Molecular Dynamics Simulation and Continuum Shell Model for Buckling Analysis of Carbon Nanotubes

Abstract

Carbon nanotubes (CNTs) have potential applications in various fields of science and engineering due to their extremely high elasticity, strength, and thermal and electrical conductivity. Owing to their hollow and slender nature, these tubes are susceptible to buckling under a compressive axial load. As CNTs can undergo large, reversible post-buckling deformation, one may utilize this post-buckling response of CNT to manufacture mechanical energy storage devices at the nano-scale, or use it as a nano-knife or nano-pump. It is therefore important to understand the buckling behavior of CNTs under a compressive axial load. Experimental investigations on CNT buckling are very expensive and difficult to perform. As such, researchers often rely on molecular dynamics (MD) simulations, or continuum mechanics modeling to study their mechanical behaviors. In order to develop a good continuum mechanics model for buckling analysis of CNTs, one needs to possess adequate experimental or MD simulation data for its calibration. For “short” CNTs with small aspect ratios (≤10), researchers have reported different critical buckling loads/strains for the same CNTs based on MD simulations. Moreover, existing MD simulation data are not sufficiently comprehensive to allow rigorous benchmarking of continuum-based models. This chapter presents extensive sets of MD critical buckling loads/strains for armchair single-walled CNT (SWCNTs) and double-walled CNTs (DWCNTs), with various aspect ratios less than 10. These results serve to address the discrepancies found in the existing MD simulations, as well as to offer a comprehensive database for the critical buckling loads/strains for various armchair SWCNTs and DWCNTs. The Adaptive Intermolecular Reactive Bond Order (AIREBO) potential was adopted for MD simulations. Based on the MD results, the Young’s modulus, Poisson’s ratio and thickness for an equivalent continuum cylindrical shell model of CNTs are calibrated. The equivalent continuum shell model may be used to calculate the buckling loads of CNTs, in-lieu of MD simulations.

Appendix: AIREBO potential

Adaptive Intermolecular Reactive Bond Order Potential (AIREBO) is one of the various interatomic potential available in LAMMPS. AIREBO potential developed by Stuart et al. (2000) is used in the present MD simulations. AIREBO potential comprises the second generation REBO potential $U^{2nd - REBO}$, the torsion potential $U^{Tors}$ and the Lennard-Jones potential $U^{LJ}$.

1.1 Second generation REBO potential

The second generation REBO potential $U^{2nd - REBO}$ accounts for covalent bond interaction and its function form is given by

$$ U^{2nd - REBO} = \sum\limits_{i} {\sum\limits_{j( > i)} {\left[ {V^{R} (r_{ij} ) - b_{ij} V^{A} (r_{ij} )} \right]} } $$

(A1)

where, $V^{R}$ is the repulsive term, $V^{A}$ is the attractive term and $b_{ij}$ is bond order term. The repulsive term is given by

$$ V^{R} (r_{ij} ) = w_{ij} (r_{ij} )A_{ij} (1 + Q_{ij} /r_{ij} )e^{{ - \alpha_{ij} r_{ij} }} $$

(A2)

The function form of the attractive term is given by

$$ V^{A} (r_{ij} ) = w_{ij} (r_{ij} )\left[ {B_{1} e^{{ - \beta_{1} r_{ij} }} + B_{2} e^{{ - \beta_{2} r_{ij} }} + B_{3} e^{{ - \beta_{3} r_{ij} }} } \right] $$

(A3)

In Eqs. (A1)–(A3), $w_{ij} \left( {r_{ij} } \right)$ is a switching function which automatically dictates whether covalent interaction needs to be considered between any two atoms. The switching function $w_{ij} \left( {r_{ij} } \right)$ is given by

$$ w_{ij} (r_{ij} ) \, = \, \Uptheta ( - t) + \Uptheta (t)\Uptheta (1 - t)\left[ {1 + \cos \pi t} \right]/2, \, t \, = \, \frac{{r_{ij} - r_{ij}^{\hbox{min} } }}{{r_{ij}^{\hbox{max} } - r_{ij}^{\hbox{min} } }} $$

(A4)

where $\Uptheta (t)$is the Heaviside step function.

Values of the parameters for carbon–carbon (C–C) bond in Eqs. (A1)–(A4) are summarized in Table A.1.

Table A.1 Parameters in AIREBO potential for C–C bond

In the 2^nd generation REBO potential, the influence of neighboring atoms on the considered atom is incorporated by using bond order function $b_{ij}$ in Eq. (A1). The term $b_{ij}$ adjusts attraction force between atoms based on the position of other neighboring atoms and thus considers multi-body interactions.

The bond order term $b_{ij}$ for a bond i-j, depends on the neighborhood bond angles $\theta$ and dihedral angles $\varphi$. Function form of the bond order term is given by

$$ b_{ij} \approx \frac{1}{2}\left[ {p_{ij}^{\sigma \pi } + p_{ij}^{\pi \sigma } } \right] + \Uppi_{ij}^{DH} + \Uppi_{ij}^{RC} $$

(A5)

where

$$ p_{ij}^{\sigma \pi } = \left[ {1 + \sum\limits_{\alpha \ne i,j} {w_{i\alpha } (r_{i\alpha } )g_{C} (\cos (\theta {}_{ji\alpha })) + P_{CC} (N_{i}^{C} ,N_{i}^{H} )} } \right]^{ - 1/2} $$

(A6)

and $p_{ji}^{\sigma \pi }$ is similar to Eq. (A6) except the indices i and j are interchanged. The pair-angle cross interaction is incorporated by function $g_{c} \left( {\cos \left( {\theta_{ji\alpha } } \right)} \right)$ which depends on the angle between bond i-j and neighboring atom $\alpha$. In a pristine CNT, $g_{c} \left( {\cos \left( {\theta_{ji\alpha } } \right)} \right)$ is a function of two angles$\theta_{ji\alpha }$, where $\alpha = k$, m (see Fig. A.1a).

Fig. A.1

(a) Typical C–C bond in a CNT and its neighborhood (b) Definition of dihedral angle

The function $g_{c} \left( {\cos \left( {\theta_{ji\alpha } } \right)} \right)$ in Eq. (A6) is fitted to a quintic spline. The coefficients of the quintic spline function are evaluated by LAMMPS using the values given in Table A.2.

Table A.2 Interpolation points for quintic spline $y(x) = \sum\limits_{m = 0}^{5} {C_{m} (x - x_{k} )^{m} }$ Stuart et al. (2000)

In each interval of cos($\theta$), six values of interpolation points are reported in Table A.2. These six values are used to derive the coefficients of the quintic spline. For pristine CNT $P_{\text{CC}} (N_{i}^{C} ,N_{i}^{H} )$ in Eq. (A6) is −0.027603.

The function $\Uppi_{ij}^{DH}$ in Eq. (A5) represents the cross interaction between pair i-j and dihedral angle $\varphi_{\alpha ij\beta }$. In a pristine CNT, a bond pair i-j is affected by four dihedral angles. These four dihedral angles can be represented as $\varphi_{\alpha ij\beta }$ where $\alpha \, = k,\,m$ and $\beta = l,\,n$ (please refer to Fig. A.1a). Function $\Uppi_{ij}^{DH}$ is equal to the expression given by Eq. (A7).

$$ T_{ij} (N_{i}^{t} ,N_{j}^{t} ,N_{ij}^{conj} )\sum\limits_{\alpha \ne i,j} {\sum\limits_{\beta \ne i,j} {\left( {1 - \cos^{2} \left( \varphi \right)} \right)w_{i\alpha }^{{}} w_{j\beta }^{{}} \Uptheta (\sin (\theta_{ji\alpha } - s^{\hbox{min} } ))\Uptheta (\sin (\theta_{ij\beta } - s^{\hbox{min} } ))} } $$

(A7)

where $w_{\alpha \beta }$ is the switching function to be calculated as per Eq. (A4). In a pristine CNT, $T_{ij} (N_{ij}^{{}} ,N_{ji}^{{}} ,N_{ij}^{\text{conj}} )\,{ = - 0} . 0 0 4 0 4 8$. The radical term $\Uppi_{ij}^{\text{RC}}$in Equation (A5) does not contribute to force on an atom in CNT.

1.2 Torsion term

The torsion term $U^{Tors}$ in AIREBO potential is a valence term which adds stiffness to the dihedral rotation about a bond pair i-j as shown in Fig. A.2a.

Fig. A.2

(a) Dihedral bond rotation, (b) Typical neighborhood of an atom with id 1 in pristine carbon nanotube

The torsion potential $U^{Tors}$ is given by

$$ U_{\alpha ij\beta }^{tors} = \, w_{\alpha i} (r_{\alpha i} )w_{ij} (r_{ij} )w_{j\beta } (r_{j\beta } )V^{tors} (\omega_{\alpha ij\beta } ) $$

(A8)

$$ V^{tors} = \frac{256}{405}\varepsilon_{\alpha ij\beta } \cos^{10} (\omega_{\alpha ij\beta } /2) - \frac{1}{10}\varepsilon_{\alpha ij\beta } $$

(A9)

The value of $\varepsilon_{\alpha ij\beta }$ for CNT is 0.3079. The number of dihedral angles in CNT which contribute to the forces on an atom i depends on the neighborhood of that atom. There are 25 dihedral angles in a pristine CNT, which contribute to the forces on atom 1 shown in Fig. A.2b. The switching functions w(r) in Eq. (A8) is used by the MD code to determine all the dihedral angles contributing to force on an atom. Switching function w(r) in Eq. (A8) are calculated by using Eq. (A4).

1.3 Non-bonded potential

The non-bonded potential $U^{LJ}$ given by Eq. (A10), takes into account a pair type interaction between atoms which have interatomic distances ≥2 Å. In LAMMPS, one needs to specify a cutoff distance beyond which the non-bonded potential does not contribute to interatomic forces. In the case of C–C bond, this cutoff is taken as 10.2 Å but the user may set a longer cutoff distance at an expense of more computational time. The non-bonded potential is given by

$$ \begin{gathered} U_{ij}^{LJ} = \, S(t_{r} (r_{ij} ))S(t_{b} (b_{ij}^{*} ))C_{ij} V^{LJ} (r_{ij} ) + \left[ {1 - S(t_{r} (r_{ij} ))} \right]C_{ij} V^{LJ} (r_{ij} ) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,w_{ij} (r_{ij} ) \, = \, S^{'} (t_{c} (r_{ij} )), \, t_{c} (r_{ij} ) \, = \, \frac{{r_{ij} - r_{ij}^{LJ\hbox{min} } }}{{r_{ij}^{LJ\hbox{max} } - r_{ij}^{LJ\hbox{min} } }} \hfill \\ \end{gathered} $$

(A10)

where $\begin{gathered} S^{'} (t)\; = \, \Uptheta ( - t) + \Uptheta (t)\Uptheta (1 - t)\frac{1}{2}\left[ {1 + \cos \pi t} \right] \hfill \\ t_{b} (b_{ij}^{*} ) \, = \, \frac{{b_{ij}^{*} \, - \, b_{ij}^{\hbox{min} } }}{{b_{ij}^{\hbox{max} } - b_{ij}^{\hbox{min} } }} \hfill \\ \end{gathered}$

In Eq. (A10), $b_{ij}^{*}$ is a hypothetical bond-order term which is evaluated at $r_{ij}^{ \hbox{min} }$. To understand the previous statement, let us consider the non-bonded interaction between atom i and j. Since the distance between these two atoms typically exceeds the covalent bonding distance $r_{ij}^{\hbox{max} }$(for C–C bond $r_{ij}^{\hbox{max} } = 2.0$ Å), there is no actual bond-order term $b_{ij}$ (Eq. A5). Consequently a hypothetical bond order term $b_{ij}^{*}$ is evaluated by assuming the distance between atoms i and j to be 1.7 Å. In Eq. (A10), $V^{LJ}$ is a 12-6 Lennard-Jones potential which is given by Eq. (A11).

$$ V_{ij}^{LJ} = \, 4\varepsilon_{ij} \left[ {\left( {\frac{\sigma }{{r_{ij} }}} \right)^{12} - \left( {\frac{\sigma }{{r_{ij} }}} \right)^{6} } \right] $$

(A11)

The parameters required to evaluate the non-bonded interaction in CNT are $\sigma = { 3}. 4 {\text{\AA}}$, ${{\upvarepsilon}}_{\text{ij}} { = 0} . 0 0 2 8 4 {\text{ eV}}$, $r_{\text{ij}}^{\text{LJ max}} { = }\,{{\upsigma}}$, $r_{\text{ij}}^{\text{LJ min}} { = }\, 2^{{{1 \mathord{\left/ {\vphantom {1 {}}} \right. \kern-0pt} {}}6}} \,\sigma$, $b_{\text{ij}}^{ \hbox{max} } { = 0} . 8 1$ and $b_{\text{ij}}^{ \hbox{min} } { = 0} . 7 7$.