*3.2. Flux and Di*ff*usion*

Flux is created by the pressure that is applied to the water molecules inside the pool to push them through the nanotube. Water flux is calculated as the difference between the sum of the number of water molecules that move from the left side to the right side of the nanotube and the molecules which cross in the opposite direction [24]. Figure 6 shows the variation of flux based on different deformations.

**Figure 6.** Variation of flux in perfect and different deformed nanotubes: (**a**) carbon nanotube (CNT) and (**b**) boron nitride nanotube (BNNT).

For both the carbon nanotube and boron nitride nanotube, the XY-distortion of 2 showed higher flux when compared to that of other nanotubes. XY-distortions of higher value showed much less flux when compared to the perfect nanotube. The number of molecules that cross the pore gradually decreases with an increase in the degree of deformation. The screw distorted nanotubes did not show any variations; they behave almost similarly to that of the perfect nanotube.

The diffusion coefficient of water through the nanotube is the measure of the mass of the water that diffuses through a unit surface of the nanotube in a unit time at a concentration gradient of unity. The axial diffusion coefficient of water molecules can be computed from the mean squared displacement (MSD). The MSD of the water molecule's center of mass can be calculated by using the relation used by Barati Farimani et al. [25]:

$$<\left| (\mathbf{r}(\mathbf{t}) - \mathbf{r}(0)) \right|^2 > = \mathbf{ADt}^{\mathbf{n}}$$

where 'r' denotes the center of the mass coordinate of the water molecule. The angle bracket used in the equation defines the average over all the water molecules; 't' denotes the time interval; 'D' denotes the diffusion coefficient; 'A' denotes the dimensional factor values of 2, 4, and 6 for 1-, 2-, and 3-dimensional diffusion, respectively; and 'n' defines the type of the diffusion mechanism. The value of 'n' can be 0.5, 1, and 2, depending on how the MSD varies with time. These values of 'n' represent Single-File diffusion, Fickian diffusion, and ballistic diffusion, respectively. MSD of all the molecules in that direction with A = 2 is used to compute the average axial diffusion coefficient in Z-direction. In this simulation, the single-file diffusion is observed in all cases other than the XY-distorted nanotubes. In the XY-distorted nanotube of distortion value 2, a change from single-file to near-Fickian diffusion is observed. Hence, there is a significant increase in diffusion and flux.

Figure 7 shows the variation of diffusion coefficient for nanotubes with different types of deformation. The XY-deformation of 2 shows a relatively higher diffusion coefficient due to the increase in the pore volume of the nanotube when compared to other XY-distortions of higher value. This is in good agreemen<sup>t</sup> with the Hilder et al. [26] and Corry et al. [27]. It can be seen that there are significant values for diffusion for XY-distortions of 4 and 6 when compared with that of the flux. This is due to

the filling of water molecules near the entrance and exit of the nanotube, while the number of water molecules that completely travel from one end to the other is very low.

**Figure 7.** Pressure dependence of diffusion coefficient in perfect and different types of deformed nanotubes: (**a**) carbon nanotube (CNT) and (**b**) boron nitride nanotube (BNNT).
