Template-directed growth and mechanical properties of carbon nanotube–graphene junctions with nano-fillets: molecular dynamic simulation

Abstract

The template-directed growth process of 3D carbon nanotube–graphene junctions with nano-fillets was simulated with classical molecular dynamic (MD) simulation. The carbon nanotube, graphene and their seamlessly C–C bonded junctions were formed simultaneously on amorphous alumina templates without catalysts. The C–C bonded junctions with various fillet angles were made, and the molecular structures and tensile strength of the “as-grown” C–C junctions were determined by the MD method. Among these junctions, the 135° fillet shows the most stability and improved mechanical strength, which is consistent with the experimental observations. While the fillet is a common technique that is widely used in large-scale engineering structures to reduce the stress concentration, here the nano-fillet enhances both the stability and mechanical properties of carbon nanotube–graphene junctions. This work provides a theoretical base for synthesizing high-quality carbon nanotube–graphene nanostructures via template methods.

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1 Introduction

Carbon nanotubes (CNT) and graphene (G) are two well-known one-dimensional (1D) and two-dimensional (2D) carbon allotropes, respectively, due to their unique structures. A CNT is a cylindrical carbon network with a high aspect ratio, while graphene is a one single layer carbon network in a plane direction. CNTs and graphene possess excellent mechanical, electric and thermal properties along their axis or in-plane direction, but low out-of-plane properties. The combination of CNTs and graphene into three-dimensional (3D) CNT–G nanostructures could significantly improve their properties in all directions, and even lead to novel properties because of their unique integrated nanostructures. Theoretical studies on 3D CNT pillar–G plane nanoarchitectures have revealed outstanding mechanical, thermal and electrical properties of these 3D materials, which are potential in various applications such as aerospace, integrated circuit, energy generation and storage.^1–5

Some experimental strategies have been developed to synthesize the 3D nanostructures. Zhao et al.⁶ produced covalently C–C bonded junction through one-step catalytic growth. Du et al.⁷ used the method of intercalated growth of vertical aligned CNTs into thermally expanded highly ordered pyrolytic graphite. Yan et al.⁸ obtained hybrid materials with the aid of Al₂O₃ porous film. However, most 3D nanostructures were fabricated with aid of metal nanoparticle catalysts, in which the metal particles remained in the junction structures even after subsequent treatments. The CNT–G junctions are one of the most important components in the 3D nanostructure; the metal catalysts embedded in the junctions could significantly reduce the mechanical properties of the 3D CNT–G nanostructures.⁹ To avoid the problem of catalyst remaining in the 3D CNT–G nanostructures, it is highly desirable to synthesize the nanostructures without catalysts. It has been shown that graphene and CNT can be grown on Al₂O₃ substrates. In our previous work, the seamlessly C–C bonded CNT–G junctions were synthesized by template methods.¹⁰ Specifically, 3D graphene–CNT hollow fibers with aligned CNTs seamlessly sheathed by a graphene layer were synthesized through a one-step chemical vapor deposition using an anodized aluminum wire template. The length and diameter of the CNTs could be tuned by controlling the anodization time. Interestingly, TEM images of the cross-section view of the 3D CNT–G nanostructures counterintuitively show a connecting angle about 135° at the CNT–G interface. These nanostructures, with a controllable surface area, meso-/micropores, and superior electrical properties, are excellent electrode materials for all-solid-state supercapacitors, exhibiting high surface-specific capacitance and electrochemical properties.

Although the seamlessly C–C bonded CNT–G junctions have been synthesized, it remains unclear why the junctions take a fillet angle of 135° instead of 90°, and it is challenge to reveal molecular structures and mechanical properties of the junctions because of their extremely small size. In this paper, the synthesizing process of single CNT–G junctions with fillets was simulated via molecular dynamics (MD) simulations. The stability of the junctions with different fillet angles was analysed by the MD simulations. The mechanical properties of the seamlessly C–C bonded junctions were predicted by using the same simulation methods.

2 Computational methodology

2.1 Template-directed growth of 3D CNT–G junctions

In the process of template-grown CNT–G junctions, carbon sources were progressively deposited on templates with nano-fillets, simultaneously forming graphene, CNT and their junctions with nano-fillets. To understand the CNT–G junction growth mechanism and molecular structures, a MD method was employed to simulate the growing process of the 3D nanostructures. The template was made of an amorphous Al₂O₃, which is similar to that used in the experiment.¹⁰ To generate the amorphous alumina templates, an alumina crystal structures was heated at a temperature of 5000 K for 350 ps and then quenched to room temperature. The amorphous alumina density is controlled to 3.28 g cm⁻³ according to literatures.¹¹ The alumina was trimmed to form a tunnel with a fillet. Periodic boundary conditions were applied in x and y (graphene plane) directions to represent an infinite large system.

The MD algorithm (LAMMPS) was used to simulate the chemical vapour deposition of carbon atoms on the template by progressively releasing carbon atoms near the template surface (Fig. 1a and c). To reduce the complexity and computational time, several layers of carbon atoms were pre-distributed as carbon sources over the template at a distance of >0.5 nm, and the spacing of the carbon atoms in the layers is 0.3 nm, at which no chemical bonds are formed between the carbon atoms before they are released. These pre-generated carbon atoms were kept away from the template surface such that the releasing carbon atoms have space to freely fly to the surface and interact with alumina template. To feed the growth of the junction on a template, the constraints on the carbon atoms were removed progressively from inner to outer layers. These released carbon atoms were then heated to the processing temperature (1037 K) by assigning the initial velocity according to the gauss distribution. The heated carbon gas continuously diffused to the surface of the template. With progressively releasing the carbon atoms, a seamless junction was gradually formed on alumina template. The growth speed of the junctions can be controlled by adjusting the carbon atom releasing speed. Periodic boundary conditions were applied to the system in x and y (graphene plane) directions. Since the junctions grow from carbon gas at the temperature, the process and resulting structures should be close to realistic chemical vapour deposition (CVD).

Fig. 1 Schematics of (a) Al₂O₃ template with a filleted hole in the center, (b) CNT–G junctions, (c) deposition of carbon on template, and (d) measurement of fracture strength of as-grown CNT–G junctions.

The interactions in Al₂O₃ templates were calculated using the transferable potential of Matsui,¹² while the forces of the C atoms were computed using the second-generation reactive empirical bond-order potential (AIREBO).^13–16 These many-body potentials have been used to study the growth of carbon materials.¹⁷ The interactions between C and Al(O) atoms were described by a Lannerd-Jones potential: E = 4ε[(σ/r)¹² − (σ/r)⁶], where r is the distance between atoms, and the parameters ε and σ are:^18,19 ε_Al–C = 0.0315 eV, ε_O–C = 0.00326 eV, σ_Al–C = 2.976 Å and σ_O–C = 3.19 Å, where subscripts Al–C and O–C represent the interactions between aluminium and carbon, and oxygen and carbon, respectively. With a rescaling thermostat to control temperature (1073 K), the equations of motion were integrated with a time step of 0.25 fs. The grown junction was then exposed to 3000 K for 2.5 μs.

2.2 Mechanical properties of the CNT–G junctions

The mechanical properties of the as-grown CNT–G junctions were determined by the same MD method used in growth of the 3D junctions. The interactions between atoms were calculated using an AIREBO potential, with a modified cutoff scheme. With a rescale thermostat to control temperature, the equations of motion were integrated with a time step of 0.25 fs. The edges of graphene layer were held fixed in the z-direction (pullout direction) (Fig. 1b and d). Simulation was performed by holding the several carbon rings of the CNT end as a rigid body, moving along CNT axial (z) direction at a constant speed of ∼4 m s⁻¹, achieving near equilibrium. The stress of the junctions was defined as the pulling forces divided by the cross-section area of the CNT. The maximum stress of the stress–strain curves is the tensile strength of the junctions.

3 Results and discussions

3.1 Formation of 3D CNT–G junctions

The growth processes of graphene, CNT and their junctions on the alumina template were simulated. In the initial stage, carbon nucleates and grows on the template surface to form continuous layers full of defects and nanopores. Gradually, these defects and nanopores are sealed during annealing, resulting in relatively high-quality graphene, CNT and their junctions. Generally, both single and multi-layer nanostructures can be formed on the surface of the template, with the simultaneous formation of CNT, graphene and a seamlessly C–C bonded CNT–G junction. The number of layers of the junctions depends on the amount of carbon provided in the growth process, while fillets with different angles can be formed by changing the template configuration. In this work, double-layered CNT–G junctions with fillet angles of 90°, 120°, 135, and 150° were made (Fig. 2). The junction with angle 180° has the same configuration as that with angle 90°. The structures of the junctions basically show graphene-layered structures parallel to the template surfaces with sp² dominated in the carbon nanostructures, but sp³ bonds are found between the layers, making the layers rough at the surface (Fig. 2). The carbon layers in the cross section (Fig. 2e) are disordered, which is similar to those observed in the experiment.¹⁰ After long heat treatment at high temperature, the disordered graphene layers can become more ordered structures.

Fig. 2 Side view of atomistic models of 3D CNT–G filleted junctions. (a) A 120° filleted junction, (b) a 135° filleted junction, (c) a 150° filleted junction, (d) a 180° (or 90°) filleted junction, and (e) cross-section of the 135° filleted junction.

It is reported that sp³ bonds have strong effects on the mechanical properties of multiwall carbon nanotubes, including load transfer, buckling strength, and energy dissipation.²⁰ The fraction of atoms with sp³ bonds was calculated and plotted in Fig. 3 for a 135° filleted junction. The fraction is defined as the ratio of the number of atoms with sp³ bonds (four neighboring atoms bonded to this atom) to the total number of atoms in the junction. The number of sp³ bonds evenly distributes between the walls of CNT, fillet and graphene, suggesting that the junction is composed of multi-layer graphene with a uniform defect distribution.

Fig. 3 Fraction of atoms with sp³ bonds versus position in z direction for 135° filleted junction.

3.2 Analysis of stability of filleted CNT–G junctions

Various junction structures with different fillet angles can be generated using the template-directed growth methods. However, only one structure (135° fillet) was observed in the experiment.¹⁰ To explain the experimental results, we calculated the potentials of the as-growth junctions with different fillet angles at near zero temperature (T = 0.05 K). The potentials per atom as a function of fillet angles are shown in Fig. 4. The potential achieves its lowest value at 135°, indicating the junction with 135° fillet is the most stable structures. This phenomenon can be attributed to the fillet that results in the better relaxation of structures, and thus reduction of stress concentration in the junction. This result is consistent with the experimental observation. Thus, the 135°-fillet is naturally formed because of the minimum energy required to form the junctions. While the fillet is a common technique that is widely used in large-scale engineering structures to reduce the stress concentration, the nano-fillet and its function have not been understood yet. Here we show that the observed nano-fillet is the most stable nanostructures, which can be applied to the design of nanostructures to enhance the reliability of nano-devices.

Fig. 4 Potential energy calculated by MD, and strain energy predicted by the theory (n = 2, L = d) as a function of the fillet angles.

To further understand the stability of the junctions, we analyzed the energy change of a prefect single-wall junction due to curvature of graphene in the junction area. As schematically shown in Fig. 5, a CNT–G junction has two fillet angles, α between the graphene and fillet plane, and θ between CNT and the fillet plane. These two angles are related by

α + θ = 3π/2 (1)

where the angles can vary in the range of π/2 to π. It is reasonable to assume that the radius of the curvatures at points A and B are related to the fillet angles byWhere r₀ and n are unknown parameters. It is known that, relative to the ground state of flat graphene, the deformation of a curved graphene involves both the bending curvature κ (or radius r) and in-plane strain ε.²⁰ As the tube radius increases, the bending energy (W_b = Dκ²/2 = D/(2r²)) decreases and the inplane strain energy (W_s = Cε²/2) increases, where D and C are the elastic moduli for bending and in-plane stretch, respectively.²⁴ For a fully relaxed curved graphene and under pure bending conditions, the total strain energy of the filleted junction with respect to the bending curvature, can be approximately expressed as

W_t = (W₁A₁ + W₂A₂)/(A₁ + A₂) (4)

where W₁ and W₂ are the bending energy, and A₁ and A₂ are the area of curvatures at points A and B, and are approximated as

A₁ = πdr₀ (5)

where d is the diameter of CNT and L is the length of fillet. Substituting the bending energy expression (W_b = D/2r²), eqn (5) and (6) into (4) yields

Fig. 5 Schematic of a single-wall CNT–G junction.

Therefore, the total energy is a function of the angle θ, and there may be a critical angle with minimum energy for the junctions. According to eqn (7), the energy is also related to a ratio of fillet length to CNT diameter, L/d. Since L usually increases with increasing d, the CNT diameter has less effect on the energy. The energy is independent of the CNT diameter if L is equal to d. Although eqn (7) is derived for single-wall junctions, it could be applied to multi-wall junctions since the curvature and fillet length of the layers are almost the same in multi-layer junction.

We have calculated the strain energy due to the bending curvature within the junctions. Fig. 4 shows the strain energy normalized by (D/2r₀²) as a function of the angle θ. With increasing the angle from 90° to 180° (the possible angles for the junctions), the energy initially reduces and then increases. There is a critical angle at which the energy reaches a minimum value. From energetic point of view, the junction with this critical angle will be most stable. Since n in eqn (2) and (3) is unknown, we have adjusted the value of n from 1–5, and calculated the critical angles which is in the range of 120° to 130°. These predicted values are very close to those angles (135°) observed in the experiment, and predicted by MD simulation. The discrepancy between the analytical model and experiment may be attributed to the number of factors such as residual stresses, defects and multi-layer walls. Anyway, the simple analytical model captures the major feature of the junctions with fillets.

3.3 Mechanical properties of the filleted junctions

Deformation and fracture of junctions with different fillet angles were simulated by pulling the CNT while fixing the edge of graphene. Fig. 6 shows the stress–strain curves for filleted CNT–G junctions. Overall, the relationship between stress and strain is almost linear before fracture occurs for all the junctions, indicating a brittle behavior. The fracture of most junctions takes place in CNTs near the junctions, where local defects exits, resulting in stress concentrations. The strength of the junctions is relatively higher than that of single layer junctions or the junctions embedded with catalyst nanoparticles.⁹ A plausible explanation for the high strength is the effect of interlayer sp³ bonds existing between the layers in the nanostructures. Experimental work has shown that modification of the interwall coupling in multi-wall CNTs by introducing sp³ bonds between walls can enhance the load transfer and thus the strength.^20–23 sp³ bonds existing between the layers in the nanostructures could lead to the improved mechanical properties due to the better load transfer between the layers.

Fig. 6 Stress–strain curves for CNT–G junctions with different fillet angles (the junctions with fillet angles 90° and 180° are identical).

Although all the junctions with 90°, 120°, 135°, and 150° fillets have similar mechanical behaviors, the fracture strength is different. The junction with a 135° fillet has the highest fracture strength compared with other fillet angles, as shown in Fig. 7, indicating that the fillet angles do affect the mechanical behaviors of the junctions. We have calculated the Young's modulus of the junctions based on the stress–strain curves shown in Fig. 6, and the results are plotted in Fig. 7. The Young's modulus has the same trend with the fracture strength. The higher Young's modulus of the junctions with 135° fillet angle can be attributed to its unique structure that has smallest portion of graphene compared with other type of junctions with the same size under the same boundary conditions. When the junctions are loaded along the CNT axis (z-direction), the graphene is subject to bending. Those junctions with more graphene portion (e.g., 90° fillet angle) are more complaint than that with 135° fillet angle. Consequently, the junction with 135° fillet angle shows the highest Young's modulus. Interestingly, these results are consistent with the potential energy calculations that the 135° filleted junction has the lowest formation potential energy and the most stable structure. These results suggest that engineering junction structures could increase both strength and stability of the 3D CNT–G nanostructures. It is expected that such nano-fillets would also enhance the thermal and electrical properties of the nanostructures, and are worth exploring in the future.

Fig. 7 Young's modulus and tensile strength for various fillet junctions. The junctions with fillet angles 90° and 180° are identical.

4 Conclusions

Growth processes of 3D CNT–G junctions on alumina templates were simulated via classical MD simulations. Multi-layer CNT–G junctions with fillets were formed on the alumina templates. Under real growth conditions, there are sp³ bonds between the layers, which make the nanostructure rough and disordered. The model predicts that there is a critical fillet angle (135°), at which the system becomes the most stable. An analytical solution based on bending energy of graphene also supports the conclusion. The predictions are consistent with the experimental observations. Such nano-fillet (135°) in 3D CNT–G junctions also leads to the highest strength. Therefore, the nano-fillet could be applied to form high-quality 3D CNT–G nanostructures with improved mechanical, thermal and electrical properties.

Acknowledgements

We acknowledge support by the Air Force Office of Scientific Research (AFOSR) MURI program (FA9550-12-1-0037) and National Science Foundation (NSF) CMMI program (CMMI-1212259, CMMI-1266319).

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