Water Flow through Interlayer Channels of 2D Materials with Various

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C: Physical Processes in Nanomaterials and Nanostructures

Water Flow through Interlayer Channels of 2D Materials with Various Hydrophilicities Fang Xu, Yang Song, Mingjie Wei, and Yong Wang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04719 • Publication Date (Web): 17 Jun 2018 Downloaded from http://pubs.acs.org on June 17, 2018

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The Journal of Physical Chemistry

Water Flow through Interlayer Channels of 2D Materials with Various Hydrophilicities

Fang Xu, Yang Song, Mingjie Wei*, Yong Wang*

State Key Laboratory of Materials-Oriented Chemical Engineering, Jiangsu National Synergetic Innovation Center for Advanced Materials, and College of Chemical Engineering, Nanjing Tech University, Nanjing 210009, Jiangsu (P. R. China)

* Corresponding author. Tel.: +86-25-8317 2247; Fax: +86-25-8317 2292. E-mail: [email protected] (M. Wei); [email protected] (Y. Wang)

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Abstract Water

transport

through

laminated

membranes

composed

of

two-dimensional (2D) materials has gained considerable attention because of their great potential in filtration and separation applications. However, water transport

between

interlayers

formed

by

2D

materials

cannot

be

comprehensively described by traditional transport theory because different 2D materials have varying hydrophilicity, which strongly affects water transport. Herein, we build interlayer channels formed by 2D sheets with various hydrophilicities and investigate the pressure-driven water transport via nonequilibrium molecular dynamics simulations. The influence of channel hydrophilicity on water transport is dominant, especially at the nanoscale. To model the water transport phenomena through channels with various hydrophilicities, we define a new slip length to derive the appropriate equation. By two methods of calculating the slip length based on the simulation results, we validate our derived equation, which predicts the water flux in interlayer channels with a large range of hydrophilicities from relatively hydrophobic (large slip lengths) to extremely hydrophilic (negative slip lengths). This work deepens the understanding of water transport through interlayer channels and assists in the design of 2D-material membranes for water treatment.

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Introduction Two-dimensional (2D) materials, as exemplified by graphene1, are a fascinating yet challenging topic in the development of high-performance filtration and separation membranes.2-3 The family of 2D materials comprises different types, including graphene and its derivatives, transition metal chalcogenides and halides, oxides and hydroxides.4-5 Since the interlayer nanochannels between layers could yield fast and selective transport of water molecules, interest in these new materials tends to be focused on their laminate structures, which are characterized as having a large lateral dimension and small thickness.3, 6-8 Among these laminated membranes, graphene and graphene oxides (GOs) are mostly applied in liquid-water-related systems.6, 9-11 For example, the exceptional water transport performance of graphene-based membranes, which is far better than that predicted by traditional continuum mechanics, has been demonstrated by both experiments and simulations.10, 12-13 Nair et al.6 first reported laminate GO membranes that had an average interlayer spacing of less than ~0.7 nm and were obtained by vacuum filtration. Interestingly, water molecules can permeate through these highly confined spaces with surprisingly high permeability. The regions in GO sheets can be divided into two types: oxidized graphene and pristine graphene, which correspond to the hydrophilic and hydrophobic parts, respectively. Joshi et al.14 attributed the high water permeability of GO membranes to the weak interaction between 3 / 31



water molecules and hydrophobic regions in GO sheets, which allows the nearly frictionless flow of monolayer water. Additionally, by molecular dynamics (MD) simulations, Wei et al.9 found that fast water transport through the pristine regions can be significantly hindered by a prominent side-pinning effect of the oxidized regions, which correspond to the hydrophilic regions in GO sheets. Chen et al.15 investigated water transport through GO interlayers composed of various concentrations of hydroxyl groups on the GO surface and found that the volumetric ﬂux is negatively correlated with the hydroxyl concentration. In that study, the hydroxyl concentration represented the hydrophilicity of the GO sheets. Therefore, all the described studies indicate that the hydrophilicity of 2D materials has a significant influence on water transport inside their interlayer channels. In general, 2D materials have their own unique hydrophilicities. For example, the composition, crystalline phase and electronic structure of transition metal dichalcogenides all influence their sheet chemistry and hydrophilicity.16 Moreover, 2D-material van der Waals (vdW) heterostructures can be formed by creating stacked layers of 2D materials with different hydrophilicities.4, 17 Currently, the interlayer spacing in laminated membranes made of different 2D materials can be tuned from several angstroms to nearly 3 or 4 nanometers.7-8, 18-19 Furthermore, these laminated membranes exhibit flux differences of several orders of magnitude even when they have a similar membrane thickness and channel size. This difference in flux indicates that the 4 / 31


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transport of water molecules inside these interlayer channels is more complicated than described by the Hagen-Poiseuille (HP) equation, which does not take the pore hydrophilicity into consideration. Therefore, it is crucial to develop a reliable equation for predicting water transport in 2D slit channels with various hydrophilicities. In the investigation of water molecules subject to nanoscale confinement, experimental methods cannot easily observe the molecular details of fluid properties, especially in dynamic flow processes. Instead, MD simulations can be used to directly obtain the molecular details of fluids and investigate their viscosity and slip lengths.20-22 To determine the influence of channel hydrophilicity on the transport of water molecules, it is necessary to adjust the hydrophilicity of channels by a simple means. Since water has an electrical dipole consisting of partial negative charges on the O atom and partial positive charges on the H atoms (net neutral), one could increase the partial charge of atoms in an originally nonpolar sheet to model a polarized channel. Unlike an unmodified channel, a polarized channel can affect water via electrostatic interactions. Hence, the hydrophilicity of channels can be easily tuned by adjusting the partial charge of atoms in sheets. Similarly, some 2D materials, such as β-silicene, β-germanene, silicon carbide, hexagonal boron nitride and α-transition metal chalcogenides, have similar atomic structures to graphene but different atom charges.5 In this work, we construct an interlayer channel by placing two 5 / 31



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graphene-like sheets. We conduct steady-state nonequilibrium molecular dynamics (SS-NEMD) analysis to investigate the water molecules flowing through modified channels with various hydrophilicities. By comparing the behaviors predicted by the HP equation, we investigate the influence of channel hydrophilicity on water transport through the interlayer channels. In addition, we analyze the boundary slip conditions to study the origin of this influence. Furthermore, we develop a flux equation linked with the slip length, which is newly defined here according to the Navier-Stokes equation. The developed equation can exactly describe water transport through interlayer channels in laminated membranes made of different 2D materials with various hydrophilicities.

Theory Driven by a pressure gradient dP/dx, the steady-state flow of simple incompressible fluids with a viscosity of η in a slit channel with a width of 2h can be described by the Navier-Stokes equation. The velocity profile can be given by22:

z =

1 d

+ ℎ − 2 d

(1)

The schematic velocity profile is plotted in Figure 1. During the process of deriving this profile, we find that the definition of Ls in Equation 1 should be based on an extension of the velocity profile rather than the gradient at the interface (b in Figure 1); the latter is usually defined in Couette shear flow.23 6 / 31


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That is, the slip length in this work is the solution to Equation 1 in the condition that ux = 0; i.e., Ls can be described as:

= | − ℎ

(2)

Typically, the flux for steady flow in slit pores can be computed as24:

=

1 2ℎ

! d

(3)

Here, ρx(z) is the local density of water molecules along the z direction flowing in the x direction. It is obvious that when Ls = 0, this equation is the traditional HP equation. The flux enhancement factor, which is defined as the ratio of the slip flux to the nonslip HP flux, can be calculated as

"=

# ! ; d # ! ; = 0d

(4)

Assuming that the local density at the corresponding position is essentially constant, in this case, the flux enhancement factor can then be estimated as

"=

# ; d # ; = 0d

(5)

By substituting the velocity profile in Equation 1 into the above equation, followed by mathematical integration, the flux enhancement factor can be easily obtained:

"=

3 3 + +1 2ℎ ℎ

(6)

When the channel size is under confinement at the nanometer scale, at which the vacuum space resulting from the channel thickness cannot be ignored, h represents the hydrodynamic thickness. 7 / 31



Figure 1. Schematic of fluid flow in a slit channel and its slip length

Simulation details As shown in Figure 2(a), the model channel used in the simulation contained 7872 carbon atoms that constituted two 2D graphene sheets, with 6245 water molecules confined between the sheets. The location of the sheet plane defined the xy plane. The two sheets were set to be parallel to each other, and both had a size of 10×10 nm2. The distance between the sheets was fixed at 2.2 nm with the center at z = 0 Å. Previous works have demonstrated that water can form distinct layered structures, including monolayer and bilayer ice and different amorphous/ordered ice structures, 8 / 31


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under strong confinement, even at room temperature25-27. Moreover, the viscosity of water has also been reported to be profoundly affected by layered structures of confined water when the confined size is less than 2 nm.28 Therefore, in order to eliminate the effect of water structure and viscosity and to investigate the impact of channel hydrophilicity alone, the channel width in our system was chosen to be 2.2 nm, a distance at which the bulk phase of liquid water was present and the physical properties of water were not altered dramatically with the channel size. The simulation box lengths in the x and y directions were set equal to the size of the sheets, and the length in the z direction was set to be 6.2 nm, which was large enough to prevent interactions between the water molecules and those in the periodic images in z direction. The interlayer distance of the sheets was denoted by d, while the channel lengths in the x and y directions were denoted by L and W, respectively.

(a)

(b)

Figure 2. Geometry of the simulated system and the charge pattern used for modified channels. 9 / 31



All molecular dynamics simulations were performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) package. The SPC/E model29 was used for water molecules, and the SHAKE algorithm30 was used to constrain the bonds and angles of the water molecules. The vdW interaction between water and carbon was described by the 12-6 Lennard-Jones (L-J) potential 4ε[(σ/r)12 - (σ/r)6], with parameters ε = 0.1043 kcalmol-1 and σ = 3.302 Å. To obtain the hydrophilic channels, the interaction between the water molecules and sheets was increased by assigning a partial charge of the opposite type but the same magnitude, i.e., positive or negative (the net charge of the entire sheet is zero), to adjacent atoms in 2D sheets, as shown in Figure 2(b). The absolute values of the applied partial charges (0.0 to 3.0 e) were used to identify the different channels; e.g., the channel modified with 3.0 e was denoted the 3.0 e channel, and the 0.0, 1.0 and 2.0 e channels were designated similarly. The higher the number of partial charges assigned was, the larger the polarity and the higher the degree of hydrophilicity generated. A similar method was adopted in Wang’s work31, which assigned opposite charges to certain atoms of a hexagonal lattice to simulate a superhydrophilic surface. Goldsmith et al. also employed charge patterns similar to our method to investigate flux and salt rejection in their asymmetric nanopore model.32 The long-range Coulomb interactions were computed by using the standard Ewald summation with a root-mean-square accuracy of 10 / 31


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10-5. The cutoff distances for the vdW interactions and electrostatic interaction were set as 10 Å and 12 Å, respectively. A time step of 1 fs was used in all simulations. To establish the simulation model for SS-NEMD simulations (shown in Figure 2(a)), the channel was initially connected to two reservoirs. A pressure of 1 atm was applied to the two reservoirs to cause the water molecules to fully fill the channel at a temperature of 300 K. The two reservoirs were removed when the system reached an equilibrium state, and then periodic boundary conditions were applied in all three dimensions. Then, SS-NEMD method was used to establish a pressure drop (∆P) through the channel and maintain a steady-state water flux. For this purpose, a constant force in the positive x direction was applied to all water molecules. To give a desired ∆P between the two ends of the channel, the force f was calculated as33:

' = Δ)*/,

(7)

where n was the number of water molecules; W and d represented the lateral width and thickness of the channel, respectively; and their product Wd was the cross-sectional area of the channel. ∆P was varied from 1.6 to 200 MPa along differently modified channels. It was a very common approach in MD to apply a large ∆P to enhance the signal/noise ratio within a nanosecond time scale and to save computational costs.9, 34-36 The system undergone a constant energy input when an external force was applied to the water molecules. To maintain the dynamics of the water molecules in the channel, a Nosé-Hoover 11 / 31



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thermostat was applied to atomic 2D sheets so that the sheets would remove the excess energy. This is preferred to the method that directly thermostat the fluid atoms and constrain the sheet atoms to be rigid, which would influence the mechanical and dynamical properties of confined nanofluids.37-38 This method has been acknowledged and used by previous research that investigates the water transport through graphene-based membrane.39 In this work, the temperature of the sheet atoms was set to room temperature (300 K). According to the different hydrophilicity of the sheets, the simulations were conducted for 4 - 10 ns, during which the data were collected for at least 3 ns, which was long enough to obtain the steady-state streaming velocityv.

Results and Discussion 1. Flux and enhancement factor In this work, the water flux per unit time per cross-sectional area Q is defined as11:

=

-ρ ρ./0

(8)

where ρ and ρbulk are the density of water in the channels (the total mass of water molecules divided by the volume of the channel) and in the bulk phase (1 g/m3), respectively. For a given channel size and a given number of water molecules inside the channel, ρ is also constant; hence, the water flux only depends on the average streaming velocity,v. As shown in Figure 3, the water fluxes in the four channels appear to increase linearly with ∆P, indicating a 12 / 31


Page 13 of 31

stable pressure-driven flow. Care should be taken in selecting the ∆P values to obtain the observed fluxes while avoiding an unexpected temperature rise in the system. Although the applied ∆P values are significantly larger than those used in actual membrane processes, the linear relationships between fluxes and ∆P values validate the reliability of extrapolating the fluxes at low ∆P values.

20

20

QSimulated Qnonslip

QSimulated Qnonslip

15

PWF (m/s)

PWF (m/s)

15

10

5

0

10

5

0

1.0

1.5

2.0

2.5

3.0

0

5

∆P (MPa)

10

15

20

∆P (MPa)

(a)

(b)

10.0

10

QSimulated Qnonslip

QSimulated Qnonslip

8

PWF (m/s)

7.5

PWF (m/s)

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


5.0

6

4

2.5 2

0.0

0

35

70

105

140

0

0

60

120

180

240

300

∆P (MPa)

∆P (MPa)

(c)

(d)

Figure 3. Water fluxes in the four channels with different hydrophilicities, (a) 0.0 e, (b) 1.0 e, (c) 2.0 e, (d) 3.0 e.

13 / 31



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The simulated fluxes were compared with that predicted from the nonslip HP equation based on continuum mechanics9, in which the flux, Qnonslip, is expressed as

1213456 = −

2ℎ7 Δ) 12

(9)

where 2h denotes the hydrodynamic thickness (excluding the vacuum space from the channel thickness), ∆P represents the pressure drop between two points and η is the shear viscosity of water, which is reported to be close to the value for bulk water when the channel thickness is more than 2 nm.28 Each Qnonslip value is also normalized by cross-sectional area and plotted in Figure 3. It can be seen from Figure 3 that the actual simulated permeability gradually approaches that predicted by the HP equation as the hydrophilicity increases. When the degree of hydrophilicity reaches 3.0 e, the simulated value is even lower than the predicted value. This implies that the boundary condition of the HP equation can only represent a certain degree of hydrophilicity. Moreover, this trend can qualitatively demonstrate the influence of hydrophilicity on water flux; i.e., a higher hydrophilicity causes a greater drop in water flux. Alternatively, to quantitatively display this influence, the flow enhancement factor ε, which can be calculated by ε = Qsimulated/Qnonslip, is introduced. As shown in Table 1, the calculated enhancement factor is approximately 156.8 for the hydrophobic channel (0.0 e), indicating that the hydrophobic interactions between the 2D sheets and water molecules enhance the flux significantly more than predicted by the HP equation, which has also been 14 / 31


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observed in previous work on graphene nanosheets.35 In contrast, the enhancement factor is only approximately 0.6450 for the hydrophilic channel (3.0 e), suggesting that this degree of hydrophilicity actually exceeds the default hydrophilicity of the HP equation and that the strong interactions between the 2D sheets and water molecules hinder water transport inside the channel. The 1.0 and 2.0 e channels represent transition states from hydrophobic to hydrophilic; ε decreases progressively with rising channel hydrophilicity. Accordingly, we assume that the default degree of hydrophilicity of the HP equation should be a value between 2.0 and 3.0 e, where the enhancement factor is equal to 1. Thus, the hydrophilicity range of 0.0 - 3.0 e that we chose initially spans from extremely hydrophilic to significantly hydrophobic conditions.

Table 1. Enhancement factors and friction coefficients in four channels with different hydrophilicities Hydrophilicity ε λ (104 Ns/m3)

0.0 e

1.0 e

2.0 e

3.0 e

156.8±7.056 25.17±1.596

1.738±0.2084

0.6450±0.01543

1.33±0.038

236.10±14.02

-

8.72±0.14

As mentioned above, the driving forces were applied in the x direction, and the channels were placed in the z direction; therefore, the water molecules were still free in the y direction. Hence, in addition to the x-direction water flux, 15 / 31



the diffusion mobility of water molecules is also characterized by the mean-square displacement in the y direction (MSDy), as shown in Figure 4. In accordance with the water fluxes, the diffusion coefficients also rely on the channel hydrophilicity: a higher hydrophilicity corresponds to slower diffusion. In addition, we simulated the contact angles of water on sheets with four degrees of hydrophilicity to give an intuitive exhibition of hydrophilicity; the selected values were 88, 48, 25, and 17° (Figure S1). Although these values are all in the hydrophilic region according to conventional experimental observations, the contact angle measured in MD simulations is usually different from those measured in experiments due to the loss of the gas phase and the limited size scale. From the snapshots, visible differences among the four surfaces could be observed, and the surfaces do show increased hydrophilicity with rising charge magnitude, which verifies the reliability of this hydrophilic channel modification.

16 / 31


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Figure 4. Diffusion coefficients in the y direction of the water molecules in four channels with different hydrophilicities. The inset shows the MSDy of the water molecules.

2. Boundary slip condition The above results show that it is unreliable to describe the water transport processes in nanochannels with varying hydrophilicities by the HP equation, which contains only two variable parameters, ∆P and channel size. To further investigate the reason for the influence of hydrophilicity on flux, the velocity distributions along the z direction in the four channels with various hydrophilicities are also shown in Figure 5. As the HP equation predicts, the maximum value of the velocity profile increases for each channel as ∆P 17 / 31



increases. In the hydrophobic 0.0 e channel, the velocity profile is almost a flat line, which is quite different from the parabolic profile of the HP equation. This means that the water molecules in the boundary layers interact weakly with the 2D sheets, which represents a significant slip condition. In contrast, as the hydrophilicity increases, the velocity profile gradually varies from a small and near-flat curvature curve to a distinctly parabolic one. In the 1.0 and 2.0 e channels, the velocity of the water molecules in the boundary layer appears to decline somewhat compared with that of the bulk phase in the middle of the channel, indicating that the channel hydrophilicity affects the transport of water molecules inside the channel, especially those in the boundary layer. Although there are still some degrees of boundary slip in these two channels, the slip becomes increasingly less clear with increasing hydrophilicity. Finally, when the degree of hydrophilicity reaches 3.0 e, the channel hydrophilicity exceeds the default degree of the nonslip boundary condition assumed by the HP equation, which makes the velocity profile of the water molecules in the boundary layer deviate from parabolic conditions, corresponding to a negative slip case. In other words, an adhered water layer forms at the inner surfaces of the 2D sheets (Figure S3), and the water molecules in this layer exhibit no obvious macroscopic transport rate. This result explains why the simulated fluxes in the 3.0 e channel are all lower than that predicted by the HP equation and why their enhancement factors are less than 1.

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24

18

18

vx (m/s)

24

12

12

6

2.4 MPa 2.2 MPa 2.0 MPa 1.8 MPa 1.6 MPa

6

15 MPa 12.5 MPa 10 MPa 7.5 MPa 5 MPa

0

0 -10

-5

0

5

10

-10

-5

z (Å)

0

5

10

z (Å)

(a)

(b) 15

15 100 MPa 87.5 MPa 75 MPa 62.5 MPa 50 MPa

9

6

200 MPa 175 MPa 150 MPa 125 MPa 100 MPa

12

vx (m/s)

12

vx (m/s)

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


vx (m/s)

Page 19 of 31

9 6 3

3

0

0 -10

-5

0

5

10

-10

z (Å)

-5

0

5

z (Å)

(c)

(d)

Figure 5. Velocity profiles of water molecules in four channels with various hydrophilicities, (a) 0.0 e, (b) 1.0 e, (c) 2.0 e, (d) 3.0 e.

3. Verification of slip length According to Equation 6 in the theory section, the slip lengths are calculated from the enhancement factors and are listed in Table 2. The slip length decreases with increasing hydrophilicity and becomes negative when the channel hydrophilicity reaches 3.0 e, which is consistent with the velocity 19 / 31


10


profile above. The largest advantage of using the SS-NEMD method to verify Equation 6 is that we can obtain not only the fluxes but also, more importantly, the velocity distribution, as shown in Figure 5. Based on this advantage, we adopted two methods that are commonly used in simulations to verify the slip lengths calculated from Equation 6. The first method is the apparent-fitting method. A parabolic curve was used to fit all velocity distribution points under each pressure in Figure 5 and then the equation of each curve was obtained. Thereupon, the fitting slip lengths, Ls-fitting, were calculated by Equation 2 and are listed in Table 2. The profiles of the 1.0 and 2.0 e channels had clear curvatures, as a result of which Ls-fitting was easily obtained; for the 3.0 e channel, due to an adhered water layer at the inner surface of the 2D sheet, which caused the profile to deviate from parabolic behavior, the fitting curve excluded the velocity of zero at the surfaces of the 2D sheets. These values are in good agreement with the Ls values calculated from Equation 6. However, for the 0.0 e channel, the curvature of the velocity profile is too small to obtain the Ls-fitting by the apparent-fitting method. In this case, the slip length is reported to be sensitive to the external field40-41, which is a limitation of NEMD41. Therefore, we use another parameter, the friction coefficient, to estimate the slip length of the 0.0 e channel. To obtain the slip lengths based on the friction coefficient, the friction 20 / 31


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forces between the water molecules and the 2D sheets with four hydrophilicities were calculated when the system reached steady state. Then the friction coefficient was extracted as the ratio between the friction force per contact area and the slip velocity at the inner surface of the 2D sheet

42

,λ=

-F/Avslip, as listed in Table 1. The friction coefficient increases with a rising degree of hydrophilicity and a larger friction coefficient means a greater effect of friction exerted by channel walls on water molecules, which reduces the flow velocity of water molecules, especially those near the channel walls. This explains the trends of variation in flux and diffusion mentioned above; i.e., a higher hydrophilicity results in greater friction and thus reduces the water flux and diffusion. Difference of friction coefficients between 1.0 and 2.0 e is significantly larger than that between 0.0 and 1.0 e, which explains why the curves of 0.0 and 1.0 e in Figure 5 are similar while the change between 1.0 and 2.0 e is significant. When the friction coefficient increases to a critical extent (which happens to be hydrophilic conditions), an adhered layer of water molecules forms at the inner surface of the 2D sheet and cannot be driven by the applied ∆P. Although the water molecules in this layer still exhibit micromovement, from the macroscopic viewpoint, there is no obvious transport phenomenon. Based on above, the slip lengths based on the friction coefficient, Ls-λ, can be calculated by the following equation (the derivation is detailed in the Supporting Information). 21 / 31



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89 = :ℎ + 2ℎ − ℎ ;

(10)

The hydrophilicity of the 3.0 e channel exceeded this critical point, and its friction was too large for the applied ∆P to overcome the effect of friction on the water molecules in the adhered layer; thus, the velocity at the inner surfaces of the 2D sheets declines to zero in Figure 5. Therefore, referring to the equation λ = -F/Avslip, the friction coefficient in the most hydrophilic channel (3.0 e) cannot be calculated in this way, therefore, the Ls-λ cannot be obtained. If the hydrophilicity continues to increase, a larger ∆P will be required to drive the flow, which potentially has a risk of exceeding the pressure domain of the liquid water in the phase diagram. This is also the reason why we selected hydrophilicities ranging from 0.0 e to 3.0 e. Then, the Ls-λ values for the other three channels (0.0 - 2.0 e) were deduced according to Equation 10, including the 0.0 e channel, for which a velocity profile cannot be fitted. As shown in Table 2, the results are all close to those calculated from Equation 6. On the basis of comparison with these two methods of calculating slip lengths based on curve fitting and friction coefficients, Equation 6 is verified as reliable. Moreover, this method works when the degree of hydrophilicity ranges from hydrophobic to hydrophilic.

Table 2. Slip lengths of four channels with various hydrophilicities obtained by different approaches Hydrophilicity

0.0 e

1.0 e

2.0 e

22 / 31


3.0 e

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Ls (Å)

78.54

26.66

1.883

-1.074

Ls-fitting (Å)

-

26.64±1.02

1.70±0.296

-1.067±0.048

Ls-λ (Å)

70.21±7.18

23.35±0.82

1.845±0.12

-

Ls: slip lengths calculated from Equation 6; Ls-fitting: slip lengths obtained by fitting the velocity curves; and Ls-λ: slip lengths obtained from friction coefficients.

4. Flux prediction in new 2D materials Inversely, if we approach the problem in reverse, from the definition of the above enhancement factor, the actual flux for channels with boundary slips could be expressed as

3 3 = + + 1? 2ℎ ℎ

(11)

That is, the permeability of a certain material can be predicted by the slip length Ls. According to Table 2, the most hydrophobic 0.0 e channel has a slip length of nearly 8 nm, and hence, (Ls + h) will be affected by the value of Ls throughout the entire ultrafiltration range (2 - 50 nm), according to Equation 1. A larger value of (Ls + h) results in a larger value of ux(z) at a given z position. In addition, when h is small, the effect of Ls on the value of (Ls + h) plays a dominant role, so the value of ux(z) also mainly depends on the value of Ls, which indicates that the effect of channel hydrophilicity on water transfer at the nanoscale is extremely large and even exceeds the influence of the channel size. Moreover, Zhao et al.

24

recently translated the slip length Ls, which

cannot currently be directly measured experimentally, into a more intuitive surface property parameter by introducing the wetting parameter αw, which 23 / 31



was presented by Gubbins et al.43-45 Furthermore, the αw could be obtained by experiments46-47 and this provides a possible way of predicting the permeability of a material.

Conclusions In this work, a new definition of the slip length according to the Navier-Stokes equation is given, and a flux equation that includes boundary slips is derived. The SS-NEMD method is performed to investigate water molecules flowing through modified channels with various hydrophilicities, and the effect of channel hydrophilicity on water transport is systematically investigated. The pure water fluxes at steady state are systematically measured, and the flux increases linearly with ∆P as expected. It is found that the channel hydrophilicity, which is absent in the HP equation, could have a significant impact on water transport. The principle of this effect is that the water molecules in the boundary layer of the hydrophilic interface are greatly influenced by the channel wall friction, which varies remarkably with the hydrophilicity and even form an adhered layer, which decreases the slip length and impedes water flow. Hydrophilicity also caused immense variations in the equilibrium diffusivity of the water molecules. Furthermore, we validate the flux equation by two slip-length calculation methods. This equation would be useful for cases ranging from relatively hydrophobic channels (large slip lengths) to extremely hydrophilic channels (negative slip lengths). Moreover, one could 24 / 31


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translate the slip length into the wetting parameter and then relate the flux to the wetting parameter, which means that one could predict the flux of a certain material with its wetting parameter. This study is expected to improve our understanding of the effect of channel hydrophilicity on water transport behavior both qualitatively and quantitatively, which would be of value for the design optimization of 2D interlayer channels in laminated membranes for water separation applications.

Supporting Information Contact angle of a water droplet on walls with various partial charges, methodology used to calculate the slip length based on the friction coefficient and detailed water structure in channels with varying hydrophilicity. This information is available free of charge via the Internet at http://pubs.acs.org

Acknowledgement Financial support from the National Basic Research Program of China (2015CB655301), the National Key Research and Development Program of China (2017YFC0403902), the National Natural Science Foundation of China (21506091), the Jiangsu Natural Science Foundations (BK20150944, BK20150063), and the Project of Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) is gratefully acknowledged. We 25 / 31



are also grateful to the High Performance Computing Center of Nanjing Tech University and the National Supercomputing Center in Wuxi for supporting the computational resources.

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Water Flow through Interlayer Channels of 2D Materials with Various

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