Transition states and minimum energy pathways for the collapse of carbon nanotubes

Carbon nanotubes (CNTs) can undergo collapse from their customary cylindrical configurations to ribbons. The energy minima corresponding to these two states are identified using either atomistic molecular mechanics or the theory of finite crystal elasticity with reduced dimensionality. The minimum energy path between these two minima is found using the nudged elastic band reaction-pathway sampling scheme. The energetics of CNT collapse is explored for both single- and multi-walled CNTs as well as small bundles. The process has a strong diameter dependence, with collapse being more favorable for the larger diameter tubes, but is nearly independent of chirality. The saddle point always lies close to the collapsed state, and the absolute barrier energies—even for fairly short tube lengths—are sufficiently high, even when the reaction is highly exothermic, that thermal activation cannot initiate collapse via this pathway. The hydrostatic pressure required to buckle and collapse CNTs of various diameters is discussed.

Figure 1

(Color online) Fully relaxed cylindrical and collapsed configurations of CNTs. (a) [40,0] SWCNT, cylindrical configuration. (b) [40,0] SWCNT, collapsed configuration. (c) $[30,30]∕[35,35]$ double-walled CNT, cylindrical configuration. (d) $[30,30]∕[35,35]$ double-walled CNT, collapsed configuration. (e) Two [20,20] CNTs, original configuration. (f) Two [20,20] CNTs, collapsed configuration. (g) Two [30,30] CNTs, original configuration. (h) Two [30,30] CNTs, collapsed configuration.

Figure 2

(Color online) (a) Energy difference $\Delta E=E_{\text{cylindrical}}-E_{\text{collapsed}}$ per unit area between the cylindrical and the collapsed configurations for SWCNTs vs tube diameter. (b) As in (a) except for double-walled CNTs of the form $[n,n]∕[n+5,n+5]$; the diameter refers to that of the inner tube. (c) As in (a) except for two-tube bundles, where the tubes are the same size, and the diameter values refer to that of a single tube.

Figure 3

(Color online) Energy difference $\Delta E=E_{\text{cylindrical}}-E_{\text{collapsed}}$ per unit area (of the inner shell) as a function of the number of walls. The shells are $[40+5n,40+5n]$ for $n=0,1,\dots ,8$.

Figure 4

(Color online) MEPs for the collapse of selected CNTs. The zeros of energy are relative to the cylindrical configurations and the energies are per unit length. The reaction coordinate is calculated as the Euclidean distance between a given configuration and the initial (cylindrical) configuration divided by the distance between the final (collapsed) and initial configuration. The symbols represent the images from the nudged elastic band calculation and the curves are cubic spline fits.

Figure 5

(a) Cross sections for five configurations along the MEP for the collapse of a [50,0] CNT; configurations include the initial (cylindrical) configuration, the saddle point, the collapsed configuration, a configuration midway between the initial configuration and the saddle point, and a configuration midway between the saddle point and the final configuration. (b) Same as (a) except for an [80,0] CNT.

Figure 6

(Color online) Activation energy barrier per unit length as a function of the tube diameter for the collapse of SWCNTs.