modification of the bond angle term in the original model in order to account for linear bending moments instead of constant ones. The development of this modified model is presented in detail in the next section of the present paper.

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Figure 7: Schematic representation of the internal bending moments in armchair (left) and zig-zag (right) CNTs.

2.3 Development of the new, modified molecular structural mechanics model

In the present section, the necessary steps to modify the bond angle bending term of the

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original model are explained in detail. The aim of the modification is the reduction of the energy deviation given in Figure 6 by considering linear bending moments instead of constant ones in the bond angle bending.

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2.3.1 Elastic strain energy in the beam elements In a first step, the elastic strain energy occurring in the beam elements due to bond angle bending is calculated. In a single beam element a linear progression of the bending moment

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M(z) was found as given in Figure 8. It has to be noted, that the absolute values M0 at the ends of the beams are equal. Hence, the function for the bending moment is found to be  z M(z) = M0 1 − 2 l

(14)

in the interval 0 ≤ z ≤ l. With this bending moment, the elastic strain energy in the beam

WB is calculated as

WB = WB =

1 l M 2 (z) dz 2 0 Eb I M02 l , 6Eb I Z

13

(15) (16)

ACCEPTED MANUSCRIPT where l is the length of a carbon-carbon bond aC−C . Furthermore, as given in the original model, a circular cross section of the beam element is assumed, which results in the area moment of inertia I of the beam given as I=

π db4 . 64

(17)

elastic strain energy reads WB =

32M02 l . 3π Eb db4

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Here, db is the diameter of the beam element representing the covalent bond. With this, the (18)

In order to enable an analogous calculation of the beam parameters like in the original

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model, it is necessary to gain the elastic strain energy due to bond angle bending in depen-

dence of the bond angle changes occurring in the CNT structure. As a result of this, the

M0 z

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next step is the calculation of the beam deformations within the CNT structure.

M0

l

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Figure 8: Linear bending moment on the beam elements.

2.3.2 Beam deformation

The calculation of the deformation behavior of the CNT and the resulting beam deforma-

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tions is facilitated by having a closer look at just one single tetrahedron within the CNT, see Figure 9. The lengths and angles in the tetrahedron are also given in detail in Figure 10. The tetrahedron defined by the mentioned measures can be found in armchair as well

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as in zig-zag CNTs. Compared to the orientation in the zig-zag CNTs one can notice that the orientation of the tetrahedron in the armchair CNTs is rotated. The deformations in the CNTs are from now on described solely by the deformation of the tetrahedrons. For a detailed description of the occurring deformations and their qualitative as well as quantitative analysis, the interested reader is referred to [16]. The tetrahedron given in detail in Figure 10 is folded into the plane in Figures 11 a)-d) in order to create a clearly arranged sketch. Figure 11 a) shows the undeformed state, where ˚ and the bond angles are distinguished into the all the bond lengths are l = aC−C = 1.421 A bond angles θ1 and θ2 . Due to the force applied on the CNT during the virtual tensile test, the tetrahedron will be subjected to deformations. The generalized deformed state of the tetrahedron for armchair and zig-zag CNTs is given in Figure 11 b). The geometry of the tetrahedron is governed (i) by the bond lengths (in the deformed state r1 and r2 ) and (ii)

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q1

r2

q1

q2

q1 r1

r2

r1

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q1

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q2

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Figure 9: Tetrahedrons in unit cells of armchair (left) and zig-zag (right) CNTs.

4 r1

r2

3

q2 q1

r2 1 2

Figure 10: Detailed description and measures of the investigated tetrahedron.

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ACCEPTED MANUSCRIPT by the bond angles θ1 and θ2 , or in the deformed state by θ˜1 and θ˜2 , respectively. A conceivable influence of the bond torsion is neglected in the present work. The application of this assumption is permissible since bond torsion in zig-zag tubes was found to be small in general and furthermore decreases for increasing CNT diameters. Furthermore, in armchair CNTs bond torsion is not present at all. The deformed tetrahedron is given again in Figure 11 c). Here, the deformations are described with the deformation variables applied in the

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mechanical truss-beam element representation of the CNT. The deformations include the beam deflection w as well as the the beam stretching ∆l. The deformed state of the tetrahe-

dron is now investigated in detail. In particular, the bond angle change of the angle θ1 into the new bond angle θ˜1 is investigated. The deformed angle θ˜1 can be described with the

r2

aC-C q1

q2 q2

b)

aC-C

aC-C

~ q2

~ q1

~ q2

aC-C

d)

~ q1 ~ ~2 w q1

aC-C w q1

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aC-C

Dl ~ q1 2

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w q1

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aC-C

r1

undeformed

r2

c)

deformed

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a)

aC-C

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help of both, the deflection w and the beam stretching ∆l as depicted in Figure 11 c).

aC-C

Dl ~ q1 2

~ q1 ~ ~2 Dl ~ q1 2

Figure 11: Deformations of the tetrahedrons.

However, since the general idea of the original approach shall be followed, it is mandatory to describe the bond angle change only in dependence of the deflection w. This is the result of the circumstance, that the original approach requires a strictly separable description of the deformation energies due to stretching, bending and torsion. Hence, the new bond angle θ˜1 is approximated as given in Figure 11 d) in order to use only the deflection w for the description of the angle. Consequently, the next step is the calculation of the corresponding beam deflections. This can be done with the help of Figure 12. In accordance with the

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Figure 12: Beam deformation

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the deflection: ′′

w =

d 2 w M(z) . = dz2 Eb I

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basic theory of bending, the following differential equation is applied in order to calculate

(19)

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Since bending occurs only in the slanted beams of Figure 11 which are subjected to the linear bending moments M(z) (equation (14)) discussed above, equation (19) becomes: ′′

w =

M0  z 1−2 . Eb I l

(20)

The differential equation is solved by integration. After this integration and after the con-

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sideration of the boundary conditions

w(z = 0) = 0 ′

w (z = 0) = 0

(21) (22)

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the result for the deflection w(z) is

M0 w(z) = Eb I



 1 2 1 z3 z − . 2 3 l

(23)

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This yields the deflection at the end of the beam (z = l): w = w(l) =

1 M0 l 2 6 Eb I

(24)

where I is the area moment of inertia of the beams cross-section. In order to relate the bond angle change ∆θ1 to the deflection w, small angle changes are assumed. The assumption is reasonable since the application of the harmonic potentials allows only the investigation of small deformations in the CNTs anyway. The bond angle change can be found in the deflected beam as given in Figure 12. As a result of this, the relation between the deflection and the bond angle change under the assumption of small

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∆θ1 w w = = 2 aC−C l

(25)

which can be used together with equation (24) to calculate the maximum value of the bendM0 = 3

Eb I ∆θ1 . l

(26)

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ing moment M0 , see Figure 8:

Inserting equation (26) into equation (16) yields the elastic strain energy due to bending in the beam element in dependence of the bond angle change ∆θ1 : WB =

3 Eb I (∆θ1 )2 . 2 l

(27)

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2.3.3 Calculation and comparison of the deformation works due to bond angle bending

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The overall target of all the considerations given above is the calculation of a modified relationship between the beam parameters Eb (Young’s modulus of the beam material), I (area moment of inertia of the beam) and l (undeformed beam length) and the chemical force constant kθ . This can be achieved by comparing the deformation works due to bond angle bending calculated with the help of the chemical force fields on the one hand and the elastic strain energies on the other hand. In the original approach this was done by considering a single bond angle change. In contrast to the original approach, in the present

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work the comparison of the deformation works due to bond angle changes is done for a single tetrahedron within the CNT structure. The situation is depicted in Figure 13. The left hand side of the Figure depicts the calculation of the deformation work using the elastic strain energy of the beams. To do this, the beams in the tetrahedrons are considered, keeping in mind, that only the slanted ones are subjected to bending deformations.

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The elastic strain energy for the bending of a single beam was computed in the previous section. In order to get the total deformation work in the tetrahedron, the contributions of the beams are summed up which results in the factor 2 given in equation (28) in Figure 13.

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However, since one beam is connected to two carbon atom positions, one has to consider that a single beam takes part in bond angle changes at both positions. In the present work it is assumed, that the deformation work can be split up in equal parts between these two bond angle changes. This consideration leads to the factor 1/2 also given in equation θ (28). This results in the given deformation work Wmech (equation (29)) in the mechanical

representation. The calculation of the deformation work by using the chemical force field representation is done by applying the harmonic potentials and the corresponding bond angle changes ∆θ1 and ∆θ2 . Again, all the contributions have to be summed up. Keeping in mind, that two bond angle changes ∆θ1 and ∆θ2 can be found in the structure which are different, but not independent from each other. Hence, a compatibility condition (equation (31)) is introduced

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ACCEPTED MANUSCRIPT mechanical calulation

chemical calculation

aC-C q2 q2

aC-C

θ Wmech

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1 2

1 1 θ Wchem = kθ ∆θ12 + 2 kθ ∆θ22 (30) 2 2

(28)

3 Eb I 2 = ∆θ 1 2 l

(29)

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θ Wmech = 2WB

aC-C

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q1

∆θ 1 = −∆θ2 2

(31)

3 θ = kθ ∆θ12 Wchem 4

(32)

Figure 13: Comparison of the deformation works due to bond angle changes. relating ∆θ1 to ∆θ2 . It has to be noted, that this condition is only exact if the tetrahedron is

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perfectly flat. However, the results show, that this approximation is reasonable, see section 3.

The earning of all the calculations given in this section is a replacement of the bond angle part of the original model, and in particular a replacement for the bond angle bending equation given in Figure 1 connecting the beam properties and the chemical force constant. 3 Eb I 2 2 l ∆θ1

θ = (equation (29)) to Wchem

(equation (32)) given in Figure 13 which yields:

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3 2 4 kθ ∆θ1

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θ This new equation is obtained by equating Wmech =

kθ = 2

Eb I . l

(33)

2.3.4 Comparison of the original and the modified bond angle bending approach Compared to the bond angle part of the original approach, the present work introduced two major changes, see Figure 14: 1. The consideration of linear bending moments instead of constant ones. 2. The calculation of equation (33) is based on the calculation of the deformation work due to bending of all contributing beams in a tetrahedron. In contrast to the new approach, the original approach used a single bond angle change only.

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new approach aC-C q2 q2

q1

Dq

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aC-C

θ Wchem = 43 kθ ∆θ12

Uθ = 12 kθ ∆θ 2

UM =

1 Eb I 2 l

Eb I l

kθ = 2 ElbI

θ = Wmech

(2∆α )2

M Da

Da

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M

3 Eb I 2 2 l ∆θ 1

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kθ =

aC-C

Figure 14: Comparison of the bond angle parts in the original model by Li and Chou [14] with our new model.

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This new, modified bond angle bending approach is now integrated into the framework of the original approach. Since the bond stretching and the bond torsion part remain unchanged, see Figure 1, the three equations connecting the beam properties with the chemical

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force constants now read:

Eb A l Eb I = 2 l Gb J = . l

kr =

(34)

kθ

(35)

kτ

(36) (37)

With the introduction of the beams cross-sectional (circular) properties like the area A=

π 2 d , 4 b

(38)

I=

π 4 d , 64 b

(39)

the area moment of inertia

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π 4 d , 32 b

(40)

it is possible to calculate the beam properties Eb (Young’s modulus of the beam material), Gb (shear modulus of the beams material) and db (diameter of the beam circular cross

kr2 l 4π kθ kτ lkr2 = 4 8π kθ2 r 1 kθ = √ 4 . kr 2

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section) by solving the system of equations given above: Eb = 2

db

(42)

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Gb

(41)

(43)

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This means, that the proportions of the original parameters to the new ones are the following:

= 2

(44)

= 4

(45)

=

1 √ . 2

(46)

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Eb modified Eb Li and Chou Gb modified Gb Li and Chou db modified db Li and Chou

The evaluation whether and how the modified parameters contribute to an improvement of the model is done in the next section.

Results and discussion

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3

In this section, the modified model developed in section 2.3 is applied in order to calculate

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the elastic constants (Young’s modulus and Poisson ratio) of the CNTs. The principle of the determination is the same as used in section 2.1. This means, that a virtual tensile test is conducted, and the results of this tensile test are used to calculate the elastic constants. Furthermore, the energy deviation introduced in section 2.2 is determined. The energy deviation serves as benchmark, whether the model of the CNTs is improved due to the modifications done in section 2.3 or not.

3.1 Young’s modulus Figure 15 and Tables 1 and 2 show the results obtained for the Young’s modulus. Given are both, the results obtained with the original as well as the results obtained with the modified approach. One can recognize, that the Young’s modulus calculated with the modified

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ACCEPTED MANUSCRIPT model is smaller compared to the original calculations. For tubes with larger diameters the Young’s modulus converges to approximately 0.8 TPa. While figure 15 shows the Young’s ˚ table 1 and table 2 also modulus for CNTs with diameters from approx. 2.5 to 16.5 A, include SWCNTs with diameters larger than 3 nm ((24,24,), (36,36), (40,0), (60,0)). These additional results in the table are given to (i) ensure that the Young’s modulus is convergent, and (ii) provide results SWCNTs accessible for experimental investigations.

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The graphs of the corresponding armchair and zig-zag tubes modeled with the original or the modified approach, respectively are basically only shifted, while the general progression is maintained. 1200

(20,0)

(2,2)

800

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Young‘s modulus [GPa]

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(12,12)

1000

(2,2) 600

(3,0)

400

(3,0)

0

2

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200

4

6

8

(12,12)

(20,0)

armchair (n,n) original model zig-zag (n,0) original model armchair (n,n) modified model zig-zag (n,0) modified model 10

12

14

16

18

CNT diameter [Å]

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Figure 15: Results for the Young’s modulus for armchair and zig-zag nanotubes for the original and the modified model. This means, that the CNTs calculated with the modified model behave less stiff than the

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ones calculated with the original approach. Since both models use the same geometrical representation (exact polyhedral model), the reason for the deviations can solely be found in the changed parameters for the representation of the covalent bonds. Using the relations in equations (44)-(46) one can prove that the axial as well as the torsional rigidity of the covalent bond stay the same EAmodified = EALi and Chou

(47)

GJmodified = GJLi and Chou .

(48)

However, the comparison of the flexural rigidity shows, that the modified flexural rigidity

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ACCEPTED MANUSCRIPT is reduced to be the half of the rigidity in the original approach EImodified =

1 EILi and Chou . 2

(49)

This explains why the Young’s modulus of the investigated CNTs is reduced. The increase of the Young’s modulus with increasing CNT diameters is due to geometrical

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reasons. The discrete CNT geometry, including its diameter, governs the load distribution

in the CNT. As a result of this, it was found, that the portions of the deformation works

spent on (i) bond stretching, (ii) bond angle bending and (iii) bond torsion depend on the

CNT diameter and are therefore not constant. One may assume, that the more energy is used for bond stretching, the stiffer the CNT and the higher the Young’s modulus. Hence,

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we calculated the portions of the deformation works spent on bond stretching, bond angle

bending and bond torsion and observed that the bond stretching contributions increase with

3.2 Poisson ratio

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rising diameters resulting in the increased Young’s modulus.

Figure 16 and Tables 1 and 2 give the results for the Poisson ratio of the investigated CNTs. Again, the results from the original as well as from the modified approach are given in order to compare them. It can be noticed, that the Poisson ratio, compared to the original model,

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is increased and can be found at approximately ν = 0.275. Figure 16 shows the Poisson ˚ Again, table 1 and table 2 ratio for SWCNTs with diameters from approx. 2.5 to 16.5 A. provide extended results for the Poisson ratio obtained for SWCNTs with diameters larger than 3 nm. The results are given to ensure the convergence of the Poisson ratio. While the armchair CNTs (original and modified model) show basically no dependence on the CNT diameter, the Poisson ratio of the zig-zag CNTs (modified model) increases

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very fast from smaller diameters towards larger ones. After the maximum for (5, 0) zig-zag CNTs, the Poisson ratio decreases slowly towards its converging value. This shows that the behavior of the zig-zag tubes changed significantly, since the original results for the zig-

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zag CNTs show a decreasing progression of the graph. The graph for the armchair CNTs is only shifted when comparing the original and the modified results. The fundamental reason for this behavior has not been identified yet. The overall increase of the Poisson ratio compared to the results obtained with the original model can be explained by the reduced bending rigidity of the beam elements representing the covalent bonds. This results in a change of the load distribution in the CNTs which needs, to be investigated in more detail.

3.3 Energy deviation The results of the energy deviation discussed in section 2.2 of the present work are given in Figure 17 as well as in Tables 1 and 2 for both, the original as well as for the modified

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(2,2)

(12,12) (20,0)

0.25

(3,0)

armchair (n,n) original model zig-zag (n,0) original model armchair (n,n) modified model zig-zag (n,0) modified model

(3,0)

0.15

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Poisson ratio [-]

0.2

0.1

(20,0)

(2,2)

(12,12)

0

2

4

6

8

10

12

14

16

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CNT diameter [Å]

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0.05

Figure 16: Results for the Poisson ratio for armchair and zig-zag nanotubes for the original and the modified model. model. The figure shows that the energy deviation is strongly reduced with the application of the modified theory. While for CNTs with smaller diameters still a deviation, yet strongly reduced, can be found, one can notice that for larger diameter CNTs the energy

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deviation converges to zero. We assume, that the deviation for smaller diameter CNTs are, at least partly, due to the geometrical simplifications we introduced in section 2.3. This is indicated by the fact, that the errors due to the geometrical simplifications decrease for larger diameters of the CNTs. In order to verify the results obtained for SWCNTs, also

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a plane zig-zag graphene sheet with the dimensions 3.13 nm by 37.51 nm was subjected to the virtual tensile test. Again, the energy deviation is close to zero when applying the modified MSM model. The very small remaining energy deviation of 0.55 % is due to edge

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effects in the graphene sheet on the one hand, and on the other hand due to the calculations of the deformation works which include assumptions specifically made for tubular structures.

In conclusion, the results in the present section show, that the drawbacks of the original model are resolved by the modifications introduced in section 2.3 of the present work. Hence, the modified parameters achieved a substantial improvement of the CNTs’ mechanical model.

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armchair (n,n) original model zig-zag (n,0) original model armchair (n,n) modified model zig-zag (n,0) modified model

(2,2) 25

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20

(12,12)

(20,0)

15

10

(2,2)

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energy deviation DW [%]

(3,0)

5

0

2

4

6

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(3,0) 8

10

12

14

(20,0) (12,12) 16

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CNT diameter [Å]

original model E [GPa] ν [-] ∆W [%] 925 693 0.062 26.04 989 103 0.061 21.12 1 011 871 0.060 19.42 1 022 509 0.060 18.63 1 028 315 0.060 18.20 1 031 825 0.060 17.93 1 034 106 0.060 17.76 1 035 672 0.060 17.64 1 036 793 0.060 17.56 1 037 623 0.060 17.50 1 038 254 0.060 17.45 1 040 747 0.059 17.26 1 041 209 0.059 17.22 1 041 371 0.059 17.21

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˚ d [A] 2.9022 4.1956 5.5211 6.8593 8.2038 9.5519 10.9022 12.2540 13.6068 14.9603 16.3145 32.5824 48.8607 68.1416

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(n, m) (2, 2) (3, 3) (4, 4) (5, 5) (6, 6) (7, 7) (8, 8) (9, 9) (10, 10) (11, 11) (12, 12) (24, 24) (36, 36) (48, 48)

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Figure 17: Results for the energy deviation ∆W for armchair and zig-zag nanotubes for the original and the modified model.

modified model E [GPa] ν [-] ∆W [%] 683 411 0.274 7.97 749 038 0.274 3.77 773 010 0.274 2.13 784 278 0.274 1.32 790 445 0.274 0.86 794 179 0.274 0.59 796 609 0.274 0.41 798 278 0.274 0.28 799 473 0.274 0.19 800 358 0.274 0.13 801 031 0.274 0.08 803 691 0.273 0.12 804 182 0.273 0.15 804 354 0.273 0.16

Table 1: Results of the numerical simulations for armchair nanotubes.

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modified model E [GPa] ν [-] ∆W [%] 431 899 0.250 3.90 571 743 0.289 3.30 648 422 0.293 2.54 693 411 0.291 1.94 721 692 0.288 1.50 740 515 0.286 1.18 753 633 0.284 0.95 763 124 0.282 0.78 770 203 0.281 0.65 775 619 0.280 0.56 779 854 0.279 0.48 783 227 0.278 0.42 785 955 0.278 0.36 788 194 0.277 0.32 790 053 0.277 0.29 791 613 0.276 0.26 792 935 0.276 0.24 794 065 0.276 0.22 801 940 0.274 0.08 803 401 0.274 0.06 803 912 0.274 0.06

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original model E [GPa] ν [-] ∆W [%] 605 892 0.149 23.37 778 011 0.139 20.35 867 866 0.119 19.11 919 155 0.104 18.50 950 872 0.094 18.16 971 762 0.086 17.95 986 219 0.081 17.79 996 625 0.077 17.69 1 004 359 0.074 17.62 1 010 260 0.072 17.56 1 014 865 0.070 17.51 1 018 525 0.069 17.48 1 021 483 0.068 17.45 1 023 907 0.067 17.42 1 025 918 0.066 17.40 1 027 604 0.065 17.39 1 029 032 0.065 17.37 1 030 252 0.064 17.35 1 038 742 0.061 17.28 1 040 317 0.060 17.26 1 040 869 0.060 17.25

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˚ d [A] 2.6795 3.3797 4.1130 4.8632 5.6231 6.3890 7.1588 7.9314 8.7060 9.4820 10.2592 11.0373 11.8161 12.5955 13.3754 14.1556 14.9363 15.7171 31.3617 47.0224 62.6872

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(n, m) (3, 0) (4, 0) (5, 0) (6, 0) (7, 0) (8, 0) (9, 0) (10, 0) (11, 0) (12, 0) (13, 0) (14, 0) (15, 0) (16, 0) (17, 0) (18, 0) (19, 0) (20, 0) (40, 0) (60, 0) (80, 0)

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Table 2: Results of the numerical simulations for zig-zag nanotubes.

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4

Conclusions

In the present work a new, modified molecular structural mechanics model was developed. The model was based on the representation of the covalent bonds by beam elements. The mandatory geometrical and material parameters defining the beams were obtained with the help of a chemical force field representation. At first, the motivation for the development

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of a new model was given. This included the detailed investigation of the original MSM model of Li and Chou [14]. The investigation revealed a major drawback of the model,

which is, that it is not consistent to the underlying chemical force field description in terms of energy. This fact was shown by the energy deviation calculated for a virtual tensile test.

The reason for this energy deviation was detected in the not properly considered linear in-

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ternal bending moments.

As a result of the considerations given above, a new, modified molecular structural mechanics model was developed. In a first step, independent on the energy deviation, the

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geometrical model of the CNTs in the original approach (roll up) was replaced by the more accurate exact polyhedral model developed by Cox and Hill [24]. Afterwards, the representation of the covalent bonds, in particular the bond angle bending part of the original model, was modified by an approach which took the linear internal bending moments into account. Furthermore, in contrast to the original approach, all the bond angle changes in a tetrahedron were considered for the calculation of the deformation works instead of a single bond angle change. Those modifications led to changed beam parameters and thus

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to different results for the elastic properties of the CNTs. In order to obtain those elastic constants, a virtual tensile test was conducted. The Young’s modulus calculated with the modified model was found in the range of approximately 800 GPa, and the Poisson ratio was found to be approximately 0.275.

The effectiveness of the modifications was verified by calculating the energy deviations

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occurring in the modified model. It was found, that the energy deviation was strongly reduced. In fact, the energy deviation for nanotubes with larger diameters was almost zero,

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while for CNTs with smaller diameters the energy deviation was not zero, but small compared to the results given by the original approach. The reason for this small deviation was assumed to be caused by the not exactly considered spatial geometry of the nanotubes during the tensile test. Despite the modifications, the advantages of the original model, like the straightforward accessibility of the general approach and the simple applicability, were maintained.

In conclusion, the present paper showed that the newly developed, modified model is - unlike other MSM models - consistent to the underlying chemical force field description in terms of energy. This fact leads to a better prediction of the elastic parameters of CNTs, since a major drawback of the original MSM approach was resolved.

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Acknowledgement The support of the German Science Foundation (DFG) within the grant WA2323/6-1 is

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gratefully acknowledged.

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