BUCKLING BEHAVIOUR OF SINGLE-WALLED CARBON NANOTUBES UNDER AXIAL LOADING

We investigate the behaviour of a single-walled Carbon Nanotube under axial compressive line load applied at both edges. The expected buckling response is studied by application of a molecular computation model. We formulate a global potential and search for its minimum to obtain the equilibrium configuration. The critical nanotube diameter, when local shell buckling occurs, is measured with two parameters: the value of compression loading and tube diameter.


INTRODUCTION
Carbon nanotubes (CNTs), after their discovery in 1991 by Iijima [6], attracted a lot interest from researchers working mainly in the new field of nanotechnology.Among unusual properties of these new structures there are mechanical properties [3,8].For instance the Young's modulus of a carbon nanotube is expected to be several times that of the diamond (~ 1 TPa) reaching the extreme values compared to any other material.CNTs show various electrical properties as metallic conductors or semiconductors, depending on their physical properties [12].
In this note, motivated by potential applications of these interesting materials [11], we examine a critical value of axial loading which leads to local buckling instability for a single walled carbon nanotube.

MODEL ASSUMPTIONS
In the first step we define interactions between carbon atoms forming a tube.They can be defined by a simplified inter-atomic phenomenological potential V of Brenner-Teresoff type [1,10]: where i and j indicate the atomic sites, while r ij is the distance between i-th and j-th carbon atoms while B ̅ ij = (B ij +B ji )/2.The related nearest neighbours configuration and the applied axial forces are presented in Figure 1.The system parameters are included in Table 1.
Figures 2a and 2b show atomic bond configuration under loading for armchair carbon nanotubes of two different diameters.We show the deformation of a short piece of a carbon nanotube under uniaxial compression preserving the cylindrical symmetry.Such deformation is connected with small changes in nanotube geometry in both diameter and length.

STABILITY ANALYSIS
In the next step we analyse the atomic potential and stability of the deformed structures (Fig. 2).The repulsive and attractive parts of the potential: V R , V A are defined as: (2) (3) where:  Symbol ϕ ijk defines the angle between the lines, connecting i-th with j-th atom and i-th with k-th atom.R ij (e) is the distance between atoms at equilibrium.All the potential parameters appearing in Equations 1-7 are given in Table 1, and were adjusted from Ref. [1].
To determine the stability of the given equilibrium structure under the loading we analyze the Jacobian matrix of the total potential V TOT (r 1 , r 2 , ..., r N ) composed of the inter-atomic potential V (Eq. 1) and the work of the external axial forces (Fig. 1b) F=-F'.
To analyze the stability of atom configuration we used the condition that for a stable configuration of atoms eigenvalues of the Jacobian matrix calculated from V TOT with the following matrix elements: where ξ i , η j = x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ,..., x n , y n , z n (8) are Cartesian coordinates of all N atoms defined in the studied carbon nanotube.For a stable conall eigenvalues of the Jacobian matrix should be positive.In the present approximation we restricted the analysis to a single carbon atom displacement (Fig. 3a) where all surrounding atoms positions were fixed.The results of the critical force F c are shown in Figure 3b.Note that the critical force F c , above which the system is unstable, is growing with the size of the unloaded nanotube diameter D n .This represents a general trend expected to occur in any cylinder structure, which is usually more stable on buckling for larger D n .

CONCLUSIONS
We investigated stability of a single-walled (armchair) Carbon Nanotube under axial compressive line loading applied at both of its edges.By analyzing the Jacobian matrix eigenvalues we successfully estimated the critical force leading to local bifurcations.To obtain more appropriate results of F c of a local buckling instability, one should go beyond a single atom displacement.The results of such an investigation are in preparation and they will reported in a separate paper.

Fig. 3 .
Fig. 3. Schematic plot of a single atom displacement (a) and critical values of the axial force versus the unloaded nanotube diameter D n = 3n/π_ (for the armchair structure of the size n × n) where the unit value is distance of carbon nearest atoms R ij