General Rolled-Up and Polyhedral Models for Carbon Nanotubes

Abstract

In many computational studies of carbon nanotubes, the minimum energy configuration frequently settles on a structure for which the bond lengths are distinct. Here, we extend both the rolled-up and the polyhedral models for SWCNTs to produce general models incorporating either distinct bond lengths and the same bond angle, or distinct bond lengths and distinct bond angles. The CNTs considered here are assumed to be formed by sp 2 hybridization but with different bond lengths so that the nanotube structure is assumed to comprise irregular hexagonal patterns. The polyhedral model with distinct bond lengths and distinct bond angles is based on the two postulates that all bonds lying on the same helix are equal in length and that all atoms are equidistant from a common axis of symmetry. The polyhedral model with distinct bond lengths and the same bond angle has the additional postulate that all the adjacent bond angles are equal. We derive exact formulae for the geometric parameters and we present asymptotic expansions for the polyhedral model with distinct bond lengths and distinct bond angles to the first two orders of magnitude. Good agreement is demonstrated for the predictions of the polyhedral model compared with the results obtained from other computational studies.

Geometry and Self-stress of Single-Wall Carbon Nanotubes and Graphene via a Discrete Model Based on a 2nd-Generation REBO Potential

“…The main findings of these studies can be summarized as follows: whatever the chirality, (a) 'relaxed' and 'unrelaxed' bond lengths and bond angles are different; (b) 'relaxed' CNTs have larger radius. Interestingly, in [31] geometrical relationships alternative to those of RUM are derived, in order to obtain the radius of a CNT once its bond lengths and bond angles are known.…”

Carbon-nanotube geometries: Analytical and numerical results

“…An extension of the Cox-Hill model to these situations has been considered in [14] where it is remarked that two bond angles can be expected to be equal in the armchair and in the zigzag geometry (while three different angles seem in general to be needed for modeling chiral nanotubes). This finds confirmation for a specific choice of the interaction energy in [17,18,23] where the contribution to the energy of a single carbon atom (plus the three nearest neighbors) is numerically investigated.…”

“…Eventually, if α ∈ (α ∨ (2π/3 − ε), 2π/3), then β z ∈ I ε : since the composition of v 3 (convex and strictly decreasing on I ε ) and β z (strictly concave) is strictly convex, it follows from (12) that E z is strictly convex in (α ∨ (2π/3 − ε), 2π/3). We use this strict convexity together with (13) and (14) to infer that…”

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule

“…Consequently, by a standard continuity argument we see that λ µ 1 and λ µ 2 increase continuously for µ ∈ M , possibly passing to a smaller (not relabeled) open interval M containing µ us . The proof of the fact that µ > µ us implies λ µ 1 , λ µ 2 > 1 is already contained in the proof of Proposition 4.3, see particularly (51) and (52). The fact that µ < µ us implies λ µ 1 , λ µ 2 < 1 can be proved along similar lines.…”

“…Secondly, we rigorously check that, also in presence of small stretching, the geometrical model obtained via local minimization corresponds neither to the classical rolled-up model [18,19,45], where two out of three bond angles at each atom are 2π/3, nor to the polyhedral model [14,15,52], where all bond angles are equal. The optimal configuration lies between these two (Proposition 3.4), a fact which remarkably corresponds to measurements on very thin carbon nanotubes [80].…”