Water in Narrow Carbon Nanotubes: Roughness Promoted Diffusion

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C: Physical Processes in Nanomaterials and Nanostructures

Water in Narrow Carbon Nanotubes: Roughness Promoted Diffusion Transition Wei Cao, Liangliang Huang, Ming Ma, Linghong Lu, and Xiaohua Lu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02929 • Publication Date (Web): 30 Jul 2018 Downloaded from http://pubs.acs.org on July 31, 2018

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The Journal of Physical Chemistry

Water in Narrow Carbon Nanotubes: Roughness Promoted Diffusion Transition Wei Cao1, Liangliang Huang2, Ming Ma1, Linghong Lu3*, and Xiaohua Lu3* 1. State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China 2. School of Chemical, Biological and Materials Engineering, University of Oklahoma, Norman, Oklahoma 73019, United States 3. State Key Laboratory of Materials-Oriented Chemical Engineering, Nanjing Tech University, Nanjing, Jiangsu 210009, China

ABSTRACT: Diffusion of single-file water confined in carbon nanotubes (CNT) have been studied extensively via molecular dynamics simulations in literatures. However, very often the simulation results do not provide a satisfactory insight to the fundamental mechanism, failing to explain the experimental trend. Sometimes, simulation results are even contradicting to each other. In this work, we first summarize factors that affect the diffusion calculation of single-file confined water in narrow CNTs. In particular, we investigate how the roughness of the CNT model affects diffusion properties of the confined water. A transition of ballistic to Fickian diffusion shall be observable when water is in confined in (6, 6) CNTs with roughness. Our simulation results also

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reveal the single-file diffusion (SFD) mechanism of confined water when the CNT has significant roughness. This is because the single-file water molecules confined in smooth CNTs move in a collective way, while the motion becomes random in CNTs with roughness. In larger CNTs, even with the roughness change, only the Fickian type diffusion mechanism is observed. Our study provides a new insight into the argument on how confined water diffuses in narrow CNTs.

INTRODUCTION Due to the ultra-fast water transportation, carbon nanotube (CNT) is considered to be one of the materials for next-generation membranes for water desalination and purification1. The exceptionally high rate water transportation in CNTs has been recognized for a long time by means of both experiments and computer simulations2-6. For instance, vertically aligned CNT-based membranes with average inner diameters of 1.6 nm demonstrated that water permeability was three to five orders of magnitude higher than that predicted by continuum flow models3. Recently, Secchi et al. measured water flow rate and revealed an unexpected large and radius-dependent slippage of water transport through individual nanotubes7. The fast water transport was attributed to the frictionless interfaces of CNTs8-9. In addition, it is also reported that the confined water structure departs significantly from what is observed of bulk water, indicating a strong modification of the hydrogen bond (H-bond) network due to the confinement10-11. Fluids transport through CNT membranes occurs mainly by diffusion. For bulk water, the diffusion mechanism follows the Fickian type where the mean square displacement (MSD) scales linearly with diffusion time. Water molecules can pass each other in the direction of motion. When the diffusion is restricted under confinement, anomalous modes like single-file diffusion (SFD,


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MSD proportional to the square root of time) and ballistic (MSD proportional to the square of time) diffusions emerge. The diffusion mechanism of fluids can be expressed based on the scaling factor between MSD and diffusion time, lim 〈[𝑧(𝑡) − 𝑧(0)]2 〉 = 2𝐷𝑡 𝛼

𝑡→∞

(1)

where z(t) represents the coordinate of fluid molecules; bracket is the ensemble average; D is the diffusion coefficient; t is diffusion time; α represents the type of diffusion mechanism. Fluids diffuse in SFD, Fickian and ballistic modes have α values of 0.5, 1 and 2, respectively. In experiments, limited results have been reported on water diffusion in narrow CNTs. For example, Das and co-workers applied the pulsed field gradient NMR method to study water diffusion in CNTs, and they observed the SFD mode for water confined in CNTs with a diameter of 1.4 nm12. In a CNT with a diameter of about 1.2 nm, a ballistic diffusion mode was reported to be the dominant mechanism of ultrafast water diffusion13. Other experimental work concluded that water confined in CNTs (diameter is about 2.3 nm) diffuses in a Fickian diffusion mode, similar to the bulk water14. Despite those measurements, it is still very challenging to measure the diffusion of water confined in small CNTs. The difficulty is partly due the fact that the confined single-file water chain diffusion is non-continuous, which makes it hard to detect the diffusion. In addition, due to the nonhomogeneous nature of those narrow pores (diameter difference, roughness, curvature, open vs capped pores, induced local structural flexibility, etc.)15, it is almost impossible to directly compare experiments from different resources for a complete understanding of those control parameters. From the molecular simulation perspective, most calculations are using a single pristine CNT. Similar to experimental measurements, the calculated water diffusion also varies from one paper


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to the other. Hummer and co-workers reported that water molecules diffuse as a single-file structure in the (6, 6) CNT with a diameter of 0.8 nm2. One can intuitively explain this SFD mode since water molecules cannot pass each other20. However, with the same (6, 6) CNT, all the three diffusion mechanisms have been reported: SFD16, ballistic17 and Fickian18. It is not surprising that the calculated diffusion coefficients do not provide a satisfactory explanation to available experimental results. As shown in Table 1, calculated water diffusion in (6, 6) CNTs are summarized. In addition, water diffusion characteristics in some larger CNTs and in other nanopores are also listed for comparison. It is worth nothing that Table 1 is an incomplete list of reported studies. But one can easily notice that different combinations of simulation model, ensemble, thermostat and simulation time have been used in the literature. The simulation systems in Table 1 can be divided into two groups, based on the calculation setup. For the first group, the CNT with a finite length is immersed in a box of water molecules. During the simulation, water molecules can enter or leave the CNT from both ends. For the second group, water molecules are initially placed inside the CNT. With the periodic boundary condition applied along the axial direction, the CNT is of infinite length to those confined water molecules. It is interesting to note that if a finite CNT is used, namely, the Group 1, Case 1 to Case 5 in Table 1, the confined water diffusion mode is always the Fickian type, regardless the water model adopted in the simulation. Despite the same diffusion mode, the calculated diffusion coefficients vary from case to case, which is mainly due to the different force field parameters adopted in literatures19. In addition, for the Group 1, the calculated diffusion coefficients are all smaller than that of bulk water. For the Group 2, different diffusion mechanisms have been reported, which could result from a few factors. The simulation time is one of those factors. It has been reported that the diffusion mechanism can transform from one mode to another, with a typical transition


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time larger than 100~500 ps20. A longer simulation is thus required for the diffusion calculation. The thermostat is another parameter that significantly affects the diffusion calculation. In general, the thermostat used in MD simulations will modify the equation of motion by tuning the fluid velocities. Such a treatment will alter the realistic trajectory of fluids and leads to an unintended effect on the diffusion. It is accepted that diffusion coefficients are more accurate when NVE ensembles are applied or when a system is controlled by a weak coupling strategy21. For example, water flow in CNTs is reported to be greatly affected by thermostat22. Even for the widely used Langevin thermostat, the algorithm depends on the random modification of water velocities, which can favor the SFD mode for confined water23. Adding random forces mimics diffusive collisions, but it is not the real effect one observes from perfect cylindrical pores. The aforementioned discussions were about water diffusion in narrow (6, 6) CNTs via MD simulations. For water in a larger CNT, it is intuitive that water molecules can pass each other and diffuse in the Fickian type. However, we found some results that reported the abnormal diffusion for water in a large CNT, especially for the subcontinuum water flow when the diameter is less than 1.39 nm24-25. For example, as shown in Table 1, Striolo showed that the Fickian diffusion mechanism of water prevails after a nearly 500 ps ballistic phenomenon, as a result of the longlasting H-bonds, for water confined in infinitely long CNTs with a diameter of 1.08 nm20. There is no SFD mode in the study though the single-file like water structure exists. By the diffusion calculation of water in a wide CNT (diameter > 2 nm), Farimani and co-workers observed a dual mechanism, that is, the ballistic diffusion for water near the CNT wall and the Fickian diffusion mode for water in the tube center26. They showed the subdiffusion (α < 1) of water in CNTs with the diameter less than 1.3 nm. Besides CNTs, other nanopores were also found to host abnormal diffusion (α  1) mode of confined water. For instance, the diffusion was reported to follow an


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SFD when there were two or more water chains, for water in a carbon nanoring made up of (6, 6) CNTs. It goes over to normal and ballistic diffusion as long as the water molecules formed a single chain27. SFD was also observed for water in a single-walled silicon carbide nanotube28. Nevertheless, more research is needed into the mechanism of these abnormal diffusion calculations.


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Table 1 The factors influence the calculation of water diffusion in CNTs Case

Water model

CNT model

aDensity/nm-1

Ensemble

Thermostat

Time/ns

Diffusion

Diffusion

mechanism

Coefficient

Reference

Diffusion mechanism and coefficient of water inside (6, 6) CNT 1

TIP3P

2

SPC/E

3

TIP3P

4

TIP3P

5

SPC

6

7

TIP4P -EW TIP3P

bundle, external bath length = 1.34 nm external bath length = 1.42 nm external bath length = 1.4~5.6 nm external bath length = 2 nm external bath length = 1.34 nm infinite long

infinite long

10

Fickian

9.1 E-10 m2/s

29

40

Fickian

1.16 E-9 m2/s

30

-

8~20

Fickian

(2.2~2.8) E-9 m2/s

18

NPT

Langevin

4~5

3 E-9 m2/s0.96

31

NPT

Berendsen

50

1.15 E-9 m2/s

32

5.4 E-10 m2/s0.5

16

2.72 E-8 m2/s

33

3.73

NPT

-

NPT

-

NPT

3.5

3.7 3.65 (862 molecules) 3.1 (38 molecules)

NVT

NVT

NoseHoover

NoseHoover Langevin

8.5

α = 0.96 ~ Fickian Fickian from ballistic to single-file

10

Fickian 1~3 molecules:

8

SPC

infinite long

0.31~4.08 (1~13 molecules)

NVT

NoseHoover

100

Fickian

1~3 molecules:

4~13 molecules:

(1.5~0.29) E-6 m2/s

17

ballistic b

Anomalous diffusion mechanism of water inside larger CNTs


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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

9

diameter = 1.086

0.68~14.3

nm

(10~208

infinite long

molecules)

SPC/E

10

11

TIP3P

SPC/E

external bath length = 2 nm

NVE

diameter = 1.0 nm

NPT

no thermostat

Langevin

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18

4~5

from ballistic to Fickian α = 0.78 subdiffusion

-

20

4.4 E-9 m2/s0.78

31

diameter = 0.95 nm

α = 0.7

1.2 E-9 m2/s0.7

diameter = 1.086 nm

α = 0.5

0.53 E-9 m2/s0.5

α = 0.3

0.14 E-9 m2/s0.3

α = 0.9

1.8 E-9 m2/s0.9

At center α=1.0

2.56 m2/s

Near tube α=1.9

2.9 m2/s1.9

external bath

diameter = 1.221 nm

length ≈ 2 nm

diameter = 1.357 nm

NVT

Nose-

6~10

Hoover

diameter = 4.071 nm

26

Anomalous diffusion mechanism of water inside other nanopores 1 molecule

12

TIP3P

(6, 6) carbon nanoring

40 molecules 81 molecules

-

150 ns

40 uncharged

single-file

-

ballistic to single-file to Fickian

8.45 E-9 m2/s

ballistic to single-file to Fickian

-

single-file

-

molecules c

13

TIP3P

SWSiCNT

external bath length ≈ 2~3 nm

14 aThe

SPC/E

diameter = 0.7~0.86 nm

(6, 6) dBNNT

diameter = 0.82

external bath

nm

NPT

NPT

Berendsen

NoseHoover

27

3

single-file

-

28

28

Fickian

1.29 E-9 m2/s

34

density was defined by number of molecules inside CNT divided by the length of CNT fragment for simplicity. bAnomalous diffusion is defined as

diffusion that α in equation (1) is not equal to 1. cSWSiCNT is short for single-walled silicon carbide nanotube. dBNNT is short for boron nitride nanotube.


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So far, there is no unanimous agreement on the diffusion mechanism of confined water in narrow CNTs, especially in CNTs where water exits as a linear chain structure. Here in this work, the aim is to propose a fundamental explanation of how these water molecules diffuse. The difficulties in characterizing single-file water diffusion in narrow CNTs are from the approximate reflection between water and CNT walls, and the strong water-water interactions. To reveal the diffusion mechanisms, sinusoidal CNT models were applied to investigate the wall roughness effects on water diffusion. It has been reported that when the CNT wall is rough, the effective shear stress increases and the transport of water is suppressed35-37. The sinusoidal CNT model can be represented by a kind of acoustic phonon mode, verified by the phonon dispersion curve of CNTs38. The rough model has been found to be the coupling of the bending mode and breathing mode39. These rough models have been also adopted to study the Rayleigh traveling waves on the CNT surface, which were verified by both the experiments and theories40-42. We want to note that this manuscript by no means tries to propose a sinusoidal carbon nanotube, not even to suggest the sinusoidal models we adopt represent the actual synthesized carbon nanotube with roughness. In our computational study, we mainly point out that: (a) the reported confinement diffusion studies (experiments and computations) do not always agree to each other, sometimes are contradictory; We understand that the CNT samples used in the experiments vary from one case to the other, and that the computational studies do not always use the CNT model/force field/computational details consistently. But the current status is more or less like, there is no agreed recipe for confined diffusion studies. (b) the use of a pristine smooth CNT as a narrow pore model could be troublesome, and a proper implementation of the roughness (or curvature, or local structural dynamics) is critical to reveal how confined water actually diffuse in those narrow carbon pores. (c) the sinusoidal CNT model studied in this manuscript seems to compensate the roughness nature


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missing from a pristine smooth CNT, and therefore provides predictions agreeable to available experiments and other computations.

MODELS AND METHODS The rough CNT was constructed with a sinusoidal surface morphology, as shown in Figure 1. The CNT diameter varies periodically and is expressed as a function of z: R = R0 + Asin(2πz/λ), where R0 is the radius of the pristine CNT; R is the local radius at z; A is the amplitude of roughness, and λ is the wavelength. In this work, λ/R0 was set to 1.0 in the calculations, and A/R0 was varying from 0.01 to 0.1. The amplitude is chosen to be around the diameter changes characterized by the radial breathing oscillation of CNTs43. In our work, we studied a serial of A/R0 values to gain a complete view of the roughness effect on confined water diffusion. The CNT models were held rigid during the simulation. However, if the flexibility is allowed, the surface roughness would deviate from the pristine sinusoidal geometry, which makes it difficult to quantitatively correlate the roughness and diffusion. The diameters for (6, 6) and (10, 10) CNTs are 0.8 and 1.4 nm, respectively. Figure S1 shows the models used in this work. We also calculated the pore size distribution (PSD) for all models (see Figure S2). Water molecules were described by the SPC/E model, which is known to accurately predict the diffusion of water44-45. Carbon atoms of CNTs were modeled as unchanged Lennard-Jones (L-J) particles with parameters of εc = 0.07 kcal/mol and σc = 0.355 nm46. All simulations were performed using the LAMMPS software packages47. The short-range van der Waals were computed with a cutoff distance of 1.0 nm and long-range electrostatic interactions were calculated with particle mesh Ewald (PME) method. The initial configurations of CNTs with water inside were obtained in two steps (seen from Figure 1): the CNT with a finite length (~ 2.0


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nm) was solvated in a cubic box and water would saturate the CNT pore via the NVT ensemble; after that, water molecules which are outside of the CNT are removed, and the CNT with confined water is used to duplicate along the CNT axis direction for a few hundred times, so that there are adequate water molecules ( > 1000) to enable reliable statistics48. The initial density of confined water was about 0.2 g/cm3. The initial velocities were assigned according to the Boltzmann distribution. The dynamics of Newton’s equation were iterated using a 1.0-fs time step. After minimization, the system was equilibrated at 300 K and fully coupled to a Nose–Hoover thermostat for 10 ns. Afterwards, the thermostat was removed from water and a microcanonical (NVE) ensemble was applied to the system for another 15 ns, of which the last 10 ns was collected for analysis at an interval of 0.1 ps. The total energy in the last step is a constant, and there is no energy drift. The temperature and energy for water in a smooth (6, 6) CNT in simulations are shown in Figure S3. To obtain reliable statistics, three independent simulations have been performed for each system.


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Figure 1. (a) The snapshot of the simulation system where the CNT with confined water immersed is placed between two water reservoirs; the green dashed lines represent the periodic boundary condition applied in the MD calculations; (b) The CNT model with roughness is defined by the wavelength (λ) and the amplitude (A); (c) A rough CNT filled with confined water molecules.

RESULTS AND DISCUSSIONS Figure 2a shows the calculated mean-square displacement (MSD) of water confined in (6, 6) CNTs with roughness. The dashed lines represent typical diﬀusion modes of ballistic (red), Fickian (black), and single-file (blue). In a pristine smooth (6, 6) CNT, water diffuses in the ballistic mode: the calculated MSD is proportional to the square of diffusion time, as shown in Figure S4. This is attributed to the fact that water molecules are highly correlated and move as a water chain under confinement. The MSDs of the center of mass (com) of water molecules in CNTs were also calculated and found to be similar to the total MSDs shown here (see Figure S5). This indicates that water molecules hardly exhibit self-diffusion in a smooth (6, 6) CNT. In a larger pristine (8, 8) CNT, Striolo and co-workers also observed that confined water molecules diffuse via the ballistic mode for a few hundred picoseconds, which he concluded as an effect of strong and stable H-bonding network20. Compared with water in (8, 8) CNTs, the H-bonds of water in a smaller (6, 6) CNTs are even stronger, which prevents individual movement of water molecules and ultimately leads to a ballistic diffusion mode. In another MD simulation study, diffusion of water in a (6, 6) CNT was also reported to be the ballistic mode. It was accepted as a result of the unidirectional velocity of the highly correlated water molecules49. When the loading is gradually decreasing, the water cluster chain can no longer be maintained. Thus, the diffusion transits from the ballistic mode to the Fickian mode. The collective diffusion of monolayer water is also reported


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when water is confined between graphene. Such collective diffusion also lead to a fast transportation under confinement50. The diffusion mechanism is sensitive to the roughness of the CNTs. For (6, 6) CNTs with a small roughness, 0.01 < A/R0 < 0.06, as shown in Figure 2a, the MSD slopes (red, blue and pink solid lines) decrease from the pristine case (black solid line), suggesting the diffusion mode transits from ballistic to Fickian. When the roughness is increased to A/R0 = 0.1, an SFD mode is observed. In a rough CNT, the interaction between water and CNT depends on specific position along the z direction, leading to a more random change of water velocity. In other words, compared with the pristine CNTs, water confined in rough CNTs experiences an additional force in the z direction51. Thus, water molecules move faster in the valley but slow down when they are near the peak. Here the valley and the peak correspond to the positions from the roughness sine function. It is obvious that by tuning the roughness, we are able to change the diffusion mechanism, and observe the transition from ballistic to Fickian and then to the SFD mode. By comparing the total and com MSDs (see Figure S5), the com movement of water molecules could be neglected, and water molecules diffuse randomly in a highly confined space in CNTs with A/R0 = 0.1. To further corroborate the diffusion mechanism, we traced water molecules by calculating the displacement along the z direction. As shown in Figure S6 of the supplementary document, for water confined in smooth CNTs, there is a uniform motion, which leads to the calculated MSD proportional to the square of diffusion time. With the increase of roughness, the uniform motion gradually disappears. For example, in the rough CNT with A/R0 = 0.1, the displacement varies among confined water molecules. In addition, water molecules in this rough CNT have a much smaller accessible volume. Such enhanced confinement effect also partially explains why the SFD mode is preferential and is stable. The different wavelength of a rough CNT is not addressed in this work, since the decreasing


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wavelength and the increasing amplitude can both alter the collective movement of confined water. It is important to note that, in the larger (10, 10) CNT, the studied roughness magnitude does not produce a noticeable effect on the water diffusion. As shown in Figure 2b, only Fickian diffusion has been identified in the rough (10, 10) CNTs.

(a)

(b)

Figure 2. The logarithmic plot of MSDs of water confined in pristine and rough CNTs: (a) (6, 6) CNTs; (b) (10, 10) CNTs. Four rough CNTs have been studied, namely, the wavelength and the amplitude ratio (A/R0) of 0.01, 0.03, 0.06, and 0.10. Random force is implemented via the stochastic Langevin thermostat. Black, red, and blue dashed lines represent the Fickian, ballistic, and single-file diffusion modes, respectively.

In order to understand how easily the diffusion mode be transitioned from one to the other, we performed structural relaxation analysis to the confined water, as shown in Figure 3. The relaxation is characterized by calculating the probability of one water molecule in any water cluster using 𝑡 𝑄(𝑡) = ∑𝑁 𝑖=1〈∏𝑡0 =1 𝑝𝑖 (𝑡0 )〉 , where 𝑝𝑖 (𝑡0 ) equals to 1 when the distance between two water

molecules at a given time 𝑡0 is less than 0.5 nm; otherwise 𝑝𝑖 (𝑡0 ) is 0.27 The Q(t) value is then


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normalized to unity and the characteristic time can be obtained from the fitting according to 𝑄(𝑡) ~ 𝑒 −𝑡⁄𝜏 . A small relaxation time suggests that water molecules are easily breaking away from local clusters (such as a single-file water chain), thus the diffusion transition from the ballistic mode to the Fickian or SFD mode shall be more favorable. Our calculation results reveal that the characteristic relaxation time for water confined in rough CNTs models is generally smaller than that from a pristine CNT. For the model with the most significant roughness, A/R 0 = 0.1, the characteristic relaxation time is around 584 ps, much smaller than that in the pristine smooth (6, 6) CNT, 1257 ps. This analysis provides a fundamental insight and confirms that the diffusion mode transition is easier in CNTs with roughness. The transition of different diffusion modes has been also reported for water confined in (6, 6) carbon nanorings, another rough CNT-based material27. Water in carbon nanorings initially diffuses in the ballistic mode. But after about 100 ps, water molecules can move from one water cluster to another cluster, thus enabling the diffusion transition to the Fickian mode. Furthermore, if the water-water interaction is weaker than the water-nanoring interaction, the SFD mode is preferred. It has been also reported that the diffusion mechanism can be tuned by adding functional groups (such as carbonyls) to the inner pore surface52. Those oxygenated sites reduce the effective pore size, and also interact strongly with confined water, both of which favors the SFD mode.


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Figure 3. The characteristic relaxation time of confined water in studied (6, 6) CNTs models. The curves were fitted by an equation of y = 𝑒 −𝑥⁄𝜏 .

The single-file structured water diffusion in smooth and rough (A/R0 = 0.1) CNTs at different water density were also simulated to concern the water content effect. The original density of water in smooth and rough CNTs is 0.34 and 0.33 nm-1, respectively. Here, the water density is defined by the number of water molecules inside CNTs divided by the length of CNT fragment for simplicity. Two smaller water contents are calculated, say 0.20 and 0.07 nm-1 for smooth CNTs, and 0.16 and 0.05 nm-1 for rough CNTs. The MSDs as a function of the simulation time are shown in Figure 4a. For water in smooth CNTs, the calculated MSDs for all the three cases are proportional to the square of diffusion time, showing the ballistic mode. While for water in rough CNTs, the MSDs are proportional to the square root of diffusion time, suggesting the SFD mode. Thus, the density of single-file water in narrow CNTs would not affect the diffusion mechanism. We could also find a faster movement of water with a smaller water density. The structural relaxation of the confined water is analyzed to understand the density effect (see Figure 4b). The


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water chains are more stable for a smaller water density in smooth CNTs. Inversely, water chains intend to break more frequently for a smaller water density in rough CNTs.

Figure 4. (a) The logarithmic plot of MSDs of water confined in smooth (solid lines) and rough (dashed lines) CNTs. The different color represents different water content. (b) The characteristic relaxation time of confined water in CNTs.

The diffusion transition of single-file water in CNTs with roughness is illustrated in this work. According to the wave equation, the diffusion for water in smooth (6, 6) CNTs can be ballistic in nanosecond time, as water chains move along one direction and turn to the opposite after some time. While in macroscopic time scale, the diffusion would follow Fick’s laws, showing the normal Fickian mode53. The SFD mode is also found for single-file particles diffusion in nanochannels by other approaches, such as Langevin motion dynamics54. Here, the SFD has been observed in a pristine smooth CNT with the use of the Langevin thermostat, as seen in Figure 2a. Langevin thermostat is to regulate the temperature by randomly modifying the velocities of water molecules.


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Generally, when the single-file fluid is constrained in a pristine CNT, the fluid-fluid collisions do not create a random walk. This can be verified by the fact that the self-diffusion hardly shows up for water in a smooth (6 ,6) CNT, derived from the similar total and com MSDs as aforementioned. Thus, the absence of a random walk from collisions results in the absence of the square root of time dependence of the MSD, which is the characteristics of an SFD. Although adding random forces to the fluid molecules can mimics diffusive collisions, it is not a real effect for perfectly cylindrical pores54. In this work, CNT’s roughness plays a similar role as that of the Langevin thermostat. The diffusion mechanism is influenced by altering the approximately specular reflections between water and CNT walls, enabling to an extent the random movement of water. Therefore, when the roughness increases, the diffusion transits from ballistic to Fickian, and then to the SFD mode. To understand the diffusion of the water in more detail, the radial density profile (RDP), axial density profile (ADP) and dipole orientation profile (DOP) have been analyzed and shown in Figure 5. The snapshots verify the fact that water molecules cannot pass each other in the studied rough (6, 6) CNTs, showing the single-file structures. However, they are different in static properties, especially in a CNT with A/R0 = 0.1. It is obvious to find the distribution of DOP of water molecules in the roughest CNT is more dispersed. This finding demonstrates that the correlated water diffusion hardly happens in this tube, resulting in an SFD mechanism. The zigzag single-file structure is found to successively smooth with the increasing roughness. The periodic forces acted on the water molecules by roughness seem to press the water to the center and destabilize the chain-like water. From the ADP figures, the density fluctuation is larger in rougher CNTs. Suffering from the periodic forces, water molecules would distribute regularly along the axial direction. These results also support the aforementioned point that collective water motion


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would disappear gradually with increasing roughness. Thus, the slop of MSDs for water on the log–log scale decreases and the diffusion changes from ballistic to Fickian and single-file modes in rough CNTs. We also analyzed water properties as an effect of water density, there are no much differences between water in smooth and rough CNTs with the decrease of water content (see Figure S7).

Figure 5. Structure, radial density profile (RDP), axial density profile (ADP), and dipole orientation profile (DOP) along the axial direction of water in CNTs. Unites of abscissa for RDF, ADF, and DOP are nm and degrees, respectively.

Finally, we present in Figure 6 the characteristics of the interaction energies of water-CNT and water-water. All the data of interaction energy can be found in Figure S8 in the Supporting Information. As the results show, water-water interactions are generally stronger than that of water-CNT. This indicates that the water-water interaction plays a greater role in the diffusion of


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confined water. It is worth to note that compared with confined water, interactions between gas molecules confined in CNTs are weaker than that of gas-CNT55. Such difference explains the varies diffusion mechanisms observed in narrow CNTs: the ballistic for water vs. the Fickian for gases. With the increase of wall roughness, water-water interaction decreases but the averaged water-CNT interaction hardly changes. However, the local water-CNT interaction varies with roughness, revealed by the probability distribution function of interactions. As shown in Figure 6a, the interaction energy between water and the smooth CNT has a sharp peak with a large magnitude. When it comes to the rough CNTs, such peak transforms into two lower peaks. With the increase of roughness, the height of the two peaks decreases and the gap between the two peaks also increases, where the high and low peaks represent water at the peak and valley, respectively. Thus, the two-different value of interactions would lead to the random change of the velocity of water. While for water-water interaction, the distributions for all cases are nearly the same. In particular, the water-water interaction in the roughest CNT (A/R0 = 0.1) resulted a different distribution from other cases, see Figure 6b, which confirms the different configuration of water molecules under confinement. Similar distribution of interaction energy was found for water in CNTs with different density (see Figure S9). H-bonding network is another property that is sensitive to the confinement environment. In this work, we adopted a geometric criterion to define the H-bond when the distance between the acceptor (Oxygen atom) and the donor (Hydrogen atom from another water) is less than 0.35 nm and the angle of O-H with the donor Hydrogen atom is less than 30°. The energy of H-bond formed by two confined water molecules was found to be about 7 kcal/mol for single-file water in CNTs56. Thus, we can estimate the probability for a water molecule to break from the water chain, using 𝑒 −Δ𝐸⁄𝑘𝑇 , is about 8E-6 at T=300 K. The H-bond dynamics (relaxation time, tR) was also analyzed


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by the H-bond autocorrelation function (c(t)).57 The result in Figure 6c reveals that the average number of H-bonds, namely , decreases with tube roughness. It seems that the reduced number of H-bonds leaves more degrees of freedom for the water molecules to diffuse near the surface of the CNT. However, the geometrical confinement and periodic force generated by the tube hinder the water movement, leading to a slow diffusion in rough CNTs. Figure 6d shows that the relaxation time tR decreases when the roughness increases. For water in a smooth (6, 6) CNT, H-bonds hardly break and can persist for a long time. They can move in a collective way, indicative of the ballistic diffusion. For water in a small rough (6, 6) CNT, such as A/R0 = 0.01, or 0.03, and tR are almost the same as those in a pristine smooth CNT, although the diffusion mechanism changes. This indicates that, to some extent, H-bonds itself cannot determine the diffusion mechanism of confined water. For water in a rougher (6, 6) CNT (A/R0 = 0.1), tR decreases significantly, leading to a more random movement of confined water molecules. Thus, water molecules diffuse in the SFD mode. We also found the decreasing averaged number of H-bonds and relaxation time with the decreasing water density, as seen from Figure S9.


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Figure 6. The distribution of water-CNT (a) and water-water interactions (b) of water in various (6, 6) CNTs; (c) Average number of hydrogen bonds (black) and relaxation time (red) per molecule as a function of tube roughness; (d) the intermittent hydrogen bond correlation function of water as a function of time in various CNTs.

CONCLUSIONS Various diffusion mechanisms have been reported for water confined in narrow CNTs. In this work, we designed a serial of rough CNTs and calculated diffusion coefficients and mechanisms of confined water. Our results demonstrate that the diffusion mechanism depends strongly on the roughness. The diffusion follows the ballistic mode in smooth CNTs. The diffusion mode transits to the Fickian type, when small roughness was added to the CNT model. In CNTs with great roughness, the single-file diffusion mode was observed. No diffusion transition was observed in a larger (10, 10) CNT, even with various roughness levels: only Fickian diffusion model was


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observed. In addition, our calculations support that the interaction between water and rough (6, 6) CNTs is heterogeneous, so that some water molecules change their velocities randomly. When the roughness is large enough, the hydrogen bonding network will break, and the diffusion will transit to the single-file diffusion mode. The theoretical calculations reveal the roughness assisted diffusion transition in narrow CNTs, which can inspire future materials design for controllable diffusions in nanoporous mediums.

ASSOCIATED CONTENT Supporting Information. Additional details of the simulation models, the calculation of pore size distribution, the temperature and energy of water, the repeated calculations, the center of mass MSDs, the displacements of water molecules, the water properties as a function of water density, and the interaction energy analysis. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] (Xiaohua Lu), [email protected] (Linghong Lu). ORCID Wei Cao: 0000-0001-5227-7632 Liangliang Huang: 0000-0003-2358-9375 Notes The authors declare no competing financial interest.


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ACKNOWLEDGMENT This work was supported by the National Basic Research Program of China (Grant No. 2015CB655301), National Natural Science Foundation of China (NSFC) (Grant Nos, 21676137 and 21490584). L.L.H. acknowledge the U.S. National Science Foundation (NSF) for support through grant CHE-1710102, and the support from NSFC (Grant No. 21676232). M.M. acknowledge the Thousand Young Talents Program and the NSFC (Grant Nos. 11632009 and 11772168). REFERENCES 1. Werber, J. R.; Osuji, C. O.; Elimelech, M. Materials for next-generation desalination and water purification membranes. Nat. Rev. Mater. 2016, 1, 16018. 2. Hummer, G.; Rasaiah, J. C.; Noworyta, J. P. Water conduction through the hydrophobic channel of a carbon nanotube. Nature 2001, 414, 188-190. 3. Holt, J. K.; Park, H. G.; Wang, Y. M.; Stadermann, M.; Artyukhin, A. B.; Grigoropoulos, C. P.; Noy, A.; Bakajin, O. Fast mass transport through sub-2-nanometer carbon nanotubes. Science 2006, 312, 1034-1037. 4. Whitby, M.; Cagnon, L.; Thanou, M.; Quirke, N. Enhanced fluid flow through nanoscale carbon pipes. Nano Lett. 2008, 8, 2632-2637. 5. Ma, M.; Grey, F.; Shen, L.; Urbakh, M.; Wu, S.; Liu, J. Z.; Liu, Y.; Zheng, Q. Water transport inside carbon nanotubes mediated by phonon-induced oscillating friction. Nat. Nanotechnol. 2015, 10, 692-695. 6. Chakraborty, S.; Kumar, H.; Dasgupta, C.; Maiti, P. K. Confined water: structure, dynamics, and thermodynamics. Acc. Chem. Res. 2017, 50, 2139-2146. 7. Secchi, E.; Marbach, S.; Nigues, A.; Stein, D.; Siria, A.; Bocquet, L. Massive radius-dependent flow slippage in carbon nanotubes. Nature 2016, 537, 210-213. 8. Guo, S.; Meshot, E. R.; Kuykendall, T.; Cabrini, S.; Fornasiero, F. Nanofluidic transport through isolated carbon nanotube channels: advances, controversies, and challenges. Adv. Mater. 2015, 27, 5726-5737.


24

Page 25 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


9. Thomas, M.; Corry, B.; Hilder, T. A. What have we learnt about the mechanisms of rapid water transport, ion rejection and selectivity in nanopores from molecular simulation? Small 2014, 10, 1453-1465. 10. Pascal, T. A.; Goddard, W. A.; Jung, Y. Entropy and the driving force for the filling of carbon nanotubes with water. P. Natl. Acad. Sci. USA. 2011, 108, 11794-11798. 11. Dalla Bernardina, S.; Paineau, E.; Brubach, J. B.; Judeinstein, P.; Rouziere, S.; Launois, P.; Roy, P. Water in carbon nanotubes: the peculiar hydrogen bond network revealed by infrared spectroscopy. J. Am. Chem. Soc. 2016, 138, 10437-10443. 12. Das, A.; Jayanthi, S.; Deepak, H. S. M. V.; Ramanathan, K. V.; Kumar, A.; Dasgupta, C.; Sood, A. K. Single-file diffusion of confined water inside SWNTs: an NMR Study. Acs Nano 2010, 4, 1687-1695. 13. Hassan, J.; Diamantopoulos, G.; Gkoura, L.; Karayanni, M.; Alhassan, S.; Katsiotis, M. S.; Karagiannis, T.; Fardis, M.; Panopoulos, N.; Kim, H. J., et al. Ultrafast ballistic diffusion of nanotubular water shells in carbon nanotubes. Direct experimental evidence with two-dimensional (D-T2) NMR. 2016, arXiv:1612.01161. 14. Liu, X.; Pan, X.; Zhang, S.; Han, X.; Bao, X. Diffusion of water inside carbon nanotubes studied by pulsed field gradient NMR spectroscopy. Langmuir 2014, 30, 8036-8045. 15. Tasis, D.; Tagmatarchis, N.; Bianco, A.; Prato, M. Chemistry of Carbon Nanotubes. Chem. Rev. 2006, 106, 1105-1136. 16. Zheng, Y. G.; Ye, H. F.; Zhang, Z. Q.; Zhang, H. W. Water diffusion inside carbon nanotubes: mutual effects of surface and confinement. Phys. Chem. Chem. Phys. 2012, 14, 964-971. 17. Garate, J. A.; Perez-Acle, T.; Oostenbrink, C. On the thermodynamics of carbon nanotube single-file water loading: free energy, energy and entropy calculations. Phys. Chem. Chem. Phys. 2014, 16, 5119-5128. 18. Mukherjee, B.; Maiti, P. K.; Dasgupta, C.; Sood, A. K. Strong correlations and Fickian water diffusion in narrow carbon nanotubes. J. Chem. Phys. 2007, 126, 124704. 19. Mark, P.; Nilsson, L. Structure and dynamics of the TIP3P, SPC, and SPC/E water models at 298 K. J. Phys. Chem. A 2001, 105, 9954-9960. 20. Striolo, A. The mechanism of water diffusion in narrow carbon nanotubes. Nano Lett. 2006, 6, 633-639.


25


Page 26 of 29

21. Basconi, J. E.; Shirts, M. R. Effects of temperature control algorithms on transport properties and kinetics in molecular dynamics simulations. J. Chem. Theory Comput. 2013, 9, 2887-2899. 22. Thomas, M.; Corry, B. Thermostat choice significantly influences water flow rates in molecular dynamics studies of carbon nanotubes. Microfluid. Nanofluid. 2014, 18, 41-47. 23. Gubbins, K. E.; Liu, Y. C.; Moore, J. D.; Palmer, J. C. The role of molecular modeling in confined systems: impact and prospects. Phys. Chem. Chem. Phys. 2011, 13, 58-85. 24. Qin, X.; Yuan, Q.; Zhao, Y.; Xie, S.; Liu, Z. Measurement of the rate of water translocation through carbon nanotubes. Nano Lett. 2011, 11, 2173-2177. 25. Thomas, J. A.; McGaughey, A. J. H. Water flow in carbon nanotubes: transition to subcontinuum transport. Phys. Rev. Lett. 2009, 102, 184502. 26. Farimani, A. B.; Aluru, N. R. Spatial diffusion of water in carbon nanotubes: from fickian to ballistic motion. J. Phys. Chem. B 2011, 115, 12145-12149. 27. Mukherjee, B.; Maiti, P. K.; Dasgupta, C.; Sood, A. K. Single-file diffusion of water inside narrow carbon nanorings. Acs Nano 2010, 4, 985-991. 28. Taghavi, F.; Javadian, S.; Hashemianzadeh, S. M. Molecular dynamics simulation of singlewalled silicon carbide nanotubes immersed in water. J. Mol. Graph. Model. 2013, 44, 33-43. 29. Zhu, F.; Schulten, K. Water and proton conduction through carbon nanotubes as models for biological channels. Biophys. J. 2003, 85, 236-244. 30. Won, C. Y.; Aluru, N. R. Water permeation through a subnanometer boron nitride nanotube. J. Am. Chem. Soc. 2007, 129, 2748-2749. 31. Alexiadis, A.; Kassinos, S. Self-diffusivity, hydrogen bonding and density of different water models in carbon nanotubes. Mol. Simulat. 2008, 34, 671-678. 32. Sahu, P.; Ali, S. M.; Shenoy, K. T. Thermodynamics of fluid conduction through hydrophobic channel of carbon nanotubes: the exciting force for filling of nanotubes with polar and nonpolar fluids. J. Chem. Phys. 2015, 142, 074501. 33. Chen, Q.; Wang, Q.; Liu, Y. C.; Wu, T. The effect of hydrogen bonds on diffusion mechanism of water inside single-walled carbon nanotubes. J. Chem. Phys. 2014, 140, 214507. 34. Won, C. Y.; Aluru, N. Structure and dynamics of water confined in a boron nitride nanotube. J. Phys. Chem. C 2008, 112 , 1812-1818. 35. Xu, B.; Li, Y.; Park, T.; Chen, X. Effect of wall roughness on fluid transport resistance in nanopores. J. Chem. Phys. 2011, 135, 144703.


26

Page 27 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


36. Moghimi Kheirabadi, A.; Moosavi, A. Water electrolyte transport through corrugated carbon nanopores. Phys. Rev. E 2014, 90, 012304. 37. Joseph, S.; Aluru, N. R. Why are carbon nanotubes fast transporters of water? Nano Lett. 2008, 8, 452-458. 38. Thomas, J. A.; Turney, J. E.; Iutzi, R. M.; Amon, C. H.; McGaughey, A. J. H. Predicting phonon dispersion relations and lifetimes from the spectral energy density. Phys. Rev. B 2010, 81, 081411. 39. Chen, C.; Ma, M.; Jin, K.; Liu, J. Z.; Shen, L.; Zheng, Q.; Xu, Z. Nanoscale fluid-structure interaction: Flow resistance and energy transfer between water and carbon nanotubes. Physical Review E 2011, 84, 046314. 40. Insepov, Z.; Wolf, D.; Hassanein, A. Nanopumping using carbon nanotubes. Nano Lett. 2006, 6, 1893-1895. 41. Clorennec, D.; Royer, D. Analysis of surface acoustic wave propagation on a cylinder using laser ultrasonics. Appl. Phys. Lett. 2003, 82, 4608-4610. 42. Natsuki, T.; Ni, Q.-Q.; Endo, M. Wave propagation in single-and double-walled carbon nanotubes filled with fluids. J. Appl. Phys. 2007, 101, 034319 43. Liu, K.; Hong, X.; Wu, M.; Xiao, F.; Wang, W.; Bai, X.; Ager, J. W.; Aloni, S.; Zettl, A.; Wang, E., et al. Quantum-coupled radial-breathing oscillations in double-walled carbon nanotubes. Nat. Comm. 2013, 4, 1375. 44. Berendsen, H.; Grigera, J.; Straatsma, T. The missing term in effective pair potentials. J. Phys. Chem. 1987, 91, 6269-6271. 45. Vega, C.; Abascal, J. L. Simulating water with rigid non-polarizable models: a general perspective. Phys. Chem. Chem. Phys. 2011, 13, 19663-19688. 46. Shao, Q.; Zhou, J.; Lu, L.; Lu, X.; Zhu, Y.; Jiang, S. Anomalous hydration shell order of Na + and K+ inside carbon nanotubes. Nano Lett. 2009, 9, 989-994. 47. Plimpton, S. Fast parallel algorithms for short-range molecular-dynamics. J. Comput. Phys. 1995, 117, 1-19. 48. Moore, J. D.; Palmer, J. C.; Liu, Y. C.; Roussel, T. J.; Brennan, J. K.; Gubbins, K. E. Adsorption and diffusion of argon confined in ordered and disordered microporous carbons. Appl. Surf. Sci. 2010, 256, 5131-5136.


27


Page 28 of 29

49. Garate, J. A.; Perez-Acle, T.; Oostenbrink, C. On the thermodynamics of carbon nanotube single-file water loading: free energy, energy and entropy calculations. Phys. Chem. Chem. Phys. 2014, 16, 5119-5128. 50. Jiao, S.; Xu, Z. Non-continuum intercalated water diffusion explains fast permeation through graphene oxide membranes. ACS Nano 2017, 11, 11152-11161 51. Hahn, K.; Karger, J. Molecular dynamics simulation of single-file systems. J. Phys. Chem. 1996, 100, 316-326. 52. Striolo, A. Water self-diffusion through narrow oxygenated carbon nanotubes. Nanotechnology 2007, 18, 475704. 53. Nelson, P. H.; Auerbach, S. M. Self-diffusion in single-file zeolite membranes is Fickian at long times. J. Chem. Phys. 1999, 110, 9235. 54. Mon, K.; Percus, J. Molecular dynamics simulation of anomalous self-diffusion for single-file fluids. J. Chem. Phys. 2003, 119, 3343-3346. 55. Cao, W.; Tow, G. M.; Lu, L.; Huang, L.; Lu, X. Diffusion of CO2/CH4 confined in narrow carbon nanotube bundles. Mol. Phys. 2016, 114, 2530-2540. 56. Zhou, X.; Li, C. Q.; Iwamoto, M. Equilibrium and kinetics: water confined in carbon nanotubes as one-dimensional lattice gas. J. Chem. Phys. 2004, 121, 7996-8002. 57. Cao, W.; Lu, L.; Huang, L.; Dong, Y.; Lu, X. Molecular behavior of water on titanium dioxide nanotubes: a molecular dynamics simulation Study. J. Chem. Eng. Data 2016, 61, 4131-4138.


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82x47mm (150 x 150 DPI)


Water in Narrow Carbon Nanotubes: Roughness Promoted Diffusion

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