## Abstract

The binding energy between two parallel (and two crossing) single-walled (and multi-walled) carbon nanotubes (CNTs) is obtained by continuum modelling of the van der Waals interaction between them. The dependence of the binding energy on their diameters, number of walls and crossing angles is systematically analysed. The critical length for the mechanical stability and adhesion of the CNTs is determined by the function of *E*_{i}*I*_{i}, *h* and *γ*, where *E*_{i}*I*_{i}, *h* and *γ* are the CNTs bending stiffness, distance and binding energy between them, respectively. Checking against full atom molecular dynamics calculations show that the continuum solution has high accuracy. The established analytical solutions should be of great help for designing nanoelectromechanical devices.

## 1. Introduction

The unique mechanical, electrical, thermal and optical properties of carbon nanotubes (CNTs) enable them highly potential and ideal candidates for multifarious applications [1–3]. CNT exists in several structures forms such as single-walled CNTs (SWCNTs), multi-walled CNTs (MWCNTs), bundles and networks [4,5]. The mechanical properties of SWCNTs and MWCNTs have been extensively studied in previous work [6–9]. Recently, the CNT networks have been taken as a potential energy-saving material [10], while the CNT bundles have potential applications in nanocomposites materials. In the synthesis of CNT bundles and networks, their formation is a challenge for understanding how to measure and predict the properties of such large systems [4,11]. At the nanoscale, the weak van der Waals (vdW) interactions govern the structural organization and the mechanical properties of CNT bundles and networks [12–15]. Therefore, a clear understanding of the vdW interactions in these systems is crucial for their potential applications in nanoelectromechanical systems and electronic devices. The self-folding of SWCNTs, MWCNTs and multi-layer graphene sheets has been investigated and the bundle pattern formation has also been studied in previous work [14–16]. However, all their binding energies were from full atom molecular dynamics (MD) simulation or experimental results. Girifalco *et al.* [17] obtained the cohesive energy between two parallel and same radii SWCNTs using atomistic models. In recent years, a suspended SWCNT crossbar array for both I/O and switchable, bistable device elements with well-defined OFF and ON states has been exploited by Rueckes *et al*. [18]. The crossbar consists of a set of parallel SWCNTs on a substrate and a set of perpendicular SWCNTs that are suspended on a periodic array of supports (figure 1*b*). It was found that the mechanical stability of the structures is determined by the vdW interactions between the two SWCNTs [18]. Because the vdW interactions between the two SWCNTs dominate the working reliability of the bistable device, the key issue of the binding energy and mechanical stability of two parallel and crossing CNTs has to be solved in order to clearly understand the working mechanism of the bistable device.

In this paper, the binding energy between two parallel (and two crossing) SWCNTs (and MWCNTs) is obtained from a continuum model based on the Lennard-Jones (LJ) potential. The analytical expressions are validated by comparing with our full atom MD simulations. Since the critical length for the mechanical stability and adhesion of the two CNTs dominate the working reliability of a bistable device, it is significant to derive their analytical solutions.

## 2. Analytical model of binding energy and mechanical stability

Figure 1 shows the two parallel CNTs (*b*) and two crossing CNTs (*c*) under adherent conditions, in which the two CNT radii can be different. To determine the critical and stable lengths where the two CNTs do not make contact, an analytical model is presented in this paper and the corresponding geometry of the problem is plotted in figure 1. Some assumptions are proposed to simplify the problem: (i) the two CNTs are taken as two cantilever beams and the shear deformation are ignored. (ii) The closest distance between the adherent components of the two CNTs is taken as zero (or a constant *d* which does not influence the results). (iii) The radii of CNTs and the displacement between the two CNTs are both far less than the length *L*_{0}, that is to say, *L*_{0} is approximately equal to *s*+*l* under adherent conditions.

As shown in figure 1*b*, the total energy is composed of elastic energy and adhesion energy.
*U*_{CNT1}, *U*_{CNT2} and *γ* are elastic energy of CNT1 and CNT2 as well as binding energy per unit length, respectively.

Based on the present boundary condition, the total energy of equation (2.1) can be expressed as
*E*_{1} and *E*_{2} are Young's moduli of CNT1 and CNT2, and *I*_{1} and *I*_{2} are the moment of inertia of CNT1 and CNT2, respectively.

For an SWCNT, the bending stiffness of the beam is [19]
*r* and *t* (0.34 nm is chosen here) are the radius and the thickness of the SWCNT, respectively.

For an MWCNT, the bending stiffness of the beam is [20,21]
*EI*)_{inner} and *r*_{inner} are the bending stiffness and radius of the innermost wall, respectively, *m* is the number of walls, and *σ*=0.34 nm is the interwall spacing.

In view of the equilibrium of system, the total energy should be a minimum value. The critical value of *s* can be obtained by d*U*_{T}/d*s*=0, that is given
*h*.

If *E*_{1}=*E*_{2}=*E* and *I*_{1}=*I*_{2}=*I* (that is, CNT1=CNT2), then *h*_{1}=*h*_{2}=*h* and equation (2.5) can be written as
*γ*_{crossing} is the absolute minimum of the cohesive energy between two crossing CNTs at the equilibrium distance. If we assume the *s*_{1} and *s*_{2} of the two crossing CNTs are both the same, we can obtain the stability length from equations (2.2) and (2.7),
*h*.

If *E*_{1}=*E*_{2}=*E* and *I*_{1}=*I*_{2}=*I* (i.e. CNT1=CNT2), then *h*_{1}=*h*_{2}=*h* and equation (2.8) can be written as
*γ* and *γ*_{crossing} between two parallel CNTs and two crossing CNTs is a crucial issue in this work.

For two parallel SWCNTs, the cohesive energy per unit length has been obtained [22] as
*σ* are, respectively, the depth and the equilibrium distance of the 6–12 LJ potential between two carbon atoms (∈ =2.8437 mev and *σ*=3.4 Å are adopted from the literature) [6,7], and *ρ* is the area density CNTs, and *r*_{1} and *r*_{2} are the radii of the two CNTs, and *F*_{5} and *F*_{2} can be found in our previous work [22]. The binding energy *γ* per unit length is the absolute minimum of *ϕ*_{circle} at equilibrium distance between the two SWCNTs.

For two crossing SWCNTs, the total cohesive energy has been obtained [22] as
*β* is the crossing angle between the two centre axes of the two crossing CNTs [22]. The binding energy *γ*_{crossing} is the absolute minimum of *ϕ*_{total} at equilibrium distance between the two SWCNTs.

Figure 2*a* shows the binding energy distribution with CNT radius between two parallel SWCNTs and two crossing SWCNTs using equations (2.10) and (2.11) and full atom MD simulations, in which the MD simulation is performed using LAMMPS [23] with the AIREBO potential and periodic boundary conditions are applied along the centre axis of the CNTs (the LJ cut-off radius is 60 Å, which is enough distance to get accurate results). The analytical results are in good agreement with those from our full atom MD simulations. Figure 2*b* shows the analytical binding energy between two different parallel SWCNTs and two different crossing SWCNTs.

Figure 3 shows an SWCNT parallel to an MWCNT. We assume that the distance between any two neighbour CNTs in the MWCNT is 3.4 Å.

Based on equations (2.10) and (2.11), the cohesive energy between the *i*th CNT in the MWCNT and an SWCNT should be easily obtained as

The total cohesive energy from equation (2.12) should be given as
*a*_{0} is the same as that in equation (2.11).

Similarly, the total energy between two parallel MWCNTs can be obtained
*h*_{1} is the distance between the outmost CNT in one MWCNT and the outmost CNT in the other MWCNT.

Similarly, the total energy between two crossing MWCNTs can be obtained

Figure 4 shows the binding energy between two parallel MWCNTs and two crossing MWCNTs from our analytical model, in which the innermost CNT is the (5,5) CNT. The binding energy nonlinearly increases with increasing number of walls.

## 3. Molecular dynamics simulations

Figure 5 shows the critical length and stable length for two parallel SWCNTs and two crossing SWCNTs based on our analytical results and full atom MD simulations. To obtain the bending stiffness from the MD simulations, the initial atomic structure of a (5,5) CNT is optimized by the MD method, such that the total potential energy is minimized and forces between atoms are zero before the bending deformation. To apply bending deformation, rigid body translation is applied to the atoms in both end layers of the CNT, such that both end sections remain circular and are kept perpendicular to the deformed axis in each displacement increment; the length of the deformed tube axis remains unchanged and its curvature is essentially uniform throughout deformation. The displacement-controlled loading is widely used in literature to simulate the pure bending deformation of SWCNTs in MD [24]. The bending stiffness *EI*=3.95×10^{−26} J m of the (5,5) CNT is obtained by our MD results by LAMMPS software with AIREBO potential (figure 5*a*), which is close to the available value 3.84×10^{−26} J m from previous work [25]. *U*_{bending} is the bending energy per unit length and *κ* is the 1/*r*, in which *r* is the curvature radius and the (5,5) CNT length is equal to 11.6 nm in figure 5*a* [22] and the detailed MD process is the same as previous work [24]. To obtain the critical length of figure 5*b* for two parallel SWCNTs by full atom MD simulations, we keep the initial distance around 30 Å between two parallel SWCNTs, where vdW interactions are considerably weaker and can be ignored. After the energy minimization, the left ends of the two parallel SWCNTs (see the yellow and blue ends on the SWCNTs in the inset of figure 5*b*) are always fixed, while all right ends of the SWCNTs are free. The present simulation is at 0 K and the upper SWCNT gets gradually closer to the lower one with an increment of 0.1 Å per time step based on the deformation-control method. Afterwards, the optimized structure is taken as the initial geometry for the next calculations. The right ends of the two SWCNTs will stick together when the distance between the two SWCNTs is close enough. For a given close distance between two CNTs, the right ends of the two cantilever CNTs will stick together and the length of the stuck part increases with decreasing distance. The present analytical results are in good agreement with those from MD simulations for two parallel (5,5) CNTs in figure 5*b*.

Figure 6 shows the critical length and stable length for two parallel MWCNTs and two crossing MWCNTs based on our analytical results. For a given distance (the distance between the two outmost CNTs in the two MWCNTs (figure 1)), both the critical length and the stable length nonlinearly increase with increasing number of walls.

## 4. Conclusion

In summary, the analytical expressions (such as equations (2.13)–(2.15)) of the binding energy between two parallel (and two crossing) single-walled (and multi-walled) CNTs are obtained by continuum modelling of the vdW interactions between them. The dependence of the binding energy on their diameters, number of walls and crossing angles is systematically analysed. The critical length for the mechanical stability and adhesion (such as equations (2.5), (2.6), (2.8) and (2.9)) of the CNTs has been determined by the function of *E*_{i}*I*_{i}, *h* and *γ*, where *E*_{i}*I*_{i}, *h* and *γ* are the CNT bending stiffness, distance and binding energy between them, respectively. Checking against full atom MD calculations show that the continuum solution has high accuracy. The established analytical solutions should be of great help for designing nanoelectromechanical devices.

## Authors' contributions

J.Z. performed all the calculations, interpreted the results and wrote the manuscript. Y.J., N.W. and T.R. helped to interpret the results and edited the manuscript. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

Financial support was provided by the National Natural Science Foundation of China (grant no.11302084), the Fundamental Research Funds for the Central Universities (grant no. JUSRP11529), Open Fund of Key Laboratory for Intelligent Nano Materials and Devices of the Ministry of Education (NUAA) (grant no. INMD-2015M01) is kindly acknowledged.

## Acknowledgements

We gratefully acknowledge support from the ‘Thousand Youth Talents Plan’.

- Received April 8, 2015.
- Accepted June 30, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.