Modelling and MD Simulation Method for Chiral NTHs Bundles

In this part, we intensively simulated the chiral NTH bundles with various enantiomer configurations. The enantiomers of chiral NTHs may naturally exist in the synthesisation process. It is intriguing to learn how the existence of enantiomers in the chiral NTHs ropes impacts their torsional behaviour. Figure 3.2 shows the concept of chirality in chemistry. A chiral molecule is non-superposable on its mirror image. We built the mirror image of the sample of the individual chiral NTHs above as the enantiomer and then we conducted MD simulation to examine the torsional behaviour of all the possible 7-strand bundle configurations with the NTH and its enantiomer.

Figure 3.2: Amino Acid Chirality with hands (source from Wikimedia Commons)

3.2.1 Modelling of chiral NTH bundles

To systematically investigate the torsional mechanics of chiral NTHs bundles, we needed to enumerate all the possible structures that can be established from the NTH and its enantiomer. Firstly, we labelled all the 7 positions of the NTHs in the cross-sectional view of the bundle. Starting from the right top corner, we named the positions clockwise as A, B, C, E, G, F. The character D is used to represent the inner NTH in the bundle. Meanwhile, we used binary numbers 0 and 1 to represent the chirality of the NTH. 0 indicates that the NTH has the same structure as predicted in [2], while 1 represents the chiral NTH’s enantiomer which is the mirror image seen from a cross-sectional view. Figure 3.3 gives an example of labelling the bundle structure (0_0_1_1_1_0_0).

Figure 3.3: An example of labelling the bundle structure (0_0_1_1_1_0_0)

Two different NTHs can be placed in each position. Thus, if the 7 positions are fixed or distinct, then the total structures enumerated will be ʹ^଻ (=128). However, for the bundles studied, the 6 positions in the outer layer are geometrically equivalent, and we have to find and remove all the duplications from the 128 structures. As the state space in this case is relatively small, it is possible to use a brute-force enumeration algorithm to generate all the structures. However, when dealing with a large state space (or large filament number in the bundle), more efficient and generalised algorithms need to be adopted. The enumeration algorithm we used is shown in Appendix A. To simplify the process, only the 6 positions in the outer layer are considered in the code, and the chirality of the NTH at position D will be determined after the enumeration. Firstly, we treated the 6 positions as fixed, and started to generate the structures in a list of 0 or 1 using the Python code. At the same time, we defined an array enum_no_dup to store the structures without duplications, and at each run, the rotating lists were created from the new structure and the code checked whether any elements in the rotating lists already existed in the enum_no_dup. If not, the new structure was appended to enum_no_dup. The final step was to insert the chirality-labelling number of position D into the list in enum_no_dup.

Figure 3.4 shows all the total 28 enumerated bundle structures, including 24 symmetric bundles and 4 asymmetric bundle configurations. According to the topology, the torsion behaviour of each pair of bundles in each cell is supposed to be the same while

twisting in the opposite direction. Therefore, by twisting these 28 structures in one direction, we can also examine the loading direction dependency of the carbon fibre bundles.

Figure 3.4: The enumerated structures of 7-strand bundles.

Following the previous chapter, this study focuses on PT and stiff-chiral-3 NTHs. For comparison, we also examine the torsional mechanics of (8, 3) CNT bundles. Figure 3.5 demonstrates the cross-sectional view of the 0_0_0_1_0_0_0 bundle of the three nanofibres. The chirality of PT can be identified from the layout of the hydrogen backbones, and the chirality of (8, 3) CNT can be recognised through the configuration of the carbon atoms. However, the chirality of stiff-chiral-3 NTH is visually indistinguishable. The length of NTHs was chosen as 18 nm, while for comparison, the length of CNTs was selected as 24 nm to generate a similar aspect ratio with the NTHs.

Figure 3.5: the carbon nanofibre bundles with configuration 0_0_0_1_0_0_0. (a) PT bundle. (b) stiff-chiral-3 bundle. (c) (8, 3) bundle.

3.2.2 MD simulation method for twisting chiral NTH bundles

Twisting deformation will be conducted on all the bundles established above. The setup for MD simulation of the bundles is the same as in Section 3.1.2, except that for the simulation, only clockwise torsion is considered. The widely used adaptive intermolecular reactive empirical bond order (AIREBO) potential was utilised to describe the C-C and C-H atomic interaction in NTHs. The potential includes short-range interactions, long short-range van der Waals interactions, and dihedral terms. It is shown that the potential can accurately represent the binding energy and elastic properties of carbon materials. The cut-off distance of the AIREBO potential was chosen as 2.0 Հ. To describe changes of the •’^ଶ interlayer interactions between CNTs in the bundle, we combined the AIREBO potential with the Kolmogorov-Crespi (KC) interaction potential, which is intended for interlayer interactions between two different layers of graphene. The cut-off distance of the potential was set as large as 14 Հ to smooth the force and to include all the pairs to build the neighbour list for calculating the normal. The simulation terminated whenever bond breaking was detected in the samples. represent the Cartesian components. The atomic virial stress was estimated as

Ɏ_୧^஑ஒൌ ^ଵ

ன_౪

തതതതቄെ_୧˜_୧^஑˜_୧^ஒ൅^ଵ

ଶσ^_{୨ஷ୧ ୧୨}^஑”_୧୨^ஒቅ (3.2)

where ߱തതതത is the effective volume of atom i and ȳ ൌ  σ ߱_௧ തതതത . Considering the _௧ sophisticated stress of the atoms under deformation, the atomic Von Mises (VM) stress was also calculated based on the atomic virial stress.

In term of twist deformation, the torsional energy οܧ_௧ can be calculated usingοܧ_௧ ൌ ܩܫ_௣߮^ଶȀʹܫ_଴, where ܩ and ܫ_௣ represent the shear modulus and polar moment of inertia, and ߮ and ܫ_଴ are twist angle and initial sample length respectively. The twist rate is defined as the ratio between ߮ and ܫ_଴ െ ߮Ȁܫ_଴. The gravimetric deformation energy is calculated from ܬ ൌ  οܧ_௧Ȁ݉ ; here, ݉ is sample mass. The torsional rigidities of the samples are obtained by fitting the energy density curve based on the energy relationship with the twist angle (οܧ_௧ ൌ ܩܫ_௣߮^ଶȀʹܫ_଴). The region (߮Ȁܫ_଴ ൏ ͲǤʹ) of the energy versus strain curves will be used for curve fitting. We also defined the enantiomer ratio (ER) to represent the percentage of enantiomer in the bundles, which is the ratio of the number of enantiomers to the number of total strands in the bundle.