*3.3. The Hydrogen Bonding*

The average number of hydrogen bonds per water molecule inside the nanotube for the nanotubes with different deformations is given in the figure below for both BNNT and CNT.

We know that the number of hydrogen bonds that occurs inside the nanotube is a function of the length and the pore size. From Figure 8, we can observe that there is no significant increase in the number of hydrogen bonds formed inside the pore for the nanotubes, which has a twist, as they have the same length and pore size of that of the perfect nanotube. We observed that the nanotubes with the XY-distortion of 2 show a significantly higher value of hydrogen bonding per water molecule inside the nanotube when compared to others. This is due to the increase in the pore volume of the elliptical crosses section of the nanotubes, which facilitates the accommodation of water molecules side by side inside the pore. Usually, a single-file water transport is observed in nanotubes in (6,6) nanotubes, but when they are distorted in XY-direction, it has sufficient space to accommodate two water molecules side by side, as shown clearly in the Figure 3.

**Figure 8.** Average number of hydrogen bonds per water molecule formation inside the pore for perfect and different types of deformed nanotubes: carbon nanotubes (CNT) and boron nitride nanotubes (BNNT).

For both the CNT and BNNT, we can see a significant increase in the hydrogen bond, which decreases the mobility and formation of the more-bonded system. When compared between the di ffusion and the number of hydrogen bond plots for the XY-distortion of 2, we can see that the di ffusion in the CNT is higher with a lower hydrogen bond when compared with the BNNT. This shows that the formation of a more-bonded system decreases the water di ffusion through the nanotubes. This is in good agreemen<sup>t</sup> with the results obtained by Mendonca et al. [28]. This facilitates the change of di ffusion from single file transport to near-Fickian di ffusion. The XY-distortion of 4 and 6 shows the accumulation of molecules near the entrance and exit of the pore. Due to the large energy barrier inside the pore and large interaction of the water molecules with the wall, the molecules can neither occupy the whole pore volume nor transport through them easily.

The Z-distortion of 1.2 of the nanotubes shows a noticeable decrease in the number of hydrogen bonds per water molecule when compared to other Z-distortions due to the stop-start di ffusion of molecules near the entrance of the nanotube, as shown in the Figure 9. Even though this phenomenon is observed in both the CNT and BNNT, this phenomenon is predominant in the CNT when compared with that of the BNNT. To further understand this discontinuous flow phenomenon, the friction force, radial distribution function (RDF), and potential of mean force are studied for this particular case in detail for the CNT.

**Figure 9.** Visualization of water chain formed in-between two reservoirs: (**a**) perfect CNT and (**b**) CNT with Z-distortion value 1.2.

## *3.4. Friction Force and Trajectory*

Discontinuous single-file di ffusion is found in the CNT with the Z-distortion of 1.2, as shown in Figure 9. To investigate this phenomenon, the friction force is calculated. The friction force plays a significant role in water transport through the nanotubes. The friction force is large when there is a large interaction of water molecules with the nanotube walls.

Friction force is calculated by using the method suggested by Falk et al. [29]:

$$f(\mathbf{r}) = \mathbf{d} \mathbf{U}(\mathbf{r}) / \,\mathrm{d}\mathbf{r} = 24 \,\mathrm{(\varepsilon/\sigma)} \left[ \mathbf{l} (\sigma/\mathbf{r})^{\top} - (\sigma/\mathbf{r})^{13} \right] \mathbf{l}$$

where 'r' is the distance between the oxygen atom of the water molecule and the nanotube wall atoms. Figure 10a shows the plot of friction force along the nanotube length for the carbon nanotube with Z-distortion of 1.2 and the perfect nanotube. It can be seen that the Z-distortion of 1.2 shows a higher friction force when compared to the perfect nanotube. This implies that there is less interaction between the walls of the perfect nanotube with the water molecules when compared to that of the Z-distortion of 1.2.

The discontinuity of the single-file water transport arises near the entrance of the nanotube. It is either followed by breakage of the single-file transport into short chains of molecules or filled up again at a faster pace. The molecules that follow after the broken single-file chain move relatively faster when compared to that of other molecules inside the nanotube. This can be viewed in Figure 10b, which shows the plot of the trajectory of a single molecule inside the perfect CNT compared to that of the CNT with Z-distortion of 1.2.

**Figure 10.** (**a**) Friction force inside the perfect nanotube and Z-distortion 1.2 CNT. (**b**) Comparison of water molecule trajectory inside perfect CNT and CNT with Z-distortion value 1.2.

#### *3.5. Radial Distribution Function and Potential of Mean Force*

The structural change of water molecules in the simulation cell can be described by the radial distribution function. Radial distribution function describes the atomic density variation as a function of distance from a particular atom [29]. Figure 11a illustrates the radial distribution function (RDF) of water molecules inside perfect and the Z-distortion 1.2 carbon nanotube.

**Figure 11.** (**a**) Radial distribution function of water molecules within the perfect carbon nanotube and carbon nanotube with Z-distortion 1.2. (**b**) Potential of mean force within the perfect carbon nanotube and carbon nanotube with Z-distortion 1.2.

The Z-distortion of 1.2 shows the more favorable interplay between the water and walls of the nanotube. It shows that the density increases in the vicinity of the Z-distortion of 1.2 carbon nanotube walls when compared to that of the perfect carbon nanotube walls. Therefore, this strong interplay between the oxygen atom of the water molecules and the Z-distortion of 1.2 carbon nanotube walls reduces the transport rate of water inside this nanotube. When compared to that of the Z-distortion of the 1.2 carbon nanotube, perfect carbon nanotubes show less interplay between the oxygen atoms of the water molecule with the walls of the nanotube. Thus, it favors faster water transport when compared to the Z-distortion of 1.2.

Potential of mean force (PMF) calculations inside the nanotubes help to understand the amount of energy barrier that exists inside the nanotube [30]. Water molecules have to overcome this energy barrier to move through the nanotube. The potential of mean force is calculated as follows:

#### PMF = (−kBT) ln g(*x*)

where g(*x*) is the radial distribution function (RDF), kB the Boltzmann constant, and T the thermodynamic temperature [16]. Figure 11b shows that the potential of mean force inside the Z-distortion of 1.2 CNT is significantly larger than the perfect CNT, so the water molecules should have larger energy to conduct through the nanopore. This result further supports the reason for the discontinuous flow occurring inside the Z-distortion of the 1.2 carbon nanotube.
