2.1. Choosing Pore Shape and Size
In this study, the pores’ shapes and sizes, shown in
Figure 1A,B, were chosen based on our previous investigation on gas separation membranes [
18]. The larger pore, corresponding to the removal of 14-atoms out of a graphene sheet, was proven to allow all molecules of the described gas mixture to pass through, while the smaller pore, corresponding to the removal of a 6-atom graphene ring, was shown to allow only the passage of H
2 and H
2O molecules.
An important aspect to mention when conducting investigations similar to ours, obvious when looking at
Figure 2A–C,G–I,M–O, is that some deformation of the carbon nanotube occurred during each of the simulations due to the imposed rotational movements. This aspect can be observed easiest by looking at the no gas area. The same area is specifically shown in
Figure 3. These structural deformations can lead to changes that occur in the shape and size of the pores situated on the rotating carbon nanotube and thus different gas separation behaviors. In addition, looking at
Figure 2D–F,J–L,P–R, no gas molecules ever cross back into the membrane. The small fluctuations seen in the case of H
2 molecules are due to the calculation method (the H
2 number on the graphs always goes back up quickly).
Effective pore areas A
p were determined using a Monte Carlo hit-and-miss procedure [
14,
16,
18], considering the effective carbon atom radius R
eff = R
m,c/√2 and R
m,c equal to 0.17 nm. The approximate results are shown numerically in
Figure 1A,B, meaning ~12.4Å
2 for the smaller pores located along the carbon nanotube and ~31.0Å
2 for the larger pores placed in the graphene layer at one of its ends. Blue-filled shapes for each pore type showing their effective pore areas are also shown in
Figure 1A,B (images not to scale). The two values were calculated considering the pores to be sculpted in a planar 2D graphene sheet. This approach fits very well in the case of the larger pore. For the smaller pore, however, determining the area in such a manner is less accurate due to being sculpted in a carbon nanotube with a chiral vector determined by the indices (12, 16). Nevertheless, it still provides a useful approximate value. Using the same method for calculating the smaller pore’s area, now when sculpted in the chiral nanotube, we ended up with the following approximate values: 12.26 Å
2 at 0ns before the start of the simulations, 12.51Å
2 for ω = 180° ps
−1, 13.02Å
2 for ω = 270° ps
−1 and 14.70Å
2 for ω = 360° ps
−1.
Thus, given the imposed angular velocities to the carbon nanotube, changes in the shape and size of the smaller pores took place. The shape and size of the pores, together with the angular velocity of the nanotube are the main parameters behind the distinct results obtained here for each ω case. This aspect helps explain the results obtained in this paper. The influence of an imposed angular velocity on the geometry of a carbon nanotube and pores sculpted on its surface can be investigated in future studies.
The rationale behind keeping an “open” end to the membrane, represented by the graphene layer with the large pores placed as pictured in
Figure 1A,B, was to allow for the gas mixture to be fed into the carbon nanotube. This obviously led to some H
2 molecules to be lost as they were able to exit the membrane through the pores situated at the end of the tube instead of crossing through the smaller pores into the filtrate area. However, we deemed this “open” end setup more realistic and practical. Given our previous results [
18], which confirmed that all gas molecules present in the gas mixture used here are able to pass through the 31.0 Å
2 pore, we focused the current investigation on the separation process that would occur once the gas mixture is already loaded into the rotating carbon nanotube membrane and the effect of the imposed angular velocity.
2.3. Angular Velocity ω = 180° ps−1
In the case of ω = 180° ps
−1, consistent with our previous investigations [
18], only H
2 and H
2O molecules were able to pass through the smaller pores. The process of H
2 molecules passing through the smaller pores and entering the filtrate area took place in about 7–8 ns, with ~20 H
2 molecules passing within the first nanosecond and some other 40 to 50 molecules within the next 6–7 ns. In regards to H
2O molecules, only 1–2 molecules were able to pass into the filtrate area, while most of them (6–7) were still found inside the carbon nanotube at the end of the 10 ns. The reason for this is that although the smaller pores should theoretically allow them to pass through, these are found to cluster together inside the nanotube due to hydrogen bonding. In all our simulations this prevents most of them from leaving the rotating membrane, given the pore sizes used. No other gas molecules were able to pass through the smaller pores at this angular velocity.
Taking a look at the process of separation occurring throughout our simulations, we observed that some of the gas molecules inside the nanotube position themselves as if blocking the smaller pores and remain positioned so, despite the rotating motion. This aspect is shown in
Supplementary Video S1 and in
Figure 4A with CH
4, CO, H
2O, N
2, and H
2 molecules blocking some of the pores. We also show, in
Figure 4B, that by the end of the simulation, at 10ns, pores were still blocked by the larger molecules at a time at which H
2 molecules were no longer found inside the rotating nanotube. We think that due to the imposed rotational motion and the size and shape of the chosen smaller pores, the smaller molecules such as H
2 and H
2O are able to eventually exit through the pores they are blocking. Thus, with an imposed ω of 180° ps
−1, some degree of fouling will occur, leaving fewer pores open for H
2 molecules to cross and leading more to pass through the open end of the membrane in exchange for the high selectivity offered. Between 59 and 73 out of 100 H
2 molecules were able to pass into the filtrate area throughout the three simulations with an imposed angular velocity of 180° ps
−1. Interestingly, no CO
2 molecules were found to block any of the smaller pores throughout all simulations but were instead able to pass through the larger pores as can be seen in
Figure 2I.
At ω = 180° ps
−1, as it can be observed in
Supplementary Video S1, some H
2 adsorption to the wall of the rotating carbon nanotube took place, which allowed the gas molecules to travel with the nanotube in its rotating motion. However, the phenomenon lasted for a short time (a few ps) before the adsorbed molecules were disturbed. We think both the imposed high angular velocity and the placement of the pores along the membrane, with each pore crossing disturbing the nearby gas molecules, help prevent adsorption from taking place and thus allow all H
2 molecules to leave the nanotube by the end of the simulation.
Another important aspect to discuss in the presented rotating carbon nanotube membrane design regards the placement of the pores leading to the filtrate area along the tube. In our case, as shown in
Figure 1, we have placed the 24 smaller pores in the central part of the nanotube, leaving both ends with no pores to the filtrate area. As shown in
Figure 5, at the end of all simulations (10 ns), unless found blocking one of the smaller pores, all molecules that remained in the rotating nanotube can be seen gathered at one of its ends. Without smaller pores nearby, the agglomeration of molecules becomes somewhat “stagnant” despite the rotational movement of the tube (see
Supplementary Video S2). This aspect may be beneficial to the flux of H
2 to the filtrate area as the smaller pores are then less likely to be blocked.
Additionally, inspecting
Figure 5, all H
2 molecules exited by the 10ns mark, and only CH
4, CO, CO
2, N
2, and H
2O molecules (clustered) can be seen inside. To provide an explanation for this we looked at the movement of H
2 molecules.
As shown in
Supplementary Video S2, the molecules in the mixture with a smaller mass, meaning H
2, CH
4, and H
2O were able to move slightly quicker when inside the rotating nanotube. On top of that, due to their small volume, which allowed them to “squeeze” through the larger molecules and their tendency to adsorb to the wall of the nanotube, H
2 molecules had a high number of collisions. Together with the chirality of the nanotube, we think these aspects lead to the exiting of all H
2 molecules within the 10 ns timeframe as they are more likely to travel along the nanotube and exit either towards the filtrate area or through the “open” end. The movement of H
2 molecules within rotating carbon nanotubes of different chiralities could be investigated in future studies to determine whether chirality can drive the gas molecules due to their tendency to adsorb to the wall of the nanotube.
2.4. Angular Velocity ω = 270° ps−1
For the higher imposed angular velocities, significantly different outcomes were observed compared to the ω = 180° ps
−1 case. The outcomes were consistent throughout all three repetitions corresponding to a certain ω value. In the case of ω = 270° ps
−1, H
2, CO, CH
4, N
2, and H
2O molecules were observed to pass through the smaller pores, as shown in
Figure 2G–L. Almost all H
2 molecules that passed into the filtrate area (~80–85) did so in the first half nanosecond. Again, no H
2 molecules were found inside the tube at the end of the 10 ns. Most molecules that passed through the smaller pores did so after blocking the pore for a while and exited either due to a collision with another gas molecule or due to the rotational motion of the nanotube, which allowed for slight changes in the orientation of the molecules. At this imposed angular velocity, despite that more H
2 molecules reached the filtrate area, there was far less selectivity leading to a poor separation performance.
The movement of the H
2 molecules inside the tube was slightly different compared to the ω = 180° ps
−1 case, please see
Supplementary Video S3. Due to the higher imposed angular velocity, H
2 molecules that were not blocking a pore were far less able to adsorb to the wall of the tube and travel in a synchronized manner with its rotational motion (as observed when ω = 180° ps
−1). Thus, due to the collisions with the wall, some H
2 molecules traveling in a rotational motion nearby, did so both in the sense of the rotation and in the anti-sense, but slower when compared to the rotating wall. Essentially, this should have led to a higher number of collisions taking place inside the tube, which, alongside with the enlargement of the pores due to the larger angular velocity, enabled most molecules to exit into the filtrate area. CH
4 molecules were also observed to pass through the pores after blocking it for a while and following a collision. No CO
2 molecules were able to pass through the smaller pores. The results obtained for ω = 270° ps
−1 case are somewhat intuitive in that the higher angular velocity led to the expansion of the smaller pores and plummeted its selectivity while allowing more H
2 molecules to pass into the filtrate area. From these simulations one can also see the following insight: as most remaining gas molecules at the 10ns mark are either found blocking one of the smaller pores or situated at one of the ends of the tube, the placement of the smaller pores along the center of the rotating carbon nanotube membrane could limit the passage of larger gas molecules into the filtrate area, thus improving the performance of the membrane.
2.5. Angular Velocity ω = 360° ps−1
In the final case of ω = 360° ps
−1, as shown in
Figure 2M–R, 89 to 94 H
2 molecules were able to pass through the smaller pores in less than 0.5 ns. The H
2 molecules that passed through the larger pores did so within 0.3 ns since the start of the simulations. Curiously, the only other gases which passed into the filtrate area at the end of the simulated time were CH
4 and H
2O molecules. Consistent throughout all three simulations with ω = 360° ps
−1, all 20 CH
4 molecules passed through the smaller pores and, similarly to the previously mentioned cases, ω = 180° ps
−1 and ω = 270° ps
−1, only one H
2O molecule managed to do so, due to the clustering of the water molecules inside the tube. The fact that these results were consistent throughout all three repetitions shows, counter-intuitively, that using higher angular velocities values can lead to unique and unexpected insights. The fact that almost 90–95% of H
2 molecules placed initially in the tube, together with 100% of the CH
4 molecules and only 10% of the water molecules were able to pass into the filtrate area could be useful in future studies working on improving the current design by making use of a gradual separative process. In such a case, the second step would involve only the separation of H
2 and CH
4 molecules, as, curiously, no other gas molecules were able to pass into the filtrate area.
Observing the movement of the molecules inside the rotating carbon nanotube at ω = 360° ps
−1 shows the molecules with a smaller mass, H
2, CH
4, and H
2O moving significantly faster than the larger CO, N
2, and CO
2 molecules (see
Supplementary Video S4). On top of that, due to collisions with the walls of the tube, the same smaller mass molecules rotate around their own center of mass much quicker than the larger ones, with CH
4 being a special case and rotating faster than all linear molecules due to its geometry. Further on, given the quick rotation of the tube, no gas molecules were seen blocking any of the smaller pores or being able to adsorb to the rotating walls of the membrane. Thus, we think that when a gas molecule reaches the vicinity of a pore, due to the high angular velocity of the nanotube, the right orientation has to be found quickly in order to exit, which did not happen in the case of the larger linear molecules with a higher mass, but did take place in the case of the non-linear molecules with a smaller mass, meaning CH
4 and H
2O molecules. That is, gas molecules had a very short time to match an orientation that would have allowed them to cross the pore. After nearing a pore and colliding with the rotating wall, a small mass non-linear molecule was more likely to quickly rotate around its mass center and change its orientation quickly, which increased its chances of crossing. However, despite its linear structure, H
2 molecules crossed the smaller pores easily due to their very small mass and volume as these aspects allowed them to change their orientation quickly and match that necessary to exit through the pore. More details, solely in regards to pore crossing can be reached in further investigations through ab initio molecular dynamics (AIMD) techniques.
2.6. Flux and Permeance
To further characterize the rotating carbon nanotube design for gas separation, we calculated the flux of H
2 molecules through the smaller pores. Thus, in order to determine the total flux we counted the number of crossings from both inside and outside the tube and then made use of the formula below:
where
NA represents Avogadro’s number,
Ainner_surface represents the inner surface of the tube and time corresponds to the time point of the last H
2 crossing during the simulation. In addition, given that the majority of H
2 molecules exit in a time window significantly shorter than the duration of the whole simulation, we have calculated the flux within the time frame in which 80% of H
2 molecules had left the rotating carbon nanotube membrane. The inner surface of the tube was determined to be 59.854 nm
2 using the formula 2πrh. The radius of the tube, r, was determined using the formula:
where a = 0.246 nm and m and n represent the chiral indices of the nanotube. The results are shown in
Figure 6A. Given the calculated total flux values, permeance values for our membrane design were estimated and shown both in mol/m
2sPa and gas permeation units (GPUs) in
Figure 6B.
Dividing the flux values presented in
Figure 6A to the number of smaller pores present along the inner surface of the carbon nanotube, we obtain the intervals shown in
Table 1:
Despite the very different setups, comparing the total flux values obtained here (shown in
Table 1) after division to the number of relevant pores, with the values presented in our previous research for a different design with the same gas separation purpose, the results are not significantly different from the interval 4.5–14.0 mol/m
2s observed in [
18] on the third filtration layer. The 80% flux intervals presented in
Table 1 are higher, yet that is unsurprising, but these importantly highlight the manner in which the H
2 molecules cross the smaller pores in the current design, with most of them managing to exit the rotating nanotube in the first 5.0 ns or 0.5 ns depending on the ω case, yet much quicker than the last 20%.
Previous results were exceptional in terms of selectivity and permeability due to the gradual separation through multiple graphene layers each with its own specific pore size [
18]. In the current design, we observed different behaviors, yet the exit of all H
2 molecules took place in a very short time and no H
2 molecules were left inside the nanotube at the end of the simulated 10 ns. Thus, nanopores used for filtration can be customized for certain molecules not only by selecting their size and 2D shape but also by using an imposed angular velocity and the forces involved in the consequent rotational motion, thus changing its geometry. Therefore, using a rotating carbon nanotube membrane allowed the gas molecules to be exposed to most of the surface area of the membrane, uncovering interesting distinct behaviors, dependent on the imposed angular velocity. Nevertheless, alongside the advantage of the quick exiting of all H
2 molecules within the membrane, a disadvantage of the current design is the loss of some of them due to the “open” end. Future studies may investigate different scenarios for improving the efficiency of the membrane. The easiest scenario to imagine being blocking the open end with a full graphene layer with no pores.
Comparing to Sun’s results [
16], the total flux values obtained in
Figure 6A are comparable to a graphene pore size of 10 to 12 atoms, while the 80% flux values are comparable to a 12-atom graphene pore for ω = 180° ps
−1, 14-atom graphene pore for 270° ps
−1 and 16-atom graphene pore for 360° ps
−1.
Naively comparing the flux values obtained for each ω case to each other, a clear trend can be seen showing that a higher imposed angular velocity leads to generally higher flux. However, an outlier can be seen in
Figure 6A, with one total flux value in the ω = 360° ps
−1 significantly smaller than the other two. This was also one of the reasons we decided to calculate the 80% flux values. The corresponding value to the outlier for the 80% flux is higher than the two other similar simulations, which shows that most of the H
2 molecules in the outlier simulation crossed into the filtrate area quicker and left the other 20% needing more time to cross due to, potentially, fewer collisions between molecules. Thus, with this explanation, we deem the outlier valid. One must take into account, however, that the heights of the total flux and the 80% flux values are not perfectly proportional due to events such as pore blocking, exit of other gas molecules, etc.
Similar to flux calculations, we estimated permeance values for the rotating carbon nanotube membrane, as shown in
Figure 6B. Similar conclusions can be taken out of the permeance values as with the total flux calculations. We find all our simulations to indicate a value within the following intervals: 0.0097 to 0.0215 mol
−3/m
2sPa and 0.30 to 0.65 × 10
5 GPU, and thus in the vicinity of those calculated for our previous design, with values situated between 0.005 to 0.03 mol
−3/m
2sPa and 0.20 to 0.85 × 10
5 GPU, results already highlighted as superior to state-of-the-art solutions [
18].