Driving Water through Sub-2-Nanometer Carbon Nanotubes

: The ultra-low friction observed between water and carbon nanotubes has been extensively reported recently. In this study, we delve into the factors influencing the liquid–solid friction, including surface properties such as surface wettability and roughness of carbon nanotubes, as well as the driving forces involving temperature gradient and pressure drop. Utilizing non-equilibrium molecular dynamics simulations on carbon nanotube models with a diameter of ~1 nm, we observe a significant increase in water flux within a specific range of wettability, independent of roughness. This range is expected to shift to smaller values with increased pressure drop and temperature gradient. Both the mechanical transport coefficient and the thermo-osmosis coefficient exhibit a negative correlation with wettability, and roughness further decreases these coefficients. Through this work, we provide insights into the effects of surface properties on fluid transport through nanopores, contributing valuable information for the optimization of high-performance membrane processes.


Introduction
Fluid confinement plays a pivotal role in lubrication processes.In the regime of hydration lubrication, lubricant molecules are confined between solid surfaces at distances of approximately 1-3 nm [1].Within this confined space, they can form highly tenacious layers characterized by stronger interactions between the fluid and solid surfaces compared to intermolecular interactions within the fluid itself.These layers contribute to significant normal load support and result in low friction coefficients [2,3].Remarkably, even pure water molecules confined between graphene surfaces can exhibit super-low friction.This phenomenon is attributed to the formation of structured water layers on the atomically smooth graphene surfaces [4,5].Structured water has been reported to exhibit intriguing properties, such as the observation of square ice phases at room temperature [6] and anomalously low dielectric constants [7].It is of particular interest to investigate how the properties of structured water change when confined to one-dimensional surfaces, such as carbon nanotubes (CNTs).The transition from two-dimensional to one-dimensional confinement in CNTs presents a unique opportunity to explore the behavior of structured water under increasingly restricted spatial dimensions.
Water in CNTs has been extensively investigated over the last two decades, beginning with the inaugural simulation work in 2001 [8][9][10][11][12][13][14][15][16][17][18][19][20].This research reported a swift water flux through a narrow CNT, comparable to the biological channel aquaporin-1 [8].In contrast to bulk water, water confined in CNTs exhibits a more ordered structure and anisotropic interactions, coupled with weak water-CNT interactions that result in frictionless flow [21][22][23][24].The characterization of water flow often involves the concept of slip length, with experiments indicating slip lengths ranging from nanometers to thousands of nanometers [25,26].Dedicated measurements on single-digit CNTs demonstrate that the slip length of water monotonically increases with decreasing CNT diameter, exhibiting nonlinear behavior when the diameter is less than 2 nm [13,15].Recently, theoretical work has unveiled the mechanism behind this phenomenon, attributing the size-dependent water-CNT friction mainly to quantum friction [27].Quantum friction increases with the coupling effect between water dipoles and electron excitations in CNTs.While the quantum friction is pivotal, classical friction arising from van der Waals (vdW) interactions between water and CNTs remains predominant [27].
Water-CNT friction can be tuned by manipulating the properties of CNT walls, such as size, wettability, roughness, etc. [28,29].Regarding the surface wettability, water flow enhancement (defined by the flow rates relative to the no-slip Hagen-Poiseuille equation) in CNTs increases exponentially with the contact angle, showing significant variation to about seven orders of magnitude [30,31].This observation holds for larger CNTs but behaves differently for smaller diameters [32], where the pore entrance effect plays a more significant role [33].Concerning roughness, it is reported that the friction between water and CNTs would decrease [34].Previous studies have observed alterations in the water diffusion mechanism in rough, narrow CNTs [35].For instance, diffusion transitions from ballistic to single-file due to the disruption of water chains by the rough wall have been observed [35].Despite numerous investigations documenting changes in water flow, the impact of the coupling effect of roughness and wettability on water flow in CNTs, particularly in small sizes like 1.4 nm where continuum hydrodynamics breaks down [36], remains unexplored.In a recent theoretical study, the influence of surface wettability and roughness on the slip length of the liquid-solid interface in two-dimensional systems was investigated.The study concluded that the slip length exhibits a strong dependence on both surface wettability and roughness, with the slip length inversely proportional to both the strength of the interaction and the degree of roughness [28].However, it is important to recognize that the physical mechanisms governing slip length may differ substantially when considering liquids confined within one-dimensional nanochannels.
This study investigates water transport through narrow CNTs with a diameter of approximately 1 nm using non-equilibrium molecular dynamics (MD) simulations.This research aims to elucidate the influence of tube roughness and wettability, as well as their combined effect, on water flux.Additionally, this study explores the impact of the driving force on water flow, including pressure drop and temperature difference.Furthermore, the mechanical transport coefficient and the thermo-osmosis coefficient are examined in CNTs with varying surface wettability and roughness.

Simulation Methods
The MD simulations were conducted using the LAMMPS package [37].Armchair CNTs were served as nanotube models, with the chiral index set to (8,8), resulting in a diameter (d) of 1.086 nm.Additionally, a smaller (6,6) CNT with a diameter of 0.814 nm was examined for comparative analysis.Nanotube wall wettability was manipulated by adjusting the vdW interaction between carbon in CNTs and the oxygen of water molecules.To introduce wall roughness (see Figure 1a), the tube diameter was modified using a sinusoidal function: d = d 0 + Asin(2πz/λ), where d 0 represents the diameter of the pristine CNT, d is the diameter along the axial direction, A is the amplitude of roughness, and λ is the wavelength.This type of roughness was denoted as Rough_I.Another form of roughness, referred to as Rough_II, was introduced, involving bending of the nanotube wall along the radial direction according to y = y 0 + Asin(2πz/λ).For the model of Rough_II, the diameter of the CNTs is a constant.In this work, λ/d 0 was set to 1.0, and A/d 0 = δ was varied as 0, 0.01, 0.02, 0.03, 0.04, and 0.06.Additional details on the rough models and their rationale were discussed in a previous study [35].While experimental characterization of these structures is challenging, theoretical investigations of these models can yield new insights into physics and inspire novel experimental approaches.It is noted that the change in the reactivity of carbon atoms on the curved CNTs [38,39] are not considered.
In the calculations, the TIP3P model was employed to simulate water molecules.The interaction between water and CNTs was represented by the Lennard-Jones potential, with parameters ε C−O ranging from 0.02 to 0.14 kcal/mol.Notably, these parameters have been previously utilized to illustrate the hydrophilicity effect of the tube on water permeation [32], which is usually reflected by functional groups on CNTs.Therefore, we refrained from exploring other water models, acknowledging that different water models may yield different outcomes [40].In the simulations, the carbon atoms were held fixed, which could inhibit the thermal energy transfer between water and CNTs.While flexibility can reduce water-CNT friction [41], maintaining a fixed structure ensures the preservation of rough structures, which is the focus of this work.
Lubricants 2024, 12, 220 3 of 11 with parameters  ranging from 0.02 to 0.14 kcal/mol.Notably, these parameters have been previously utilized to illustrate the hydrophilicity effect of the tube on water permeation [32], which is usually reflected by functional groups on CNTs.Therefore, we refrained from exploring other water models, acknowledging that different water models may yield different outcomes [40].In the simulations, the carbon atoms were held fixed, which could inhibit the thermal energy transfer between water and CNTs.While flexibility can reduce water-CNT friction [41], maintaining a fixed structure ensures the preservation of rough structures, which is the focus of this work.The simulation model for water transport through CNTs is depicted in Figure 1b.The CNTs have a length of approximately 2 nm.The length of the tube can indeed impact water flow through CNTs [42], although the investigation of varying tube lengths is beyond the scope of this study.Two sufficiently large reservoirs were positioned on either side of the tube, separated by two graphene walls along the tubes.The dimensions of the system are 4 × 4 × 20 nm³, containing a total of 6000 water molecules.Pressure drop was induced by two pistons at the edges of the reservoirs.Temperature differences were introduced by controlling the temperature of water molecules in the reservoirs.Initially, the energy of the simulation system was minimized, followed by dynamic simulations coupled to Nose-Hoover thermostats.After achieving equilibrium for 2 ns, the driving force was applied to the system, and an additional 10 ns of simulation were conducted to observe water flux through the CNTs.The dynamics of Newton's equation were iterated using a 2.0 fs time step.Steady-state water flow examples, represented by the linear relationship between number of water molecules transported through CNTs and the simulation time, are illustrated in Figure 1c.Water flux was determined from the slope of these curves.
The contact angles of water on a graphene model with similar  values were computed to characterize wettability, resulting in values of 125.5°, 102.9°, 81.2°, and 60.9° for  equal to 0.02, 0.05, 0.075, and 0.1 kcal/mol, respectively.For the largest  value (0.14 kcal/mol), water can spread on the surface.Experimental measurements of the contact angle on a clean graphite surface yield a value of 70° [43], corresponding to an  of 0.075-0.1 kcal/mol.The effect of roughness on wettability was also investigated using a rough graphene model (see Figure 2).Generally, for the roughness studied in this work, the roughness could increase the contact area between water and the surface, The simulation model for water transport through CNTs is depicted in Figure 1b.The CNTs have a length of approximately 2 nm.The length of the tube can indeed impact water flow through CNTs [42], although the investigation of varying tube lengths is beyond the scope of this study.Two sufficiently large reservoirs were positioned on either side of the tube, separated by two graphene walls along the tubes.The dimensions of the system are 4 × 4 × 20 nm³, containing a total of 6000 water molecules.Pressure drop was induced by two pistons at the edges of the reservoirs.Temperature differences were introduced by controlling the temperature of water molecules in the reservoirs.Initially, the energy of the simulation system was minimized, followed by dynamic simulations coupled to Nose-Hoover thermostats.After achieving equilibrium for 2 ns, the driving force was applied to the system, and an additional 10 ns of simulation were conducted to observe water flux through the CNTs.The dynamics of Newton's equation were iterated using a 2.0 fs time step.Steady-state water flow examples, represented by the linear relationship between number of water molecules transported through CNTs and the simulation time, are illustrated in Figure 1c.Water flux was determined from the slope of these curves.
The contact angles of water on a graphene model with similar ε C−O values were computed to characterize wettability, resulting in values of 125.5 • , 102.9 • , 81.2 • , and 60.9 • for ε C−O equal to 0.02, 0.05, 0.075, and 0.1 kcal/mol, respectively.For the largest ε C−O value (0.14 kcal/mol), water can spread on the surface.Experimental measurements of the contact angle on a clean graphite surface yield a value of 70 • [43], corresponding to an ε C−O of 0.075-0.1 kcal/mol.The effect of roughness on wettability was also investigated using a rough graphene model (see Figure 2).Generally, for the roughness studied in this work, the roughness could increase the contact area between water and the surface, thereby influencing hydrophilicity or hydrophobicity.It can be observed in Figure 2 that the contact angle increases for hydrophobic surfaces (0.02 kcal/mol) and decreases for hydrophilic surfaces (0.1 kcal/mol) with increasing surface roughness.The wettability remains relatively unchanged in the range of 0.05-0.075kcal/mol.It is important to note that the wettability of curved nanotubes, which is difficult to simulate accurately, may differ from that of two-dimensional surfaces.These results are provided to elucidate the role of the interaction parameter, ε C−O , in determining the measurable contact angle.
Lubricants 2024, 12, 220 4 of 11 thereby influencing hydrophilicity or hydrophobicity.It can be observed in Figure 2 that the contact angle increases for hydrophobic surfaces (0.02 kcal/mol) and decreases for hydrophilic surfaces (0.1 kcal/mol) with increasing surface roughness.The wettability remains relatively unchanged in the range of 0.05-0.075kcal/mol.It is important to note that the wettability of curved nanotubes, which is difficult to simulate accurately, may differ from that of two-dimensional surfaces.These results are provided to elucidate the role of the interaction parameter,  , in determining the measurable contact angle.

Coupling Effect of Tube Roughness and Wettability
The water flow through nanopores is typically calculated using the following formula: where  represents the flow rate,  is the nanopore radius,  is the length of nanopore,  is the viscosity,  is slip length, and  is related to the pore entrance [33].This equation is generally applicable, but its suitability is limited to narrow spaces, such as narrow CNTs with a diameter of less than 2 nm.In such cases, surface diffusion, , becomes predominant over viscosity, , in determining the flow rate [44].Surface wettability has been shown to be closely related to the slip length of water on surfaces, expressed as  ∝ 1  , where  is the contact angle.This equation, derived from two-dimensional surfaces [45], is also applicable to nanotube models [30].The contact angle is determined via interfacial interactions, yielding  ∝ 1  , leading to  ∝  .Therefore, based on the aforementioned discussions, for large CNTs, the flow flux  is expected to decrease as  increases, assuming viscosity remains relatively constant.In small CNTs ( 2 ), the nonlinear dependence of  on  is reported [32], which is attributed to the dominant role of surface diffusion (compared to  in large CNTs) and the pore entrance effect, .
The nonlinear relationship between  and  is illustrated in Figure 3 at a pressure drop of Δ 100 MPa, and this relationship hardly change in rough tubes.The water flux initially increases sharply within a narrow range of  , roughly from 0.05 to 0.075 kcal/mol, and then slightly decreases for larger  (as seen in Figure 3a for the (8, 8) CNT).This behavior suggests that the surface diffusion mechanism is retained in rough models.The presence of rough walls enhances surface diffusion by introducing randomness to water clusters through the irregular tubes [35].Generally, water flux depends on the number of molecules inside the tubes and the velocity of water movement through them.The results indicate that water continues to fill the channels as the interactions

Results and Discussions Coupling Effect of Tube Roughness and Wettability
The water flow through nanopores is typically calculated using the following formula: where Q represents the flow rate, R is the nanopore radius, L is the length of nanopore, µ is the viscosity, l s is slip length, and C is related to the pore entrance [33].This equation is generally applicable, but its suitability is limited to narrow spaces, such as narrow CNTs with a diameter of less than 2 nm.In such cases, surface diffusion, D, becomes predominant over viscosity, µ, in determining the flow rate [44].Surface wettability has been shown to be closely related to the slip length of water on surfaces, expressed as where θ is the contact angle.This equation, derived from two-dimensional surfaces [45], is also applicable to nanotube models [30].The contact angle is determined via interfacial interactions, yielding Therefore, based on the aforementioned discussions, for large CNTs, the flow flux Q is expected to decrease as ε 2 C−O increases, assuming viscosity remains relatively constant.In small CNTs (d 0 < 2 nm), the nonlinear dependence of Q on ε C−O is reported [32], which is attributed to the dominant role of surface diffusion (compared to µ in large CNTs) and the pore entrance effect, C.
The nonlinear relationship between Q and ε C−O is illustrated in Figure 3 at a pressure drop of ∆P = 100 MPa, and this relationship hardly change in rough tubes.The water flux initially increases sharply within a narrow range of ε C−O , roughly from 0.05 to 0.075 kcal/mol, and then slightly decreases for larger ε C−O (as seen in Figure 3a for the (8, 8) CNT).This behavior suggests that the surface diffusion mechanism is retained in rough models.The presence of rough walls enhances surface diffusion by introducing randomness to water clusters through the irregular tubes [35].Generally, water flux depends on the number of molecules inside the tubes and the velocity of water movement through them.The results indicate that water continues to fill the channels as the interactions between water and CNTs increase (see Figure 4a).As mentioned earlier, the friction between water and CNTs increases with increasing water-CNT interactions.This is partially reflected in the velocity of water in CNTs, as depicted in Figure 4b, which is influenced by both the solidliquid friction and liquid-liquid adhesion.In tubes with small ε C−O (<0.075 kcal/mol), water molecules are unable to form a continuous water chain, resulting in small intramolecular adhesion and fast surface diffusion.The flow velocity is primarily determined between water and CNTs increase (see Figure 4a).As mentioned earlier, the friction between water and CNTs increases with increasing water-CNT interactions.This is partially reflected in the velocity of water in CNTs, as depicted in Figure 4b, which is influenced by both the solid-liquid friction and liquid-liquid adhesion.In tubes with small  (<0.075 kcal/mol), water molecules are unable to form a continuous water chain, resulting in small intra-molecular adhesion and fast surface diffusion.The flow velocity is primarily determined by surface diffusion in these tubes.Conversely, in tubes with larger  , water velocity is governed by solid-liquid friction.Consequently, the velocity first increases and then decreases, reaching a peak at  = 0.075 kcal/mol.Combining the increase in water content with  , the relationship between water flux and  establishes a transition region at 0.05-0.075kcal/mol, as observed in Figure 3a.This transition region reflects the interplay between surface diffusion and solid-liquid friction in determining water flux through the CNTs.The physical origin of the water flux behavior can be understood through the hydrogen bond network of water.In bulk water, the average number of hydrogen bonds per water molecule is approximately 3.5.Figure 4c presents profiles of the number of hydrogen bonds and water density along the axial direction of the CNTs.Firstly, the number of hydrogen bonds decreases due to nanoconfinement, indicating a reduction in hydrogen bonding within the confined water.Secondly, at the entrance to the membrane, there is a rapid drop in the degree of hydrogen bonding, suggesting an energetic penalty as water molecules enter the membrane.Hydrogen bonds must be broken for molecules to pass through the tubes.Thirdly, the number of hydrogen bonds decreases along the flow direction in hydrophobic tubes, while it remains constant in hydrophilic tubes.This indicates that water chains continue to break into smaller clusters, introducing surface diffusion to water molecules.The roughness of the tube wall induces fluctuations in the profiles of the hydrogen bonds per water molecule, which hinder water transport through the tubes.Furthermore, the average number of hydrogen bonds and hydrogen bond lifetime are analyzed in Figure 4d.With increasing water-CNT interactions, the hydrogen bond lifetime initially decreases and then increases, while the average hydrogen bond number increases linearly.A smaller lifetime implies more frequent breaking and formation of hydrogen bonds, resulting in reduced water-water adhesion and faster water movement.The snapshots of water in CNTs after steady-state are shown in Figure 4e, which conforms to the hydrogen bond profiles discussed above.The physical origin of the water flux behavior can be understood through the hydrogen bond network of water.In bulk water, the average number of hydrogen bonds per water molecule is approximately 3.5.Figure 4c presents profiles of the number of hydrogen bonds and water density along the axial direction of the CNTs.Firstly, the number of hydrogen bonds decreases due to nanoconfinement, indicating a reduction in hydrogen bonding within the confined water.Secondly, at the entrance to the membrane, there is a rapid drop in the degree of hydrogen bonding, suggesting an energetic penalty as water molecules enter the membrane.Hydrogen bonds must be broken for molecules to pass through the tubes.Thirdly, the number of hydrogen bonds decreases along the flow direction in hydrophobic tubes, while it remains constant in hydrophilic tubes.This indicates that water chains continue to break into smaller clusters, introducing surface diffusion to water molecules.The roughness of the tube wall induces fluctuations in the profiles of the hydrogen bonds per water molecule, which hinder water transport through the tubes.Furthermore, the average number of hydrogen bonds and hydrogen bond lifetime are analyzed in Figure 4d.With increasing water-CNT interactions, the hydrogen bond lifetime initially decreases and then increases, while the average hydrogen bond number increases linearly.A smaller lifetime implies more frequent breaking and formation of hydrogen bonds, resulting in reduced water-water adhesion and faster water movement.The snapshots of water in CNTs after steady-state are shown in Figure 4e, which conforms to the hydrogen bond profiles discussed above.
Similar behavior of water flux as a function of water-carbon interactions is found in smaller (6,6) CNTs (see Figure 3c), with a more pronounced decrease at larger values for  .This enhanced decrease is attributed to the greater contribution of interfacial interactions compared to water-water molecular interactions in narrower CNTs.Additionally, the two types of roughness do not alter the mechanism; however, the Rough_I model results in a larger decline in water flux compared to the Rough_II model, primarily due to the constant diameter of the Rough_II model.Similar behavior of water flux as a function of water-carbon interactions is found in smaller (6,6) CNTs (see Figure 3c), with a more pronounced decrease at larger values for ε C−O .This enhanced decrease is attributed to the greater contribution of interfacial interactions compared to water-water molecular interactions in narrower CNTs.Additionally, the two types of roughness do not alter the mechanism; however, the Rough_I model results in a larger decline in water flux compared to the Rough_II model, primarily due to the constant diameter of the Rough_II model.
The influence of roughness on water flow in CNTs with varying hydrophilicity is elucidated in Figure 3b,d.We initially address the effect of roughness on surface wettability in Figure 2. It is found that a rougher surface enhances the hydrophobicity (or hydrophilicity) of the tube wall within the range of studied structural parameters.The results reveal that the roughness-induced change is within ±7 • , which has a negligible impact on the water flow rate.In the Wenzel state, a rougher surface leads to a smaller slip length decline on a two-dimensional surface [28].However, this situation in narrow CNTs should differ significantly from two-dimensional surfaces.The outcomes indicate that hydrophobic tubes (ε C−O ≤ 0.05 kcal/mol) show no dependence on roughness due to very-weak interfacial interaction.Conversely, within hydrophilic tubes (ε C−O ≥ 0.075 kcal/mol), water molecules are distributed closer to the walls, leading to the generation of frictional forces by corrugated walls.This can break the tight hydrogen bond network, further weakening the pulse-like transmission behavior [8].The minor fluctuations for small δ arise from the synergistic effect of roughness-induced wettability and friction.

The Effect of Driving Force
The effect of driving force on water transport through CNTs is explored, including both the pressure and temperature drop.For hydrodynamic flow through nanopores, Q depends on both ∆P and ∆T, expressed as follows: where ∆P = P h − P l , ∆T = T h − T l , T avg = (T h + T l )/2, k M is the mechanical transport coefficient, and k T is the thermo-osmosis coefficient [46].The water flux as a function of pressure drop is demonstrated.In hydrophilic tubes (ε C−O = 0.1 kcal/mol), water flux shows linear dependence on the pressure drop for both nanotube sizes ((8, 8) and (6, 6) CNTs), as shown in Figure 5a.However, this linear dependence breaks for hydrophobic tubes (ε C−O = 0.05 kcal/mol) at ∆P < 100-150 MPa.In such cases, the inertial loss from the entrance likely dominates the transport process, preventing water from entering the tubes effectively.Additionally, hydrophobic tubes are unable to accommodate enough water molecules (see Figure 5b), leading to discontinuous water clusters, where the pressure-induced momentum transfer is partially hindered.The slope of the curves in Figure 5a indicates that hydrophobic tubes have larger k M than hydrophilic ones.The results are comparable to those reported in a recent simulation work [36].It is important to note that the slope for hydrophobic tubes is calculated based on the linear part with large pressure drops.The difference in k M can be explained by the fact that when the pressure is large enough to fill CNTs with water (see Figure 5b), solid-liquid friction (or slip length) determines the flow rate.Given that the slip length, if any, is proportional to ε −2 C−O , k m thus has a negative correlation with ε C−O .Consequently, it is expected that water flux in hydrophobic CNTs will be larger than that in hydrophilic CNTs with a further increase in the pressure difference.
bic tubes ( 0.05 kcal/mol) show no dependence on roughness due to very-weak interfacial interaction.Conversely, within hydrophilic tubes ( 0.075 kcal/mol), water molecules are distributed closer to the walls, leading to the generation of frictional forces by corrugated walls.This can break the tight hydrogen bond network, further weakening the pulse-like transmission behavior [8].The minor fluctuations for small  arise from the synergistic effect of roughness-induced wettability and friction.

The Effect of Driving Force
The effect of driving force on water transport through CNTs is explored, including both the pressure and temperature drop.For hydrodynamic flow through nanopores,  depends on both Δ and Δ, expressed as follows: where Δ   , Δ   ,    /2,  is the mechanical transport coefficient, and  is the thermo-osmosis coefficient [46].The water flux as a function of pressure drop is demonstrated.In hydrophilic tubes ( 0.1 kcal/mol), water flux shows linear dependence on the pressure drop for both nanotube sizes ((8, 8) and (6, 6) CNTs), as shown in Figure 5a.However, this linear dependence breaks for hydrophobic tubes ( 0.05 kcal/mol) at Δ 100 150 MPa.In such cases, the inertial loss from the entrance likely dominates the transport process, preventing water from entering the tubes effectively.Additionally, hydrophobic tubes are unable to accommodate enough water molecules (see Figure 5b), leading to discontinuous water clusters, where the pressure-induced momentum transfer is partially hindered.The slope of the curves in Figure 5a indicates that hydrophobic tubes have larger  than hydrophilic ones.The results are comparable to those reported in a recent simulation work [36].It is important to note that the slope for hydrophobic tubes is calculated based on the linear part with large pressure drops.The difference in  can be explained by the fact that when the pressure is large enough to fill CNTs with water (see Figure 5b), solidliquid friction (or slip length) determines the flow rate.Given that the slip length, if any, is proportional to  ,  thus has a negative correlation with  .Consequently, it is expected that water flux in hydrophobic CNTs will be larger than that in hydrophilic CNTs with a further increase in the pressure difference.Roughness is also capable of increasing friction, leading to a decrease in k m , regardless of surface wettability.Water transport through (8,8) and (6,6) CNTs exhibits the same pressure dependence, with the former having a larger k M .While the smaller (6,6) CNT has a larger slip length than (8,8) CNTs, the water flux is still smaller through the single tube due to its smaller volume.It is important to emphasize once again that k M is meaningful for tubes with ε C−O = 0.05 kcal/mol at ∆P > 100 MPa (for (8,8)) and ∆P > 150 MPa (for (6,6)).Therefore, based on the discussions above, it is anticipated that under larger ∆P, the transition region (~0.05 kcal/mol < ε C−O < ∼ 0.075 kcal/mol) found in Figure 3a,c will translate to a smaller range of ε C−O .Correspondingly, the independence of flux on roughness, as shown in Figure 3b,d, will certainly happen for smaller ε C−O .
By introducing a temperature gradient, the water flux can be further enhanced, a phenomenon known as thermo-osmosis [47], which harnesses waste heat for separation.A recent experimental work reports the temperature effect on fluid transport [48], indicative of the possibility of realizing the temperature gradient for CNT nanochannels.Although thermo-osmosis through CNTs has been recently reported [49], this paper specifically focuses on the influence of wettability and roughness.To create a temperature gradient, the water molecules in the feed, situated 5 Å away from the tube, were set to T h (273-473 K), and those in the permeate, also 5 Å away from the tube, were set to T l = 298 K.No thermostat was applied to water in the middle region.The flexibility of CNTs was considered to evaluate the temperature effect on water flow, with intramolecular interactions described via the Dreiding force field [50].The pressure drop of ∆P = 200 MPa is used to calculate the water flow, where a linear dependence of Q on ∆P is demonstrated (see Figure 5).
Figure 6 illustrates the calculated (Q − k M ∆P) as a function of T h .Here we have not considered the evaporation of water at high temperature (T h > 373 K), due to the large pressure in the feed.Our results are comparable to the reported values.For instance, the flow velocity in (8, 8) CNTs is ~0.79 m/s at ∆T = 50 K and ε C−O = 0.1 kcal/mol, compared to ~0.7 m/s at ∆T = 60 K [49].The water flux is about 8.1 molecules/ns −1 at ε C−O = 0.1 kcal/mol, and ∆T = 25 K through (6,6) CNTs, compared to the 0.89 molecules/ns −1 at ∆T = 15 K reported in a recent simulation work [51].Our results are larger, possibly due to the difference in T avg and the details of their tube models (CNT bundle, length, force field, etc.) [51].
pressure dependence, with the former having a larger  .While the smaller (6,6) CNT has a larger slip length than (8,8) CNTs, the water flux is still smaller through the single tube due to its smaller volume.It is important to emphasize once again that  is meaningful for tubes with  = 0.05 kcal/mol at Δ 100 MPa (for (8,8)) and Δ 150 MPa (for (6,6)).Therefore, based on the discussions above, it is anticipated that under larger Δ, the transition region (~0.05 kcal/mol  ~0.075 kcal/mol) found in Fig- ure 3a,c will translate to a smaller range of  .Correspondingly, the independence of flux on roughness, as shown in Figure 3b,d, will certainly happen for smaller  .By introducing a temperature gradient, the water flux can be further enhanced, a phenomenon known as thermo-osmosis [47], which harnesses waste heat for separation.A recent experimental work reports the temperature effect on fluid transport [48], indicative of the possibility of realizing the temperature gradient for CNT nanochannels.Although thermo-osmosis through CNTs has been recently reported [49], this paper specifically focuses on the influence of wettability and roughness.To create a temperature gradient, the water molecules in the feed, situated 5 Å away from the tube, were set to  273 473 K , and those in the permeate, also 5 Å away from the tube, were set to  298 K.No thermostat was applied to water in the middle region.The flexibility of CNTs was considered to evaluate the temperature effect on water flow, with intramolecular interactions described via the Dreiding force field [50].The pressure drop of Δ 200 MPa is used to calculate the water flow, where a linear dependence of  on  is demonstrated (see Figure 5).
Figure 6 illustrates the calculated   Δ as a function of  .Here we have not considered the evaporation of water at high temperature ( 373 K), due to the large pressure in the feed.Our results are comparable to the reported values.For instance, the flow velocity in (8,8) CNTs is ~0.79 m/s at Δ 50 K and  0.1 kcal/mol, compared to ~0.7 m/s at Δ 60 K [49].The water flux is about 8.1 molecules/ns −1 at  0.1 kcal/mol, and Δ 25 K through (6, 6) CNTs, compared to the 0.89 molecules/ns −1 at Δ 15 K reported in a recent simulation work [51].Our results are larger, possibly due to the difference in  and the details of their tube models (CNT bundle, length, force field, etc.) [51].The thermo-osmosis coefficient k T is obtained by fitting the curves in Figure 6.Similar to k M , we also found a larger k T in hydrophobic tubes than hydrophilic ones.The previously proposed model [49] yields Q ∝ l s at a certain ∆T/T avg , thus k T has a negative correlation with ε C−O , which is similar to k M .Such results lead to the larger water flux in hydrophobic tubes than hydrophilic ones at a large ∆T.The effect of roughness and diameter effect on k T is also similar to that of k M .Additionally, we find the flexibility has a little influence on k T .

Conclusions
In summary, our non-equilibrium MD simulation methods have allowed us to investigate the impact of nanotube wall properties, including roughness and wettability, on water transport through narrow CNTs with a diameter of ~1 nm.This study also includes the exploration of the driving forces, including pressure drop and temperature gradient.The results reveal a significant increase in water flux within a specific range of wettability, independent of roughness.Interestingly, roughness reduces water flux in hydrophilic CNTs but not in hydrophobic ones.The characteristic range of wettability can be smaller when the pressure drop and temperature gradient increase.Through the regulation of the driving force, we observe that the mechanical transport coefficient and the thermo-osmosis coefficient both negatively depend on wettability, determined by solidliquid friction, and roughness can further decrease these coefficients.The diameter of the nanotubes does not influence the physical mechanisms significantly; it merely holds smaller coefficients in narrower CNTs.This work provides a detailed exploration of the interplay between surface wettability, roughness, and pressure and temperature driving forces, contributing to a better understanding of this area and offering insights for high-performance membrane processes.

Figure 1 .
Figure 1.(a) Rough CNT models, including Rough_I and Rough_II.(b) Schematic of water transport through CNTs.The arrows mean the force added to the pistons.The blue parts are water molecules, and the green atoms are CNTs.(c) The number of water molecules transported through nanotubes with various C-O interactions,  , and roughness, / .

Figure 1 .
Figure 1.(a) Rough CNT models, including Rough_I and Rough_II.(b) Schematic of water transport through CNTs.The arrows mean the force added to the pistons.The blue parts are water molecules, and the green atoms are CNTs.(c) The number of water molecules transported through nanotubes with various C-O interactions, ε C−O , and roughness, A/d 0 .

Figure 2 .
Figure 2. Snapshot of water contact angle calculation on rough graphene (/ 0.06) with  of (a) 0.1 kcal/mol and (b) 0.02 kcal/mol.(c) Contact angle as a function of water-carbon interactions,  , and roughness, / .

Figure 2 .
Figure 2. Snapshot of water contact angle calculation on rough graphene (A/d 0 = 0.06) with ε C−O of (a) 0.1 kcal/mol and (b) 0.02 kcal/mol.(c) Contact angle as a function of water-carbon interactions, ε C−O , and roughness, A/d 0 .
by surface diffusion in these tubes.Conversely, in tubes with larger ε C−O , water velocity is governed by solid-liquid friction.Consequently, the velocity first increases and then decreases, reaching a peak at ε C−O = 0.075 kcal/mol.Combining the increase in water content with ε C−O , the relationship between water flux and ε C−O establishes a transition region at 0.05-0.075kcal/mol, as observed in Figure3a.This transition region reflects the interplay between surface diffusion and solid-liquid friction in determining water flux through the CNTs.

Figure 3 .
Figure 3. Water flux as a function of water-carbon interactions,  , and the roughness,  / , through (8, 8) CNTs under a pressure drop of 100 MPa and temperature of 298 K.The flux is plotted upon  (a) and  (b) separately.The solid and open symbols represent Rough_I and Rough_II, respectively.(c,d) display the corresponding results for (6, 6) CNTs.

Figure 3 .
Figure 3. Water flux as a function of water-carbon interactions, ε C−O , and the roughness, δ = A/d 0 , through (8, 8) CNTs under a pressure drop of 100 MPa and temperature of 298 K.The flux is plotted upon ε C−O (a) and δ (b) separately.The solid and open symbols represent Rough_I and Rough_II, respectively.(c,d) display the corresponding results for (6, 6) CNTs.

Figure 4 .
Figure 4. (a) Average density and (b) velocity of water in (8, 8) CNTs as a function of water-carbon interactions,  , and roughness, / .(c) Average number of hydrogen bonds per water molecule (upper panels) and density profile (lower panels) along the tube axial direction of hydrophobic ( 0.05 kcal/mol) and hydrophilic ( 0.1 kcal/mol) CNTs.The left panels are for smooth tubes with / 0, and the right panels are for rough tubes with / 0.06.The range of CNTs is from 0 to 1.7 nm.The insides of the lower panels show the zoom-in of water density profiles in the nanotubes.(d) The average number of hydrogen bonds (solid lines, left axis) and hydrogen bond lifetime (dashed lines, right axis) of all water molecules in CNTs as a function of water-carbon interactions,  , and roughness, / .The roughness model is Rough_I.(e) Snapshot of water inside CNTs with different roughness and hydrophilicity.The red lines connecting oxygen and hydrogen atoms represent hydrogen bonds between them.

Figure 4 .
Figure 4. (a) Average density and (b) velocity of water in (8, 8) CNTs as a function of water-carbon interactions, ε C−O , and roughness, A/d 0 .(c) Average number of hydrogen bonds per water molecule (upper panels) and density profile (lower panels) along the tube axial direction of hydrophobic (ε C−O = 0.05 kcal/mol) and hydrophilic (ε C−O = 0.1 kcal/mol) CNTs.The left panels are for smooth tubes with A/d 0 = 0, and the right panels are for rough tubes with A/d 0 = 0.06.The range of CNTs is from 0 to 1.7 nm.The insides of the lower panels show the zoom-in of water density profiles in the nanotubes.(d) The average number of hydrogen bonds (solid lines, left axis) and hydrogen bond lifetime (dashed lines, right axis) of all water molecules in CNTs as a function of water-carbon interactions, ε C−O , and roughness, A/d 0 .The roughness model is Rough_I.(e) Snapshot of water inside CNTs with different roughness and hydrophilicity.The red lines connecting oxygen and hydrogen atoms represent hydrogen bonds between them.

Figure 5 .
Figure 5. (a) Water transport through nanotubes at different pressure drops, ∆P.(b) Number of water molecules inside nanotubes as a function of ∆P.The square symbols denote the hydrophobic (ε C−O = 0.05 kcal/mol) models, and circular symbols denote the hydrophilic (ε C−O = 0.1 kcal/mol) models.The red curves represent smooth models (δ = 0), and the blue curves represent rough models (δ = 0.04).The solid lines and left y-axis are for (8, 8) CNTs, and the dashed lines and right y-axis are for (6, 6) CNTs.The temperature is 298 K.The roughness model is Rough_I.

Figure 6 .
Figure 6.(a) Water flux as a function of temperature in the feed at ∆P = 200 MPa for (8, 8) CNTs.The lines represent the corresponding non-linear fitting.The flux is subtracted by that at the same ∆P without a thermal gradient.The temperature in the permeate is 298 K for all cases.The square symbols denote the smooth models (δ = 0), the circular symbols denote a rough model (δ = 0.04), and the triangular symbols denote a flexible model.The open symbols represent the hydrophobic (ε C−O = 0.05 kcal/mol) models, and solid symbols represent the hydrophilic (ε C−O = 0.1 kcal/mol) models.The roughness model is Rough_I.(b) is for (6, 6) CNTs.