The adsorption and fast transport of Xe in single walled carbon nanotubes

Abstract

Combined GCMC and MD simulations have been used to investigate the adsorption and diffusion of Xe gases in carbon nanotubes (CNTs) at different conditions. The influences of several factors, such as temperature, pressure and diameter of the CNTs, on the adsorption structure and diffusion rate of Xe atoms have been comprehensively studied. The Xe atoms form an ordered helix structure in the cylindrical radial direction of the CNT, and move in a spiral way. The mean square displacement (MSD) increases with increasing the temperature, while it increases first and then decreases with increasing the pressure. The pore size has a considerable impact on both the structure and number of adsorbed molecules in the CNT, where a smaller pore size accounts for faster diffusion. The Xe atoms diffuse much faster in CNT than zeolite. In contrast to a Fickian motion in zeolite, a ballistic diffusion mechanism is followed in CNTs.

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Introduction

In recent years, many research studies have been focused on the application of CNTs due to their unique mechanical, electronic, and structural properties. As an ideal material with one-dimensional pore, CNTs with metal or metal oxide nanoparticles confined in the pore show different catalytic activities with respect to that deposited on the exterior surface.¹ One of the most important issues in catalysis is the transport and adsorptive behavior of molecular flows in the channel, which can influence the reaction rate and selectivity directly.^2,3 It is also important for the applications in material science, such as gas separation, filtration and biotechnology.^4–6 So over the last few decades, CNT has attracted considerable attention in the field of nanofluidics.^7–10 The often studied systems include water, small gases, hydrocarbon gases and polymer molecules. The confined adsorbates have been theoretically predicted to possess significantly enhanced diffusivity in CNT in contrast to other nanoporous materials. This phenomenon has driven significant interest in assembling CNT into membranes for separation applications or ideal gas adsorbents.¹¹ The self-diffusion coefficient of ionic liquids has also been predicted to increase by 3 orders of magnitude compared to its bulk reference when confined in vertically aligned CNT membranes, which is promising for high power battery separators.¹²

The diffusion of water molecule in CNT has been the most often studied in literatures because of its importance in understanding water-mediated biological activities. Mukherjee et al. found that a single-file mechanism is suitable for a CNT (6,6) nanoring model,¹³ whereas a Fickian type was suggested for the finitely long CNT (6,6).¹⁴ Ye et al.¹⁵ found the water diffusion exhibits a super- or sub-diffusion mechanism, and then transits to the single-file type inside the CNT (6,6) and shifts to the Fickian type inside the larger CNT. Farimani et al.¹⁶ found the diffusion mechanism is Fickian in the central portion of the nanotube and ballistic near the nanotube wall. Because of the difficulties in preparing ideal material and limitations of present analytical techniques, only a few experimental studies have been reported on the diffusion confined in CNT. Das et al.¹⁷ have found the diffusion of water inside single-walled carbon nanotubes follows a single-file mode by a NMR method, while through PFG-NMR, Liu et al.¹⁸ found the water in CNT follows a normal diffusion model and the diffusivity of water in double-walled carbon nanotubes with an average inner diameter of 2.3 nm is twice that in multi-walled carbon nanotubes with an average inner diameter of 6.7 nm. In addition, the effective self-diffusion coefficient in double-walled carbon nanotubes is 1 order of magnitude higher than that in mesoporous silica materials with a similar pore size. All these studies show that the diffusion in CNT is complicated and the nanoconfinement effect of CNT on adsorbates is far from being well understood. More studies are highly desirable to reveal details how the CNT channels affect the diffusion. However, for simulation study, the water in CNT is a very complex system, where many factors have to be considered, such as the models of water cluster and the hydrogen bonds between them, since transport through carbon nanotubes have been found to be very sensitive to small changes in the structure of fluid molecules.^19,20 So in order to better understand the intrinsic effect of CNT on the fluxion of its adsorbates, inert gases (e.g. Ne, Ar, Xe) would be good objects to investigate as there is no chemical interaction and polarization between the adsorbates and CNT walls.²¹ To the best of our knowledge, very few studies have been reported on the systematical simulation of Xe diffusion in CNT, although the adsorption of inert gas mixtures have been studied.^22,23 Meanwhile, there is another merit to employ Xe as a probe to investigate the diffusion in CNT. That is, the simulation results have the potential to be compared to experimental data in the future, for ¹²⁹Xe MAS NMR has been proved to be a powerful method to characterize porous materials, such as zeolites, mesoporous and amorphous silica.²⁴ For example, Xu et al. have studied the adsorption and reaction kinetics in the nanospace of zeolite with in situ continuous-flow laser-hyperpolarized ¹²⁹Xe MAS NMR.²⁵ Clewett et al. have also employed ¹²⁹Xe and ¹³¹Xe NMR to investigate the adsorption of Xe gas in carbon nanotubes.²⁶

For the reason above, in this work, we systematically studied the adsorption and diffusion behaviors of Xe gas in CNT using molecular simulation methods. The diffusion of Xe confined in CNT with diameters ranging from 8 to 26 Å, as well as Xe in a given diameter CNT (10,10) with temperatures from 253 to 313 K and pressures from 0.01 to 2 atm, have been investigated. Xe diffusions in CNT and Si-VFI zeolite with similar pore diameters have also been discussed for comparison. This study will add fundamental knowledge about the sorption and transport properties of Xe in carbon nanotubes.

Simulation methods

A combined method of Grand Canonical Monte Carlo (GCMC) and molecular dynamics (MD) were used here to study the diffusion and mobility of Xe atoms in the confined pores. Two systems containing SWNT and siliceous zeolite were performed. The diameter of carbon nanotubes ranged from 8 to 26 Å with the length of 2.46 nm. The zeolite was modeled with the framework structure of VFI, which has the same one-dimensional pore topology as CNT. After replacing all of the Al and P atoms with Si atoms, the structure of Si-VFI was optimized within generalized gradient approximation (GGA) at PW91 (ref. 27) correlation functional with Monkhorst–Pack k-point grid at 1 × 1 × 2. All these calculations were performed using DNP basis set as implemented in DMol3.²⁸ The pore diameter of the zeolite is approximately 1.5 nm, the simulation box of which is 37.95 × 37.95 × 24.312 Å. Periodic boundary conditions were exerted in three dimensions in both of CNT and zeolite models, of which the frameworks were rigid in the simulation.

Firstly, GCMC method^29,30 was used to simulate the adsorption of Xe gases at a certain temperature and pressure. The COMPASS forcefield³¹ was applied, since many researches using it^32,33 had acquired good accuracy of calculation in siliceous system. The electrostatic potential energy was calculated by the Ewald summation technique and the van der Waals potential energy was calculated by the atom based technique. The cutoff distances were set at 8.0 and 12.0 Å for CNT and zeolite, respectively. For both of the two systems, the equilibration steps were set as 500 000 and the production steps were set as 2 000 000.

The structure with the lowest energy after sorption was chosen and minimized for each system. Then a NVT-MD method³⁴ was applied to study the diffusion behavior of adsorbed atoms. In this work, the forcefield and non-bond calculation techniques were the same as that used in the previous adsorption simulation. A simulation time step of 1.0 fs was employed, which was sufficient to ensure good energy conservation. And a total of 1 000 000 steps (1 ns) were used to analysis the kinetic properties. The temperature was controlled by Berendsen method with the decay constant of 0.1 ps. The mean square displacement (MSD) can be computed from the simulations. When analyzed using log MSD–log t plots,³⁵ it should obtain a straight line, which means the calculation has converged. All the simulations were carried out using the Accelrys Material Studio software simulation package on National Supercomputing Center in Shenzhen.

Results and discussion

The diffusion of Xe confined in CNT

Temperature influence on diffusion rate of Xe in CNT (10,10). The adsorptions of Xe in CNT (10,10) at 1 atm with different temperatures ranging from 253 to 313 K were simulated by Monte Carlo method firstly. After adsorption equilibrium and optimizing, the most stable structures at different temperature were obtained, as shown in Fig. 1. All the structures formed by Xe atoms are very similar, for clarity, only the representative ones with the lowest and highest temperatures are shown here. The Xe atoms are clearly ordered in helix in the cylindrical radial direction inside the CNT (Fig. 1a). It may be attributed to the helical arrangements of the carbon atoms in the CNT wall.³⁶ Fig. 1b shows the top view of the system (along the axis of CNT). A ring of Xe atoms, which consists of 5 spiral chains, are clearly formed with a distance of about 4 Å to the wall of CNT. On the other hand, from the side view, Fig. 1c, the helix structure can be observed more clearly with two spiral chains twisted together. From GCMC, we could also obtain the loading numbers of Xe in the periodic box at different temperatures. The loading was 27, 26, 26 and 25 atoms at 253 K, 273 K, 293 K and 313 K, respectively. It has a decreasing trend with increasing temperature.

Fig. 1 (a) The adsorption structure of Xe atoms in CNT (10,10) at 253 K, 1 atm. (b) Top view of (a). (c) Side view, the yellow coloring is just for guiding the eye. (d) The adsorption structure of Xe atoms in CNT (10,10) at 313 K, 1 atm.

A NVT-MD method was applied to study the diffusion behavior of Xe in CNT. The mean squared displacement (MSD) is a measure of the average distance a molecule travels, which is reflected by the deviation over time between the position of a particle and a reference position. The diffusion mechanism can be reflected by scaling behavior between the MSD and the time. And three diffusion mechanisms, including Fickian diffusion, single-file diffusion and ballistic motion, can be expressed as follows.^15,16

The left hand side of these equations is the MSD of a molecule in time t, which can be extracted from the MD simulations, denotes an average over all the molecules, is the distance along the axial direction of the pore. D is the familiar Fickian self-diffusion coefficient, F is the single-file mobility, and B is the ballistic mobility constant. For the three diffusion mechanisms, the slope of MSD on the log–log scale would be 1, 1/2 and 2, respectively. Clearly, the ballistic motion is much faster than the Fickian diffusion, which is on the other hand much faster than the single-file diffusion.

The MSD lines as a function of time at different temperatures are given in Fig. 2a. All of them show a good parabola relationship. Four straight lines were obtained from log(MSD) and log(t) plots with almost the same slopes (approximately equal to 2), shown in Fig. 2b. The straight line means the calculation has converged and the approximate slope of 2 means ballistic type diffusions occurred in CNT under current conditions. As seen in Fig. 2a, the mean-squared displacement increases as the temperature goes up. This implies that Xe atoms will move faster at high temperature, which is consistent with the results of Ar or Ne diffused in CNT and may be attributed to the greater thermal activation effect.²¹

Fig. 2 Axial direction mean squared displacement (MSD) of Xe in CNT (10,10) at different temperatures against the time: (a) normal plot; (b) logarithmic plot.

Pressure influence on the movement of Xe in CNT (10,10). Similar methods were used here to investigate the pressure influence on the diffusion of Xe in CNT (10,10) at 293 K. The loading numbers of Xe atoms in the same model at different pressure after simulated by GCMC increase from 7 to 27 as the pressure goes up from 0.01 to 2 atm. It means the greater the pressure, the greater the density of Xe in the system. Fig. 3a shows the MSD lines as a function of time at different pressures. All of the obvious parabola relationships indicate that Xe atoms diffuse in the same ballistic type in the current pressure range. It is also confirmed by the slopes of MSD on the log–log scale, all of which are almost 2 (figures not shown). As seen in Fig. 3a, the MSD firstly increases with increasing pressure (from 0.01 to 0.1 atm), then reduces when pressure continues to go up to 2 atm. It means that the transport velocity of Xe atoms in CNT increases at first and then decreases. The reason for this tendency may be that the loading of Xe at 0.01 atm is too low (only 7 atoms) to form the ordered helix structure and Xe atoms diffuse slowly. When increasing the loading of Xe, an ordered helix structure will be formed and the diffusion rate may be increased. But as the pressure continues to increase to 2 atm, the increased density of Xe atoms will shorten their mean free paths markedly, which will increase the chance of intermolecular collisions and slow down the movement speed. The results of radial distribution function (RDF) of Xe atoms confirm the speculation. RDF is often used to estimate the corresponding molecular distances. As shown in Fig. 3b, all RDFs present two main peaks at different pressures. At 0.01 atm, they locate at about 4.20 and 6.20 Å, separately. The position of the biggest peak (4.20 Å) keeps almost the same when pressure increases from 0.01 to 0.1 atm, while it markedly moves left to 4.07 Å when the pressure increases from 0.1 to 2 atm. Xe atoms seems to be closer to each other and the stronger interaction among them will reduce their movement speed.

Fig. 3 (a) Axial direction mean squared displacement (MSD) of Xe in CNT (10,10) at different pressure against the time. (b) Radial distribution functions for Xe atoms at different temperature.

The movement of Xe in CNT with different diameters. As reported in literatures,^15,16 the confined gases or liquids in carbon nanotubes show obvious differences to their bulk references. It is conceivable that the diameter of CNT may have impact on the diffusion of adsorbed species. In order to investigate the diameter effects, four models of CNT (6,6), CNT (8,8), CNT (10,10) and CNT (15,15) with diameters of 8.1, 10.8, 13.6 and 26.2 Å, respectively, were used here to study the adsorption and diffusion of Xe gases at 1 atm and 293 K. The stable structures of Xe in different CNT after GCMC simulation are shown in Fig. 4. The four structures are obviously different. In CNT (6,6), Xe atoms form a straight line near the axis of the nanotube. When the pore diameter increases, the Xe atoms tend to form a spiral structure at a certain distance to the wall, as the cases in CNT (8,8), CNT (10,10). In CNT (15,15), besides the helix structure in the nanotube, Xe atoms are also distributed as a straight line in the internanotube spaces which have already become large enough to accommodate them. As a result, the number of Xe atoms in CNT (15,15) is much more than that in the other three CNT. Generally, the number of Xe increases (from 6 to 50 atoms) with increasing the diameter (from 8.1 to 26.2 nm), which causes more intermolecular collisions and may slower the diffusion rate.

Fig. 4 The adsorption structures and axial direction mean squared displacement of Xe in different CNT.

As shown in the MSDs plot of Xe diffused in different CNT in Fig. 4, all of them present a well parabolic function (MSD depends on the square of time), indicating a ballistic mechanism. For the diffusion of water in small diameter CNT, Das et al. suggested a single-file mechanism,¹⁷ but Mukherjee et al. found that it was only suitable for a CNT (6,6) nanoring model,¹³ whereas for the finitely long CNT (6,6) a Fickian type was suggested.¹⁴ However, our simulations demonstrate that the diffusion mechanism for Xe atoms in CNT (6,6) is ballistic. This may be related to their special moving form. Although the stable structure of Xe atoms in CNT (6,6) was a straight line, after checking the dynamic image of the diffusion track, we found they seemed to move forward also in a very slight spiral way which was more obvious in the other three larger nanotubes.

The value of MSD decreases with increasing the diameter of CNT, which means although they have the same ballistic diffusion mode, the diffusion rate varies with different CNT diameters: the smaller the pore size, the faster the motion of Xe atoms. This is consistent with the experimental result of water diffusion in CNT, which showed that the diffusivity of water in double-walled carbon nanotubes (DWNTs) with an average diameter of 2.3 ± 0.3 nm is twice that in multi-walled carbon nanotubes (MWNTs) with an average diameter of 6.7 ± 0.8 nm.¹⁸

Simulations of Xe adsorption in CNT and zeolite

Computer simulations of adsorption phenomena in CNT and zeolite have proved to be a useful complement to experimental studies, providing insight into novel structures and dynamics of adsorbed gases.^8,16,17 In this section we compared the diffusion rates of Xe in CNT and silica zeolite with similar pore sizes. In order to contain a complete channel, a super-cell model consists of 4 cells were chosen to simulate the Si-VFI zeolite. For comparison, a similar size CNT (10,10) model with 4 channels was also used. The adsorption structures of Xe in CNT and zeolite at 293 K and 1 atm are shown in Fig. 5. In contrast to the ordered helix structure and high adsorption density (108 atoms) in CNT, the Xe atoms are distributed disordered with a low adsorption density (44 atoms with the same channel length) in zeolite. Fig. 6a shows the MSD curves of Xe in CNT and Si-VFI zeolite. The MSD value in CNT is nearly 3 orders of magnitude higher than that in Si-VFI at the end of the simulation. The big divergence may be caused by different diffusion mechanisms based on the nearly linear and parabolic shape of the two curves. To confirm this point, Fig. 6b shows the logarithmic plot of the MSD with the diffusion time. They are all nearly straight lines ultimately, indicating both of the two systems had reached equilibrium. The slope of the line of CNT system is 1.99, very close to 2, unambiguously corresponding to a ballistic diffusion; while the slope of the line of zeolite system is 0.96, close to 1, indicating the diffusion follows a Fickian mechanism. As we all know, ballistic motion is much faster than Fickian diffusion, which accounts for the huge difference of MSD value in CNT and Si-VFI zeolite systems. The two different diffusion models may be correlated to the dramatic differences in the smoothness of the molecule-solid potential energy surface in CNT and silica zeolite, as well as the different status of Xe in two systems. The disordered distribution of Xe in zeolite would increase the collisions among Xe atoms, and between the Xe and the zeolite wall.

Fig. 5 Different adsorption structures of Xe in CNT (a) and Si-VFI zeolite (b).

Fig. 6 Axial direction mean squared displacement (MSD) of Xe in CNT (10,10) and Si-VFI zeolite against the time: (a) normal plot; (b) logarithmic plot.

Conclusions

Combined GCMC and MD simulations have been used to investigate the adsorption and diffusion of Xe gases in carbon nanotubes under different conditions. The factors that could influence the adsorption structure and diffusion rate of confined molecules, such as temperature, pressure and diameter of CNT, were studied systematically. It is found that the Xe atoms prefer to form an ordered helix structure in the cylindrical radial direction inside the CNT. The Xe atoms move in a spiral way during the simulation time, which may contribute to the rapid diffusion of molecules in the nanotube. The MSD increases when increasing the temperature, while it increases first and then decreases when increasing the pressure, since there may be more intermolecular collisions at high pressure with high adsorption density. When changing the diameter of CNT, both of the structure and adsorption number of Xe atoms are affected. A straight line is formed in CNT (6,6) with the smallest pore size, and an ordered helix structure is gradually formed with increasing the diameter of CNT. The larger the pore size, the more the number of molecules is adsorbed in, and the smaller the pore size, the faster the molecule diffusion is. The adsorption and diffusion of Xe atoms in Si-VFI zeolite with one-dimensional pore was also studied for comparison. It is found that the Xe atoms display a disordered distribution in zeolite, and the adsorption density is much smaller than that in CNT. The diffusion rate is also much slower in zeolite than that in CNT. They follow ballistic and Fickian diffusion mechanism in CNT and zeolite, respectively.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (21103183), High-level Talent Foundation of Henan University of Technology (2012BS059 and 2015QNJH10), Colleges and Universities Key Research Program Foundation of Henan Province (16A150006) and the Foundation of State Key Laboratory of Catalysis (N-15-03). The authors are very grateful to Dr Shutao Xu, Prof. Xinhe Bao, and Xiuwen Han in Dalian Institute of Chemical Physics, Chinese Academy of Sciences, for valuable discussions and computing resources they provided.

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