Insight of Geometry-Controlled Mechanical Properties of Spiral

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C: Physical Processes in Nanomaterials and Nanostructures

Insight of Geometry-Controlled Mechanical Properties of Spiral Carbon-Based Nanostructures Ali Sharifian, Mostafa Baghani, Jianyang Wu, Gregory M Odegard, and Majid Baniassadi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b12269 • Publication Date (Web): 15 Jan 2019 Downloaded from http://pubs.acs.org on January 15, 2019

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The Journal of Physical Chemistry

Insight of Geometry-Controlled Mechanical Properties of Spiral Carbonbased Nanostructures Ali Sharifian,1 Mostafa Baghani,1 Jianyang Wu,2 Gregory M. Odegard3 and Majid Baniassadi1,4,* 1

School of Mechanical Engineering College of Engineering University of Tehran, P.O. Box 11155-4563, Tehran, Iran

2

Department of Physics, Jiujiang Research Institute and Research Institute for Biomimetics and Soft Matter, Fujian Provincial Key Laboratory for Soft Functional Materials Research, Xiamen University, Xiamen 361005, P. R. China 3

Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, United States 4

University of Strasbourg ICube/CNRS, 2 Rue Boussingault, 6700 Strasbourg France

Abstract: The spiral structures of carbon-based materials such as coiled carbon nanotube (CCNT) and graphene helicoid (GH) have attracted great attention for use in electrical and mechanical nanodevices. There are a couple of main reasons for this attitude such as striking properties and behavioral diversity with regard to the ever-increasing need for miniaturization of devices. In this research, using atomistic simulations, the effects of geometric parameters (e.g., cross-sectional shape, pitch angle, inner diameter, and outer diameter) on the mechanical properties of CCNT are studied. Interestingly, the results show that the mechanical properties (e.g., Young’s modulus, stretchability etc.) have a heavy reliance on CCNTs geometric parameters. The stretching of the CCNT increases with raising inner radius. Geometric changes affect the various stages that the CCNTs encounter during tensile and compression tests. The different mechanical behavior of various types of CCNTs leads to their diverse applications. Thus, these results can give an insight to design and develop new-generation nanodevices.

*

Corresponding author at: Mech, Eng. Dept. College of Engineering, University of Tehran, Iran. Tel: +98 21 88024035; fax: +98 21 8801 3029. Email: [email protected]

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1. Introduction Over the past decades, carbon-based materials such as graphene and carbon nanotubes (CNTs) have attracted a great deal of interest in different fields of nanoscience and nanotechnology due to their unique combination of thermal, mechanical and electrical properties1-11. Their distinctive properties have triggered intensive studies into a wide variety of applications. They can now be used in nano-electronic devices12-14, biological sensors4,

15,

16

, nanoswitches17-20, nano

composites21, 22 and nanoelectromechanical devices19, 23, 24. Particularly, coiled carbon nanotubes (CCNTs) and graphene helicoids (GH) are greatly used in fabricating artificially new structures due to their fantastic properties and unique morphology25-28. CCNT fibers acting as ideal nanosprings are utilized to store and release energy because of their helical 3D structure. Considering the miniaturization mission of nanotechnology, CCNT is a major candidate for a new formation of electrical and mechanical nanodevices. Theoretically, the atomic structure of (CCNTs) was first suggested by Ihara et al29, 30. So far, several researchers have described the structure of CCNTs and relationships between geometric parameters (e.g., diameter, length, and position of defects)31-35. Chuang et al36, 37 introduced a generalized classification for helical carbon nanotubes and expressed that CCNTs are composed of the hexagonal network together with non-hexagonal pairs like pentagon-heptagon or quadrilaterals-octagon pairs. These non-hexagonal pairs induce positive and negative curvatures. Experimentally, Zhang et al38 have synthesized regular shapes of CCNTs in certain laboratorysettings and described the fabrication procedure for thin-coiled nanotubes. In more recent studies, researchers have successfully synthesized high quality of CCNTs39, 40. To date, there have been several experiments that investigated the mechanical properties of CCNTs with different geometrical parameters41-44. Chen et al45 employed atomic force microscopy (AFM) cantilevers to perform tensile loading on an individual coiled carbon nanotube. They observed that CCNT behaves like an elastic spring at low strains with spring constant of 0.12 N/m. In another research, Poggy et al46 examined mechanical response of a multi-walled carbon nanospring in compression with AFM and showed that nonlinear response of the nanospring is consistent with compression and buckling of CCNT. Yonemura et al47 performed a tensile test for carbon nanocoils (CNCs) with Focused Ion Beam (FIB) apparatus to study their fractured surfaces. They 2 ACS Paragon Plus Environment

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observed that stress has mainly been generated on the inner part of the coils. Also, stretchability of the CNCs was measured around 150%. Besides, several researchers theoretically characterized the mechanical properties of CCNTs. For instance, Fonseca et al48 analyzed the elastic response of helical carbon nanotube employing Kirchhoff rod model and proposed two schemes for the measurement of Young’s modulus as well as the Poisson’s ratio of CCNTs. Liu et al49 developed a simple technique to construct atomistic models for the structures of carbon nanocoils. Performing Atomistic Quantum Simulations, they also computed Young’s moduli of 3 to 6GPa of carbon nanocoils for different tubular and coil diameters with the index of (5,5), (6,6), (7,7), and (8,8). They observed, the average bond lengths of CCNTs almost remain invariant in large strains, while the elastic energy is stored through bond angle redistributions. In 2013, using atomic simulations, Wu et al41,

50

studied the stretching

instability and reversibility of tightly wound helical carbon nanotubes (HCNTs) with the coil diameters ranging from 1.5 to 16 nm and consisting of different CNT indices. Based on the results, the stiffness of HCNTs with four turns was larger than 12 nN/nm. In the initial strain, nonlinear behavior of the stress curve was observed in small HCNTs. On the other hand, linearity was observed in large HCNTs due to van der Waals (vdW) interaction between intercoil walls. They also showed that the stretchability of HCNT was in the range of 400 to 1000%. Someother researches have investigated the mechanical properties of different types of CCNTs51-54. Recently, Zhan et al25, 55, 56 proposed graphene helicoid (GH) as a new nanoscale spring. This structure is very similar to some types of CCNTs with an ellipsoidal cross-sectional structure. They investigated the tensile and thermal properties of the graphene helicoid. Their study showed that GH has a large stretching capability (the yield strain exceeding 1000%) and full recovery of the structure under tensile loading. Moreover, Wu et al57 studied the stretching properties of CCNTs with different CNT-segments chiralities. They concluded that distinctive stretching response during tensile test of CCNTs comes from different local bond configurations. Also, they reported that the tremendous extensibility of CCNTs is due to distributed nanohinge-like plastic deformation. Despite valuable information reported so far, the effects of some geometrical parameters such as inner diameter, outer diameter, the parameters of cross-section ellipsoidal shape, rising for various kinds of CCNTs have not been evaluated comprehensively yet. In addition, the compressive behaviors of different types of CCNTs have been left to answer yet. Hence, in present study, the 3 ACS Paragon Plus Environment


general effects of geometric parameters of CCNTs in order to determine Young’s modulus, elastic limit, toughness, ultimate strength, stretchability, spring constant and primary failure mechanisms are investigated. Also, the behavior of CCNTs under compression are studied through molecular dynamics simulations.

2. Atomistic models and methods In this study, molecular models of CCNTs and other spiral shapes are constructed according to the construction procedure proposed by Chuang et al36, 37. It was showed that there are different kinds of geometrical possibilities of CCNTs with non-hexagonal defects. In general, their models consist of nine parameters to form CCNTs. CCNTs are formed from parent toroidal carbon nanotube (TCNT) and nine indices characterize CCNT related to TCNT. Also, TCNT is constructed of cutand-fold procedure of graphene plane which consists of two rectangles and two isosceles trapezoids of graphene plane. To show the relation between heptagon and pentagon defects with the hexagonal network, four indices are used, namely, (s, n77, n75, n55), where s is the topological distance between adjacent heptagon defects and stands for the strips number between them. Then, n77 and n55 are half of the strips number between heptagon defects and pentagon defects, respectively. Also, n75 denotes the strips number between heptagons and pentagons along directions, which connect them directly together in the trapezoid. Fig. 1a and b illustrates a CCNT and its parent TCNT with special indices.


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Fig. 1.(a) TCNT with indices (n75, n77, n55, s)=(2, 1, 1, 2), chiral vector (1, 0), and nrot = 6. The polygonal shape is depicted as skeletal frames encompassing the corresponding TCNT. The shaded region is a particular rotational unit cell. (b) The rotational unit cell (shaded region in (a)) is unfolded onto a planar graphene sheet36.(c) Three categories of CCNTs based on the structural characteristics which consist of double-layer GH, circular-shaped CCNT and helical arrays. Indices (n1, n2) are used for describing CCNT taken from the usual notation of CNT that is the chiral vector. In this study, the chiral vector of (n1, n2) = (1, 0) is used for all cases. Moreover, nfold, nrot, and HSP are other indices of CCNT, where nfold is the number of CCNTs turns. All models include four coils (nfold=4) similar to other works50,

57

in order to be sufficiently

comprehensive and to account for necessary interactions. nrot indicates n-fold rotational symmetry that generally describes the shape of TCNT. According to Chuang model, nrot is selected equal to 6 for all samples. Also, HSP expresses the relative defects positions which have the major effect on pitch angle of CCNTs. In this research, hexagon-shaped of TCNTs were used which are more energetically stable than other ones58. Also, HSP parameter used for construction of CCNTs on higher pitch angles in Chuang models.



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Structural parameters of CCNTs are listed in Table 1. It lists structural types, parameters of ellipsoidal shape of CCNTs tube in side view, i.e., semi-major and semi-minor, effective radius, inner radius, outer radius, potential energies and initial length of relaxed CCNTs. The effective radius of CCNTs is: 2 2 ⎡⎛ ⎛ N ⎞⎞ ⎛ ⎛ N ⎞⎞ ⎤ r = ∑ ⎢⎜ xi − ⎜ ∑ xi / n ⎟ ⎟ + ⎜ yi − ⎜ ∑ yi / n ⎟ ⎟ ⎥ i ⎢⎝ ⎝ i ⎠⎠ ⎝ ⎝ i ⎠ ⎠ ⎥⎦ ⎣ N

(1)

where N denotes the number of all atoms in CCNT, also, x and y are positions of atoms in planar directions. Generally, three categories of CCNTs based on the structural characteristics are studied here as shown in Fig. 1c. These categories consist of 1) very small semi-major of the oval-shaped crosssection CCNT, e.g., (7,4,1,4) CCNT, 2) very small semi-minor of the oval-shape of cross-section CCNT, e.g., (2,1,7,1) CCNT and 3) CCNTs with a cross-section close to the circle, e.g., (6,1,2,1) CCNT. The first category of CCNTs shows helical arrays with a high ratio of semi-minor to semimajor. Notably, samples of the second category are very similar to GH. One may conclude that they are double-walled GHs. The third category of CCNTs shows circular-shaped cross section CCNTs which is called circular-shaped CCNTs. In this study, the geometry of CCNT models is constructed in accordance with changing in structural type, pitch angle, inner radius, outer radius, type of defects, effective radius and shapes of cross-section views. The heptagon and pentagon rings are located at the outer and inner edges of CCNTs due to the conformation of the curvature.

Table 1. Structural type and properties of CCNTs


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Structural Number type of atoms (s,n77,n75 ,n55) (2,0,4,0) 1536 1-(2,1,2,1) 1152 2-(2,1,2,1) 1152 (2,1,4,1) 2304 (2,1,7,1) 4752 (2,1,11,1) 9360 (4,2,2,2) 2880 1-(3,2,3,2) 3024 2-(3,2,3,2) 3024 (3,2,5,2) 4752 (3,2,7,2) 6864 (4,1,2,1) 1920 (4,1,7,1) 6480 (4,3,4,3) 5760 (4,4,1,4) 3888 (5,3,4,3) 6720 (6,1,2,1) 2688 (6,3,4,3) 7680 (7,4,1,4) 6480 (8,4,6,4) 14784

Pitch (Å)

Semimajor (Å)

Semiminor (Å)

3.5 8.8 19.6 10.58 7.6 9.81 18 12.2 31.1 15 13.6 13.1 10.38 27.3 20.52 15.6 8.7 34.9 20.17 25.17

3.185 2.75 2.6 5.55 9.185 13.725 3.05 6 6 8 8.8 2.65 7.8 4.15 1.85 5.05 2.8 4.65 1.75 8.6

0 2.65 2.35 2.55 2.85 4.45 4.28 5.05 4.2 3.5 5.05 2.8 3.6 6.45 8.35 6.7 2.8 6.35 7.65 10.2

Effective Inner Outer Potential radius diameter diameter energy (Å) (Å) (Å) (ev) 9.5611 7.1875 7.21232 9.9 14.3686 20.3208 11.4317 10.6041 10.7767 13.5693 16.6506 11.5335 18.2195 14.22 9.88625 16.1867 15.9657 17.9778 16.9223 25.5738

10.1 9.8 8.2 8.9 8.9 9.8 16.38 13.4 14 10.4 17.2 17.57 19.73 14.8 15.8 18.9 29.3 27.3 31.5 25.34

26.8 19.7 15.3 25.9 43.4 63.4 26.3 28.8 29 38.5 43.2 28.34 50.6 33.5 22.6 43.4 36.3 42.6 38.9 60.8

-10489.5 -8190.07 -7267.78 -16727.8 -34875.9 -68944.8 -20922.5 -22009.3 -19621.2 -34809.4 -50379.1 -13849.4 -47715 -40298.6 -27183 -49370.3 -19450.8 -54100.8 -47344.6 -108935

Initial length (Å) 27.91558 34.97669 65.73352 37.17291 37.08874 38.97121 49.83241 51.7157 94.59591 52.47571 57.43254 35.14853 42.21898 68.75391 83.20023 70.53959 34.73439 71.73344 79.43514 91.19415

All molecular dynamic (MD) simulations are performed under the software LAMMPS MD package (16 Mar 2018 version). The widely-used adaptive intermolecular reactive empirical bond order (AIREBO) potential59 is applied to simulate many-body short-range interactions and longrangevdw interactions. Generally, this potential includes three segments of two-body LennardJones (LJ), torsional term and the second reactive empirical bond order (REBO)60with the ability to break and re-form C-C covalent bonds. AIREBO potential form is: E=

⎡ LJ 1 TORSION REBO ⎤ Enm + ∑ ∑ Eknml + Enm ∑∑ ⎢ ⎥ 2 n m≠ n ⎣ k ≠ n l ≠ n ,m ,k ⎦

LJ

TORSION

where Enm is standard long-range 12-6 LJ potential, Eknml

(2)

is torsional interaction potential that

REBO

is dependent on dihedral angles, and Enm stands for REBO potential. The connection between these potentials occurs with switch function to preserve the reactive characters of the potential. In this work, the cutoff distance for REBO potential was set to 2 Å to eliminate unreasonable and non-physical events during tensile test61. Also, the long-range LJ cutoff distance 10.2 Å is selected to ensure the accuracy of the results of using this potential in large distance9, 41.In all calculations, 7 ACS Paragon Plus Environment


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periodic boundary condition (PBC) is employed along the tensile direction in order to avoid untrue end effects. The free boundary condition is applied in other directions. In the beginning, CCNTs energy is minimized by the conjugate gradient method. Then, the system is relaxed under an NPT (Isothermal-Isobaric) ensemble based on Nosé-Hoover thermostat and Nosé-Hoover barostat at the zero-bar pressure for 300 to 600 ps. This is related to the geometry and size of the spiral structures. Also, temperature is set on 1 K to reduce the effects of thermal fluctuations. Followed by the relaxation, the uniaxial tensile test was carried out with a strain rate of 10 8 s −1 by uniformly rescaling position of all atoms along helix axis in every 1000time-steps. During this stage, the temperature had been controlled by using a canonical NVT (i.e., constant number of particles, constant volume and constant temperature) ensemble with Nosé-Hoover thermostat. In all simulations, the velocity Verlet method and a time step of 1 fs are applied to integrate the equation of motion. Force is calculated by using the virial term of equations, which identified by computing the pressure and effective area as62: Nʹ

ri .f i NK BT ∑ i + P= V dV

(3)

where N is the number of atoms, KB represents the Boltzmann constant, T is the temperature of the system, d indicates the dimension of the system, V denotes the system volume, ri and f i are the position and force vector of atom, and N ʹ defines the periodic image atoms outside the central box in addition to the number of atoms in the system. In the equation (3), the first term is kinetic energy and the second one is the virial which is used in simulations. For stress calculation, the virial stress tensor can be used for each atom as suggested by Thompson et al62. The VMD software is used to visualize all simulations63.

3. Results and discussion 3.1. Study of the equilibrium structures of CCNTs As we expect, the pentagon and heptagon defects are energetically different from the original hexagonal network as shown in Fig. 2.Fig. 2shows top and side views of optimized CCNTs based on the effect of changing structural parameters which listed in Table 2. Table 2. Features related to studied groups of CCNTs. 8 ACS Paragon Plus Environment

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Group Variable parameter Change effect Category of CCNTs a n75 semi-major circular-shaped CCNT and double layer GH b s inner radius circular-shaped CCNT, double layer GH and helicals arrays c s , n75 semi-major and inner radius circular-shaped CCNT d n55,n77 semi-minor circular-shaped CCNT, double layer GH and GH e HSP pitch angle circular-shaped CCNT

Fig. 2. Atomistic structural of CCNTs. (a)-(e) show top and side views of CCNTs after relaxation which related to group of studied CCNT as listed in Table 2. Samples include different geometric of CCNTs. The intercoils distance is about 3.4 Å, except for 2-(2,1,2,1) and 2-(3,2,3,2) that are less structurally stable due to more defects. The color of atoms is related to potential energy. As aforementioned, the heptagon and pentagon located at inner and outer walls of CCNT. These defects were used for generating convex and concave tubes with 3D spiral structures. These defects can be energetically favorable or unfavorable than hexagons. Different shapes of CCNTs in side view and cross-section view were constructed so that defects are accommodated with hexagon 9 ACS Paragon Plus Environment


network. Notably, for CCNTs such as 1-(2,1,2,1) and 2-(2,1,2,1) (see Table 1), geometrical parameters are the same except pitch angle due to different HSP parameter. To create higher pitch angles, another type of defect is simulated. This means that some bonds are broken randomly in equilibrated CCNTs. For all CCNTs except 2-(2,1,2,1) and 2-(3,2,3,2), intertube spacing is about 3.35 Å. Approximately, carbon atoms in these structures possess so different values of potential energy than other structures with lower inter-tube spacing (see supporting information, S1). By changing ‘s’ parameter in order to increase the inner radius and consequently, increase the outer radius (without changing other specifications), changes in potential energy are observed in different structures. Generally, for special categories of CCNT e.g. helical arrays and doublewalled GH, considering the potential energy levels, it is observed that at the higher inner radius, structures exhibit a more stable state. In the following sections, this is studied by examining the stress-concentration for each case. For the circular-shaped CCNTs, no significant change with increasing the inner radius is observed. However, in very small inner radius, there is a higher potential energy in the inner part of CCNT. It seems at small inner radius of the CCNTs (e.g. (2,1,2,1) CCNT), the inner part of CCNT suffer from bond distortions which leads to higher energy than other part. Another possible explanation for this stems from vdW interaction that it clearly influences more on the inner part of CCNT. Also, by increasing semi-major of CCNTs cross-section, which means a change of ‘n75’ parameter, a higher potential energy is observed in the inner part of CCNT. This situation is aggravated at a smaller inner radius as discussed earlier. For other structural changes, there is no significant effect on the potential energy of CCNTs. 3.2. Evaluation of stretching behavior To investigate CCNTs tensile behaviors, the effect of different parameters is studied in several groups separately, which can play a key role in the application of these nanoparticles. Change of each parameter is selected as such to study the behavioral effects as illustrated in Table 2.Stressstrain and force-displacement diagrams alongside the molecular configuration motifs are used to study their correspondence. The goal is to analyze the stress-strain diagram and consequently, investigate unique stages of the tensile response of each CCNT to receive at a better physical vision. Moreover, to evaluate the importance of vdW energy and torsion energy in CCNTs structures, REBO potential has also been applied. It is noted that both REBO and AIREBO are


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known for being accurate in the breaking and formation of chemical bonds. Comparing these two potential types reveals interesting information. The role of the cross-sectional variations through changing semi-major of CCNT was investigated as the first group samples. For this purpose, the parameter ‘n75’ is changed in accordance with categories of CCNTs as illustrated in Table 2and the relevant effects are examined. Therefore, two cases were considered which one of them is approximately circular-shaped and another one as already mentioned is double-walled GH. Clearly, the structures of (3,2,3,2), (3,2,5,2) and (3,2,7,2) CCNTs have the same parameters except semi-major indices as (2,1,7,1) and (2,1,11,1) CCNTs. Fig. 3 plots stress-strain and forcedisplacement curves for these cases. Moreover, by employing REBO and AIREBO potentials, the stretching response was compared.

Fig. 3.Profile of force-displacement and stress-strain for CCNTs. (a) and (c-f) present force (stress) curves. (b) presents force curves based on REBO and AIREBO potentials. Red arrows show the behavioral change in lower strains so that as increases semi-major, CCNTs behavioral tend toward GH behavioral process. For all samples, there is an initial narrow region for stress-strain curves (stage 1). This stage shows the linear elastic response. Stage 1 arises from strong vdw interactions between two adjacent coils. Fitting the strain range of 0-0.8%, Young's modulus of (3,2,3,2), (3,2,5,2), (3,2,7,2), (2,1,7,1), 11 ACS Paragon Plus Environment


(2,1,11,1) CCNTs are computed 14.9, 9.3, 6.8, 4.9 and 4.1 GPa, while the correspondent spring constants are 38.01, 35.2, 26.72, 31.95 and 31.94

!" !#

, respectively. These springs constants are one

order of magnitude larger than those calculated in experiments45, 64.These deviations can be due to differences in size and chirality. While, these are almost compatible with Wu et al41, 50that used molecular dynamic simulation. Comparing Young’s moduli and spring constants, the crosssectional effect is well investigated so that at larger semi-majors both are reduced. However, this trend almost can be neglected for double-walled GH. In fact, this implies at higher semi-majors in this region, the slope of force versus displacement decreases, but this linear elastic response sustains in a broader span. Revisiting Fig. 3, a sharp reduction in force appears except for doublewalled GH (see also Fig. S2). To clarify this, the molecular configuration of CCNTs is depicted in Fig. 4 where the stress drops (Ɛ≈0.1). The gap space larger than usual space (about 3.4 Å) was probed, where sometimes a phase transformation emerges because of vdW intercoil interactions to accommodate with the applied strain. Double-walled GHs are under more uniform vdW force, thus, no phase transformation occurs. These observations prove the importance of the role of semimajor as a geometric parameter on the phase transformation occurrence in the first stage region. Consequently, this phenomenon affects the elastic properties.

Fig. 4. Molecular configuration of CCNTs at about 0.1 strain. A gap space greater than usual space (about 3.4 Å) happens on some CCNTs which shows by arrows. VdW interaction is caused by this gap and creates fluctuations for the force-displacement curve. Effect of geometric parameters is evident in this range of strain that can affect the elastic properties of CCNTs. The color of atoms is related to stress in the tensile direction. According to the atomic structural shown in Fig. 4, middle atoms experience compressive stress and this area grows by increasing the semi-major. It seems, most part of the tensile stress is endured 12 ACS Paragon Plus Environment

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by the inner and outer parts of CCNTs, while, a compressive stress is observed in the middle segments of CCNTs. There are four stages for (3,2,3,2) CCNT as illustrated in Fig. 5a. By going through stage 1 which already was discussed, in stage 2, the coils buckle along the strain direction in order to accommodate CCNT with further stretching in strain range of 0.1 to 0.5, without any breaking and formation of bonds. The main reason for the fluctuations of the force-displacement diagram stems from the reduction of the impact of vdW forces which leads to the phase transformation between the turns of coils. Stage 3 of this simulation includes twisting and kinking mechanisms in addition to the buckling which causes intertwining the coils. Therefore, stress increases in the inner part of the coils as shown in Fig. 5a. In the final stage, the breakdown of bonds begins in a catastrophic form. This is the reason for fluctuations and saw-tooth pattern in the force (stress) diagram. This stage continues to stand CCNT upright and finally rupture is occurred.



Fig. 5. Atomic structural of the first group samples during tensile test. (a) Deformation stages of (3,2,3,2) CCNT during the tensile test that AIREBO potential was used for simulation. (b) The figure shows stages of deformation (3,2,3,2) CCNT by using REBO potential. (c)Atomic structural of (3,2,7,2) CCNT during stretch includes 4 stages. (d) stage of (2,1,7,1) CCNT and (e) stages of (2,1,11,1) CCNT. The color of atoms is related to stress in the tensile direction. To demonstrate especially the importance of vdW energy and torsion energy in these structures, REBO potential is also employed. In comparison, only three stages are shown by using REBO 14 ACS Paragon Plus Environment

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potential. In stage 1, because of lacking vdW interactions, increasing the stress should be due to another phenomenon. Revisiting Fig. 5b, the equilibrated molecular configuration of CCNT shows covalent s𝑝% -hybridized existing between some atoms in adjacent coils. In this stage, higher stresses are applied to the nanostructure and only buckling happens. Stage 2 is characterized by kinking and twisting mechanisms while buckling is still continued. The distance between the coils which do not have any connection, is augmented due to lacking vdw interactions. Eventually, the first breakage of the bonds begins and continues up to the rupture, while some connections between adjacent coils may remain in folded coils. As expected, lower strains are observed. Next, investigating the effect of increasing semi-major, a faster drop in force is observed as shown in Fig. 3a, c, and d. As presented in Fig. 5c, the structure of (3,2,7,2) CCNT is identified during the tensile test. Closely reviewing, this CCNT has four stages similar to (3,2,3,2) CCNT, but they are a bit different. The behavior of CCNT tends toward GH behavior, due to a large semi-major together with a small semi-minor. Stage 2 shows an unstable delamination process at the small range of strain with no breaking of bonds. In general, the delamination procedure lowers force due to rescue a turn of CCNT from vdW interactions. Then, slowly other ones conquer vdW interactions without any significantly increasing in force. In further strain and with the beginning of stage 3, breaking of bonds starts along with the delamination process. Thus, the higher force is observed due to the resistance of the breaking of bonds. This continues until tearing of the bonds. Hence, breaking of bonds leads to a smaller force. For this CCNT, increasing the shear stress and normal stress simultaneously starts tearing of the bonds. The final stage demonstrates that delamination is finished completely and straight CCNT is broken. As discussed, this characteristic response is similar to that of GHs, thus, stretchability of this CCNT is 2-times larger than (3,2,3,2) CCNT. The (2,1,7,1) and (2,1,11,1) CCNTs have small semi-minors, and structurally are very similar to GHs; however, their behavior is different. Fig. 5d and e present some snapshots of these CCNTs. The stages for (2,1,7,1) CCNT are similar to those of (3,2,7,2) CCNT except stage 3in which the failure of bonds occurs faster than the delamination. Subsequently, CCNT is broken before the complement of delaminating process. This is due to significant growth of shear stress at the inner segment of CCNT. Another explanation for this goes back to the smaller area (due to a small semiminor) at the inner part where the stress acts. Therefore, stage 3 is called the last stage for this type of CCNT. 15 ACS Paragon Plus Environment


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At high values of ‘n75’ index, CCNTs show different behavior due to the increase of the area where vdW force is applied. By simulating tensile test on (2,1,11,1) CCNT, it is observed that stage 2 is different from the latter double-walled GH. As depicted in Fig. 5e and Fig. S3, stage 2was divided into two parts. The first part is similar to (3,2,7,2) CCNT. However, in the second part, vdW interactions on the adjacent layers are indicative of the phase transformation. This is shown in the form of a drop in the force curves as plotted by arrows in Fig. 3. In addition, stretching decreases due to the rise of vdW inter-coil domain. Thus, the higher force is required to overcome vdW forces. Hence, CCNT should tolerate more stress leading to faster damaging of CCNT (see Fig. 5d and e). In the second group of samples, increasing the inner radius (while fixing other parameters), mechanical properties of CCNTs including e.g., double-walled GHs, circular-shape CCNTs, and helical arrays, are studied. The stress and force response are shown in Fig. 6. Like before, stage 1 of this group of CCNTs was investigated firstly. For this goal, Young’s moduli of (2,1,2,1), (4,1,2,1), (6,1,2,1), (4,1,7,1), (4,3,4,3),(6,3,4,3),(4,4,1,4)and(7,4,1,4) CCNTs are calculated 17.5, 24, 12, 2.3, 8, 36, 33 and 20 GPa and their corresponding spring constants are calculated 39.1, 52.6, 51.7, 22.09, 21.08, 30.06, 32.04 and 31.04 strain range of 0-0.8%.


!" !#

respectively, by fitting in the

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Fig. 6. Profile of force-displacement and stress-strain for CCNTs. The results indicate that generally, elastic moduli and spring constants increase at the higher inner radius. This is due to the phase transformation phenomenon that occurs in the initial region same as before (for more details see supporting information, S14). The most important effect in this region is tilting of the coils, which is more dominant in CCNTs at the higher inner radius. Also, this effect reduces force and stress in accommodation with more strain. For example, comparison of (2,1,2,1) and (6,1,2,1) CCNT is illustrated in Fig. 7a and b.



Fig. 7. Atomic structural of (2,1,2,1), (6,1,2,1), (4,4,1,4) and (7,4,1,4) CCNTs during tensile test. In (a), stage of (2,1,2,1) CCNT and in (b) stages of (6,1,2,1) CCNT are shown. Stretching of CCNTs increases by raising in the radius of CCNTs. (c) stage of (4,4,1,4) CCNT and (d) stages of (7,4,1,4) CCNT. Helical arrays exhibit similar properties with increasing inner radius; however, stretching increases due to decreasing stress in the inner segment. The twist is the most important failure factor in helical arrays. The color of atoms is related to stress in the tensile direction.


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Tensile properties of (2,1,2,1), (4,1,2,1) and (6,1,2,1) CCNTs were investigated as the first samples of this group. Four stages are observed for (2,1,2,1) CCNT. Stage 2 involves buckling and twisting without any breaking of bonds, thus larger forces are observed. Although, some fluctuations exist in force curve caused by displacement of turns. In stage 3, breaking of bonds starts and continues with twisting. This continues until the CCNT arrives at an almost flat state. In the forcedisplacement diagram, breaking of some bonds produces some fluctuations in this region. Force is smaller at the end of this stage, which shows the importance of breaking some bonds. Eventually, taut CCNT breaks at the final stage. These breaks are mainly from inward edges of CCNT, and thus, it has a considerable effect on the force. Also, a monatomic chain is formed before failure. As inner radius increases, for instance, in (6,1,2,1) CCNT, different behavior is observed. Stage 2 shows a significant load drop due to rescuing a turn of CCNT from vdW interactions. This drop is higher than that of the previous sample. Then, the separation of the layers begins and continues until the separation is completed. The second stage is in a large range of strain (0.5 to 5.012). In higher strains, approximately, other stages are similar to the previous one. Breaking of bonds, and at the same time the process of buckling and twisting in the unfold regions, are observed in stage 3. In the final stage, straight CCNT breaks down. No monatomic chain formation is observed, and the failure happens because of the torsion at the points with higher stress. Also, its atomistic configuration is recorded (video. S1). As intuitively expected, at the higher inner radius, the expansion range strongly increases (from 4.54 to 10.26) and the behavior leans toward GHs. It is noted that the stretchability of this type of CCNT increases sharply after a certain inner radius. This depends on the cross-sectional radius of the circular-shaped CCNT as shown in Fig. 6. Again, in larger cross-sectional radius or larger semi-minor and semi-major of CCNTs e.g., (4,3,4,3) and (6,3,4,3) CCNTs, this study was carried out and similar steps were observed (Fig. S4). For these CCNTs to achieve at the same stretching range as latter samples of this group, it is necessary to have a larger inner radius. Increasing inner radius has an interesting effect on the double-walled GHs (see Fig. S5 and video. S2). (4,1,7,1) CCNT is compared with (2,1,7,1) CCNT as plotted in Fig. 6.It is shown at the larger inner radius, the tension on the inner walls is reduced. Thus, the tearing of bonds decreases and double-walled GH opens up to a larger extent. The steps in the both case are exactly the same. Notably, at the higher inner radius, we have the higher amount of maximum force. Even, it can withstand until the higher strain due to reduce the force in the inner part. Comparing to GHs, 19 ACS Paragon Plus Environment


double-walled GHs can tolerate higher stresses, while their stretchings are smaller. But as aforementioned, at higher inner radius stretching of double-walled GHs grows. For helical arrays such as (4,4,1,4) and (7,4,1,4) CCNTs, different responses were observed. The dominant behavior during the test is twist which is present from the beginning of the test until failure. Fig. 6f and i depict this behavior for different inner radii. In order to discuss more details, the relevant molecular snapshots during the tensile test are shown in Fig. 7. Firstly, examining (4,4,1,4) CCNT, one may conclude that mechanical response is accomplished in three stages, where stage 2 shows delamination of layers alongside twisting of uncoiled segments. Therefore, rise and fall occur in the force-displacement diagram. At strains beyond the second stage, the tearing occurs and CCNT becomes almost straight. The intensity of the twisting grows and eventually, failure occurs. These stages also take place with the same details for (7,4,1,4) CCNT (see the video. S4).It is noted that due to an increase in inner radius, the stress concentration in the internal parts is observed at larger strains as depicted in Fig. 7d. Therefore, failure happens in larger strains. Studying the results of REBO and AIREBO potentials for (2,1,2,1), (6,1,2,1) and (2,1,7,1) CCNTs, shows similar behavior to that of the first group of samples. Also, a comprehensive comparison between the results is given in supporting information. S6. In the third group of samples, simultaneously, the effects of increasing the inner radius and semimajor are examined. For this purpose, (5,3,4,3) and (8,4,6,4) CCNTs were selected. By studying force-displacement diagram as plotted in Fig. 8a and d, it is found that these CCNT show quite similar behaviors, except that there is only a wider range of stretching for (8,4,6,4) CCNT compared to (5,3,4,3) CCNT.


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Fig. 8.Profile of force-displacement and stress-strain for CCNTs. Reviewing the atomistic configuration (see supporting information, S7, and video. S3), four stages for these types of CCNTs are observed. Stage 1 shows vdW interactions that its strain range is small for both (around 0.1 strain). Also, no phase transformation was seen in these cases. The Young’s moduli and spring constants for the first and second CCNTs are calculated 12.5, 3.3 GPa and 51.15, 13.1

!" !#

,respectively. In stage 2, buckling of the coils happens at the strain range of 0.1

to 1 for (5,3,4,3) CCNT and 0.09 to 1for (8,4,6,4) CCNT. Stage 3 comes with twisting, bending and kinking mechanisms. Therefore, there is a significant amplification in tension. Moreover, in this stage, there are some differences in the strain range of (5,3,4,3) and (8,4,6,4) CCNTs, that are from 1 to 2.4 and 1.62 to 3.226 strain, respectively. Also, breaking of bonds was seen only in (8,4,6,4) CCNT. In the final stage, we have broken bonds for both, which create a saw-toothed 21 ACS Paragon Plus Environment


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pattern in the force diagram until the failure. As aforementioned, (8,4,6,4) CCNT has wider strain range in all stages compared to (5,3,4,3) CCNT because of having a larger cross-sectional radius. Increasing semi-minor value effects are investigated for fourth groups of samples for circle-like cross-sectional CCNTs and also for (2,0,4,0) GH and (2,1,4,1) double-walled GH as listed in Table 2. The (4,1,2,1) and (4,2,2,2) CCNTs are our circular-shaped samples, while the correspondent stages of (4,1,2,1) CCNTs have already been reviewed in the second group. Comparison of (4,1,2,1) and (4,2,2,2) CCNT, the effects of changing semi-minor were checked as shown in Fig. 8b and c. (4,2,2,2) CCNT similar to (4,1,2,1) CCNT has four stages. From 0 to 0.1 strain, vdW interaction is the dominant effect. Elastic modulus and spring constant are calculated 11 GPa and 26.08

!" !#

, respectively, in the strain range of 0-0.8%. After this step, buckling appears in the strain

of 0.1 to 1.38, similar to the behavior of (4,1,2,1) CCNT. The most important effect of increasing semi-minor is shown at stage 3 when the twisting and kinking effects are added. As expected, the twisting is more pronounced and the bonds are broken sooner. The final stage comes up earlier and the weaker upright CCNT is pulled up as given in Fig. S8. Adding thickness to GH and convert it to double-walled GH, e.g., (2,0,4,0) GH to (2,1,4,1) doublewalled GH, has a very powerful effect on its tensile response. As aforementioned, this greatly reduces its pulling out due to a higher thickness. This increases the tension on the inner part of CCNT. Therefore, the tension in these parts rises, and we face an early rupture. The details of double-walled GHs are discussed before. Moreover, a comparison between (2,0,4,0) GH and (2,1,4,1) double-walled GH is shown in Fig. S9. In order to reach a higher pitch angle in equilibrated CCNTs as the fifth group of samples, different HSP parameters and methods are used. Some bonds are broken in this method, but pentagon and heptagon defects still remain in their region. Force-displacement and stress-strain curves for 2(2,1,2,1) and 2-(3,2,3,2) CCNTs are plotted in Fig. 8e and f. Unlike previous examples, vdW interactions have no significant effects at higher pitch angles. Thus, it is expected to have a lower spring constant and lower modulus that are calculated 1.5, 1 GPa and 9,1 and 7.4

!" !#

, respectively.

For these samples, in contrary to the previous samples, the stages are not easily recognized. Therefore, with increasing the strain, bending and twisting begin until the CCNTs break apart. Fig. S10 shows the general trend of the tensile tests for these samples. 3.3. Evaluation of compressive behavior 22 ACS Paragon Plus Environment

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So far, to the knowledge of the authors, the effects of squeezing of different CCNTs have not been studied yet, while the results reveal interesting information. Fig. 9 shows force and stress diagrams for (2,1,2,1), (6,1,2,1), (3,2,3,2), (2,1,7,1) and (2,0,4,0) CCNTs under compression. In general, the behavior under compression can be divided into the three stages. At first, pressing the CCNTs, the force almost linearly rises to a point, where the force starts to damp down. Fig. S11shows the molecular structures of the CCNT coils. The movement of coils -relative to each other- occurs due to vdW interactions, which lead to the reduction in force.

Fig. 9. Profile of force-displacement and stress-strain for CCNTs under compression. Also, in order to be more precise, a comparison of force curves based on AIREBO and REBO potentials is reported in this stage. Therefore, in the absence of vdW forces, the first stage was eliminated by using REBO potential as illustrated in Fig. 9d. In stage 2, depending on the structural characteristics of CCNTs, this behavior may occur again that determines by repeating a sharp drop in the force. So far, new bonds were neither created nor broken. In the final stage, the movement of CCNTs coils has not been observed, and the compression process takes place in a steady state, which leads to a rise in force. At the higher strain of this stage, bonds may be formed or broken. Reviewing each CCNT, more details are revealed. Firstly, comparing the growth of the inner radius for (2,1,2,1) and (6,1,2,1) CCNTs is shown in Fig. 9a and b. Spring constants for the samples are 31.6 and 72 23 ACS Paragon Plus Environment

!" !#

, respectively. For the first


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sample, after stage 1, rises and falls are observed in the force. This shows that the coils move relative to each other in stage 2. In higher strains, the force increases nonlinearly. In another sample, a significant growth in the force is observed. Then, a drop in the force occurs, which represents the end of the first stage. Since the inner radius is large, no other movement of the layers occurs. Consequently, stage 2 was eliminated. Thus, at a higher inner radius of CCNT, a significant growth in the force is observed. Therefore, the first stage is gone up to a higher strain as depicted in Fig. 9. Also, by comparing (2,1,2,1) and (3,2,3,2) CCNT, it was observed at the first and second stages, that the growth of cross-sectional radius of CCNT increases the force in a wider range of strain. This is related to the movement of CCNTs coils. Considering the domain of vdW interaction, (3,2,3,2) CCNT endures a larger force. As a result, a severe drop in the force is observed, when the coils move with respect to each other. Therefore, this affects only the stages, where the movement of the coils occurs (stages 1 and 2). Interestingly, the behavior of (2,1,7,1) double-walled GH in the compressive test is different from the other samples (see Fig. 9e). Initially, the force increases with a slight slope. Dislocation of layers is not possible due to a large semi-major of the double-walled GH. So, stage 2 is eliminated. In the final stage, without changing the number of bonds, the compressive force increases sharply. This shows the considerable effect of increasing the semi-major. Also, checking the compressive behavior of (2,0,4,0) GH, after a narrow region for the first stage, catastrophic movements of layers are observed. This is due to the inability of GH to withstand the compressive stress in higher strains. The compression process continues as illustrated in Fig. 9.From the difference between (2,1,7,1) double-walled GH and (2,0,4,0) GH, it is concluded that the attachment of two adjacent layers (conversion of GH to double-walled GH) significantly affects the compressive strength as shown in Fig. S12. Also, the spring constants of GH and double-walled GH are computed 35.7 and 32.3

!" !#

, respectively.

3.4. The effect of geometry on toughness The gravimetric toughness is the ability to absorb the energy per unit mass without fracturing. It is indicated by the area under the force-displacement diagram determined as ET =

∫ Fxdx m

(4)

where m, F and x stand for the total mass of the sample, applied force, and displacement, respectively. The range of toughness is from 8539.26 to 620.37 J/g. The maximum value occurs 24 ACS Paragon Plus Environment

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for (2,0,4,0) GH and the minimum value is for the double-walled (2,1,11,1) GH as given in Table 3. Notably, results show the range of toughness variations in the GH and double-walled GH are high. The toughness in the compression until the formation or breaking of bonds is calculated. Having a smaller range of strain in the compression, the toughness in the compression is shown to be lower than that of the tensile test. The wide range of toughness may have a positive impact on the application of these materials. Table 3. The gravimetric toughness of CCNTs from the tensileand compressive tests. Also, the toughness of CCNT from compression test is calculated by area under force-displacement until the formation or breaking of bonds. Structural type (2,0,4,0) 1-(2,1,2,1) 2-(2,1,2,1) (2,1,4,1) (2,1,7,1) (2,1,11,1) (4,2,2,2) 1-(3,2,3,2) 2-(3,2,3,2) (3,2,5,2) (3,2,7,2) (4,1,2,1) (4,1,7,1) (4,3,4,3) (4,4,1,4) (5,3,4,3) (6,1,2,1) (6,3,4,3) (7,4,1,4) (8,4,6,4)

Tensile (J/g) 8539.2674 4841.7089 3974.5082 2296.3465 2123.5565 620.37228 3262.6969 3339.4917 2352.1568 2590.059 2712.1998 3236.1964 2870.6671 1169.7956 2623.458 1508.0656 2287.3416 948.48783 923.96503 1099.5403

Compression (J/g) 658.5308496 344.7719726 320.115 164.8560599 114.2344661 -

4. Summary and Conclusions In this study, the mechanical properties of the different types of CCNTs were investigated using the atomistic simulations. CCNTs were classified into different groups according to the geometric characteristics. Some of the prominent types of CCNTs included double-walled CCNT, double25 ACS Paragon Plus Environment


walled GH, helical arrays and circular-shaped CCNT. Their properties were studied during tensile and compression tests. All samples are structurally stable. However, samples with higher pitch angle (over 3.4 Å) required to randomly break some bonds in order to remain still structurally stable. Subsequently, they possessed more energy than the other samples. During the tensile test for the first group of samples, increasing in semi-major of samples led to a rise in vdW interactions. Thus, higher stress was required to be applied to CCNT; consequently, the breaking of bonds happened earlier. In the second group, increasing the inner radius are studied. Results showed at the higher inner radius, stretchability of samples was larger. Therefore, the general trend of behavior was close to those of GHs. In third group, the effects of increasing simultaneously the inner radius and semi-major led to rising stress tolerance alongside more stretchability. In other groups, increasing in semi-minor and pitch angle give the results respectively reduced the stretch in the higher stress and destruction of the initial area of the vdW interactions. During the compression test, for all samples, after the initial interaction caused by vdW interactions, the movement of CCNTs coils happened. This movement can be large or very small depending on the geometry of CCNTs. Then, a significant increase in the force was observed, albeit its amount depended to the geometry characters as discussed. Finally, the influence of the mentioned factors on the toughness was investigated. It was found that the geometric parameters can have an astonishing effect on the mechanical properties of the CCNTs. Therefore, geometrically-controlled CCNTs can be employed in optimization and enhancement of performance in a wide range of applications.

Supporting Information The Supporting Information is including: results, discussions and additional figures for tensile and compressive deformation stages of different kinds of CCNTs; potential energy versus time for 1(2,1,2,1) and 2-(2,1,2,1) CCNTs; profile of forces-displacement and stress-strain curves at low strain; distributions of C-C bond configuration for CCNTs; comparative analysis between using REBO and AIREBO potentials for CCNTs; and the guides for the videos of simulations of CCNTs under tension.


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Insight of Geometry-Controlled Mechanical Properties of Spiral

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