Journal of Membrane Science 611 (2020) 118297

Available online 3 June 20200376-7388/© 2020 Elsevier B.V. All rights reserved.

Effect of hydrophilicity on water transport through sub-nanometer pores

Fang Xu, Mingjie Wei **, Xin Zhang, Yong Wang *

State Key Laboratory of Materials-Oriented Chemical Engineering, and College of Chemical Engineering, Nanjing Tech University, Nanjing, 211816, Jiangsu, PR China

A R T I C L E I N F O

Keywords: Hydrophilicity Sub-nanometer pore Covalent organic framework Water transport Functionalization

A B S T R A C T

Pore hydrophilicity is crucial parameter for water transport through nanopores. However, the effect of pore hydrophilicity on water permeance at the sub-nanometer scale remains unclear because it cannot be described by the conventional fluid mechanics. Herein, to determine the separate relationship between water permeance and pore hydrophilicity at the sub-nanometer scale, an equation derived from interfacial friction is obtained to exclude the effects of other variables. Using non-equilibrium molecular dynamics, we investigate water transport through a series of nanopores which is constructed from 12 types of functionalized covalent organic frameworks (COFs) with various degrees of hydrophilicity but similar skeletons and sub-nanometer pore sizes. The variable- excluded permeance has a linear relationship with contact angle. However, this linear relationship is not applicable to the hydrophobic and extremely hydrophilic nanopores under experimental pressure drops. For experimental predictions, the final relationship is corrected by considering the combined effect within all-range degrees of hydrophilicity. Therefore, a range of moderate pore hydrophilicity is suggested for the application of sub-nanometer pores in water transport. This study reveals the quantitative effect of pore hydrophilicity on water permeance at the sub-nanometer scale, which can be useful for the screening of nanomaterials for various applications.

1. Introduction

Water transport through nanopores is a fundamental process related to many important fields such as geophysical processes [1,2], biological systems [3], energy storage and conversion [4–6], as well as membrane processes [7,8]. Water transport through a confined space exhibits unique characteristics that differ from those of bulk water, which are caused by many factors. Among these, pore hydrophilicity plays a sig-nificant role with decreasing confined size as it can dramatically change the water structures and dynamics inside the nanopores, particularly at the sub-nanometer scale [9–17].

However, the exact effect of pore hydrophilicity on water transport, which is crucial for designing the nanomaterials for various applica-tions, remains unclear in non-continuum mechanics. Although there have been some investigation on water transport through various nanopores [14,15,18–20], they are not competent for describing the situation in sub-nanometer pores. This is owing to certain limitations, of which one is viscosity.

Water transport through the nanopores can be described by combining the expressions for the modified Sampson and modified

Hagen� Poiseuille (HP) equations when the characteristic flow dimen-sion is larger than 1.6 nm [14,15,18–20]. However, the modified HP equation is not applicable to water transport through sub-nanometer pores. Only one or two water layers exist inside the sub-nanometer pores, which implies that almost all water molecules directly interact with the pore wall and two adjacent water layers with differences in velocity do not exist. In this case, the water viscosity in the equation is meaningless as the interaction of the pore wall instead of water on water dominates the transport at the sub-nanometer scale.

Second limitation includes the parameter describing the pore hy-drophilicity. A parameter termed as slip length is commonly used to characterize pore hydrophilicity in nanopores of pore sizes larger than 1.6 nm [14,20,21]. Slip length is calculated from the interfacial velocity and velocity gradient. However, the velocity gradient does not exist in sub-nanometer pores because hardly any water lamination is observed with different velocities. Therefore, slip length is not only obscure but also difficult to measure. Hence, it is necessary to apply a more intuitive property such as contact angle to characterize the pore hydrophilicity in transport equations.

Finally, another limitation is the relationship of pore hydrophilicity

* Corresponding author. ** Corresponding author.

E-mail addresses: mj.wei@njtech.edu.cn (M. Wei), yongwang@njtech.edu.cn (Y. Wang).

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Journal of Membrane Science

journal homepage: http://www.elsevier.com/locate/memsci

https://doi.org/10.1016/j.memsci.2020.118297 Received 12 March 2020; Received in revised form 22 May 2020; Accepted 23 May 2020

Journal of Membrane Science 611 (2020) 118297

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with water permeance at the sub-nanometer scale. In our previous study [16], it was revealed that a threshold pressure drop (ΔPT) existed particularly for the highly hydrophobic nanopores, exceeding which, a stable and constant water permeance was observed. Additionally, the interfacial velocity should be equal to zero under the experimental pressure drops (ΔPs) according to the boundary condition (completely hydrophilic) in the traditional HP equation. Accordingly, the flux of the sub-nanometer pores should be zero if the degree of pore hydrophilicity exceeds that in the HP equation. Therefore, high ΔPs values are required to obtain a measurable flux for extremely hydrophilic sub-nanometer nanopores. Based on the above discussion, the relationship between permeance and hydrophilicity varies with ΔP, implying that water transport cannot be predicted merely by one expression.

To quantitatively determine the effect of pore hydrophilicity on water permeance at the sub-nanometer scale, nanopores that can exhibit a series of hydrophilicities are needed. Fortunately, covalent organic frameworks (COFs), a class of crystalline porous polymers, satisfy the requirement of nanopores with various hydrophilicities. COFs are typi-cally synthesized via the co-condensation of organic monomers including a knot and linker [22,23]. Owing to the intrinsic porosity, COFs present a molecular platform for fabricating nanopores with desired structures and properties [22]. The pore shape and size can be determined through the topology-directed growth of COFs. In particular, their pore walls can be modified by tailor-made functionalization via predesigning [22,24,25] and post-synthetic modification [26,27]. Furthermore, COFs allow the integration of various functional units into the pore walls with the desired composition and density, affording a full spectrum of tailor-made interfaces [22]. Therefore, COFs provide un-limited potential for the development of nanopores with desired hy-drophilicities even at the sub-nanometer scale, which is well suitable for our research.

In this study, a series of 12 types of COFs are used as models to investigate the quantitative relationship between pore hydrophilicity and water permeance via non-equilibrium molecular dynamics (NEMD) simulations. These COFs are constructed from a water-stable parent COF, and modified using various functional groups in the pore walls. Therefore, these COFs possess different pore hydrophilicities but similar structures and sub-nanometer pore sizes. An equation derived from the interfacial friction coefficient is proposed to address the first limitation. The properties of 12 COFs related to water transport, including pore hydrophilicity characterized by contact angle, are analyzed to resolve the second limitation. To establish the relationship with pore hydro-philicity, water permeance is then calculated by excluding other pa-rameters affecting it through the proposed equation. Furthermore, the applicability range of pore hydrophilicity for this relationship is dis-cussed under practical experimental ΔPs values. Finally, the corrected quantitative relationship between water permeance and hydrophilicity is established within all-range degree of hydrophilicity to address the third limitation by combining the effect of hydrophobicity and completely hydrophilicity.

2. Models and methods

2.1. Model

The COF series was computationally designed based on an experi-mentally synthesized super-microporous phosphazene-based COF (MPCOF) [28]. The atomic structure of MPCOF was obtained from the CoRE COF database [29], which constructed and compiled the solvent-free and disorder-free COF structures from the experimental characterizations. According to the experimental methodology [28], MPCOF was synthesized from hexachlorocyclotriphosphazene (HECTP, knot) and p-phenylenediamine (Pa, linker). The latter was easily grafted by various functional groups affording the COFs with various degrees of hydrophilicity. Twelve functionalized MPCOF-Rs (including the original MPCOF) were constructed from the corresponding monomers, Pa-R

(Fig. 1). The unit cell of each MPCOF-R was first optimized using the Perdew� Burke� Ernzerhof (PBE) GGA exchange correlation functional by Cambridge Sequential Total Energy Package (CASTEP). Thereafter, a fragment of each MPCOF-R was selected as the model, which could be extended to an infinitely large structure via periodical boundary con-ditions. To investigate the effect of pore hydrophilicity on permeance, it was necessary to consider three processes: entry, internal transport, and leaving. Therefore, all MPCOF-R multilayers were constructed with 14 monolayers along the transport direction resulting in thicknesses of ~5 nm.

Water transport through each MPCOF-R was simulated in a system shown in Fig. 2. It consisted of two water reservoirs serving as the feed and permeate side, as well as two additional pistons to exert the pressure differences. The multilayer surface was placed parallel to the xy plane, where the periodical boundary condition was applied. Pressure differ-ences were applied in the z direction, along which the water molecules passed through the MPCOF-R multilayers.

2.2. Methods

The interactions of all atoms were represented by the Lennard-Jones (LJ) and Columbic potentials. The TIP4P-Ew [30] model was used for the water molecules. To decrease the high frequency vibrations and reduce the simulation time, the SHAKE algorithm was used to constrain the bonds and angles of the water molecules. LJ parameters for the atoms of MPCOF-R series were obtained from the Dreiding force field [31], which was successfully applied in other COF-based simulations [29,32,33]. The Lorentz-Berthelot mixing rule was adopted for all pair-wise LJ terms. The atomic partial charges were described by elec-trostatic potential (ESP) charges, which were calculated using the grid-based ChelpG algorithm based on the density functional theory (DFT) [33]. The atom types as well as the calculated atomic partial charges of MPCOF-R series are shown in Fig. S1 and Table S1. The cut-off distances of 1.0 and 1.2 nm were used to calculate the LJ and Coulombic potentials, respectively. The particle-particle particle-mesh (PPPM) al-gorithm with a root mean square accuracy of 10-5 was used to compute the long-range electrostatic interactions.

All simulations were carried out using the Large-scale Atomic/Mo-lecular Massively Parallel Simulator (LAMMPS) package [34]. Each system was initially subjected to energy minimization with a tolerance of 10-5. The temperature was set at 300 K for all simulations. The skeletal atoms of the multilayers were fixed (gray atoms in Fig. 1), whereas the functional groups (colored atoms in Fig. 1) could move freely during all simulation processes.

For the NEMD simulations, the external force f along the z/-z di-rection was applied to each atom of the two pistons to obtain the desired pressures on the pistons. The relationship is described as:

P¼fnA

(1)

where P is the desired pressure on the piston, A is the area of the piston, and n represents the atom number of the piston. During simulations, two pistons can self-adjust their positions under Pfeed and Ppermeate to pro-duce ΔP, which can be calculated using the following equation:

ΔP¼Pfeed � Ppermeate (2)

where Pfeed (50–200 MPa) and Ppermeate (1 atm) are the pressures on feed and permeate pistons, respectively. The applied ΔP varied from 50 to 200 MPa, which was much higher than the practical experimental usage. Such high ΔP is commonly used in NEMD to reduce the thermal noise and enhance signal/noise ratio within the simulation time scale of nanoseconds [35–39]. Each simulation was performed for 20–50 ns with a time step of 2 fs. The trajectory was saved every 1 ps and analyzed using the self-compiled codes after the simulation reached a steady state.

To calculate the diffusion coefficients of in-pore water molecules, the

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outside reservoirs were removed while the in-pore water molecules remained inside the nanopores. The size of the simulation box was then set equal to each MPCOF-R cell and the periodic boundary condition was applied in all dimensions to simulate the in-pore behavior of the water molecules. Simulations in this part were first equilibrated for 3 ns and then performed in the NVT ensemble for 20 ns with a time step of 1 fs to collect the trajectories.

3. Results and discussion

3.1. Theory

For water transport through a cylindrical nanopore, the total resis-tance comprises the interfacial resistance and interior resistance [20,21, 40]. The interfacial resistance can be expressed by the Sampson equa-tion [18–20], which considers it as the flow resistance of an infinitely thin orifice and is close to the exact solution within 1% error [41]. Therefore, the pressure drop at the pore entrance/exit can be described as:

ΔP1¼QCμ∞

R3 (3)

where Q is the volumetric water flux per unit time, C is the loss coeffi-cient of both the water entry and exit process, μ∞ is the viscosity of water in the reservoir, i.e. bulk water viscosity, and R is the effective radius of the nanopore. Equation (3) describes the condition for a single pore. Considering the overall surface area of the multilayer, the equation can be written as:

ΔP1¼πCμ∞

∅RF (4)

where F is the water flux based on the surface area and Φ is the surface porosity.

The interior resistance can be calculated using the modified HP expression when the pore size is larger than 1.6 nm [14,15,18–20], but it cannot describe the water flow through the sub-nanometer pore. This is because the water viscosity and slip length of the pore wall in this equation are meaningless at the sub-nanometer scale, as discussed in the

Fig. 1. Synthesis of MPCOF and molecular models of MPCOF-R series with corresponding Pa-R linkers.

Fig. 2. Schematic of the simulation system for water transport through MPCOF- R series multilayers.

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Introduction section. On the one hand, the derivation of the HP equation starts from the analysis of the forces between any two adjacent circular water layers with different velocities inside the nanopore. However, only one or two water layers exist inside the sub-nanometer pores, which implies that almost all water molecules directly interact with the pore wall and two adjacent water layers with different velocities do not exist. The interfacial layer is defined as r > R-dwater (R is the pore radius and dwater is the diameter of the water molecules) in this study. As shown in Fig. S2, the interfacial water of almost all MPCOF-Rs accounts for more than 95% of all inside water and the lowest proportion is nearly 90% for MPCOF-H. Therefore, it can be confirmed that almost all water mole-cules are inside the interfacial layer of each MPCOF-R. In this case, the analysis of force, from which the transport equation is derived, should start not from the water viscosity but from the interaction between water and the pore wall. On the other hand, no lamination with different ve-locities also renders another parameter (slip length) unusable due to its dependence on the velocity gradient of water. Therefore, the modified HP equation is not applicable for the sub-nanometer pores owing to these two problems.

Instead, the transport equation can be derived from the analysis of the forces between water and the pore wall, in which case the interfacial friction coefficient is introduced as [42]:

fdriving¼ � ffriction ¼ λAinv (5)

where fdriving is the driving force, ffriction is the frictional force along the axial direction on the contact area Ain, λ is the friction coefficient, and v is the water velocity. Then, the equation can be further transformed into a detailed expression shown below:

ΔP2 ⋅ πR2 ¼ λ⋅2πRL⋅F∅

(6a)

ΔP2¼2λL∅R

F (6b)

where L is the pore length. The pore lengths in this study are all approximately 5 nm, ensuring that all entrance and end effects can be decoupled [20].

The total pressure drop comprising the interfacial and interior parts can be expressed as:

ΔP¼ΔP1 þ ΔP2 ¼πCμ∞

∅RF þ

2λL∅R

F (7)

In this equation, the friction coefficient λ is physically relevant to the interfacial interaction that characterizes the hydrophilicity of the pore wall. However, this parameter is hardly obtained in experiments, thereby limiting it to the experimental prediction. It is important to correlate this equation with a commonly used parameter characterizing hydrophilicity (e.g. contact angle of water droplets) in the experiments. For this purpose, the permeance (F=ΔP) is calculated by excluding other factors to explore its relationship with the contact angle: �

ΔPF�

πCμ∞

∅R

�∅R2L¼ λ¼ f ðθÞ (8)

where θ is the contact angle characterizing the pore hydrophilicity.

3.2. Pore hydrophilicity

Notably, the experimentally accessible ΔPs is generally less than 10 MPa, such as capillary pressure, osmotic pressure, and operation pres-sures for the membrane processes. According to our previous study [16], the presence of large threshold pressures (ΔPT) makes the simulation results at high ΔPs unreliable for the prediction of the experimental results at low ΔPs for the nanopores with hydrophobic pore walls. Only when the objective nanopores can be wetted under experimentally low ΔPs, their simulation results at high ΔPs can be used to precisely predict

their experimental performance. As the MPCOFs are modified by several groups to tune the pore hydrophilicity, various MPCOF-R nanopores exhibit distinct wetting behaviors. Therefore, first, it is required to determine which MPCOF-R nanopores can be wetted under the experi-mentally accessible ΔPs before the large-scale NEMD simulations are performed at high ΔPs. Twelve MPCOF-Rs are then tested under 10 MPa to determine if these can be wetted. As shown in Table 1, MPCOF-(F)4, MPCOF-(Cl)2, and MPCOF-(SH)2 cannot be wetted under 10 MPa, indicating that their ΔPT values are greater than those that can be attained experimentally. This suggests their strong hydrophobicity and unsuitability in the applications of water treatment. Therefore, these are not further investigated in this work, whereas the remaining nine MPCOF-Rs are subjected to further examination.

To determine the quantitative effect of pore hydrophilicity on per-meance, the calculation of the parameters on the left (Φ, R and L) and right side (θ) of Equation (8) is required. The pore radius R listed in Table 1 is calculated using the software Zeoþþ [43]. Porosity Φ and length L are calculated based on the water content inside each multi-layer and the size of each COF unit cell, respectively.

It has been commonly reported that the dynamic random motions of the interfacial water molecules are strongly correlated to the surface hydrophilicity [44–46]. From Fig. S2, it is clear that all the in-pore water molecules can be recognized as interfacial water. Therefore, the dy-namic random motion of water molecules is significantly affected by the interaction between the pore wall and water molecules [15]. In this study, the self-diffusion coefficient along the z direction (Dz) is used to characterize the dynamics of in-pore water molecules, such that the pore hydrophilicity can be measured by Dz of the in-pore water molecules. To calculate Dz, the mean squared displacements in the z direction (MSDz) of all in-pore water molecules should be first measured, and then Dz can be calculated using the following expression:

Dz¼limt→∞

MSDzðtÞ

2t(9)

The variation in MSDz for the nine MPCOF-Rs is shown in Fig. 3a. The MSDz curves are linear, and the slopes of these lines represent their respective Dz values. The Dz values of the nine MPCOF-Rs shown in Fig. 3b are different from each other, indicating that the modification of the functional groups effectively changes the pore hydrophilicity.

It has been reported that the self-diffusion coefficients of the inter-facial water molecules have a linear relationship with the contact angle of water on a surface [44–46]. Therefore, Dz can be translated into θ by fitting it from the linear equation. In the case of water molecules on a completely hydrophilic surface (e.g. the boundary condition in HP equation where the fluid velocity is zero), the self-diffusion coefficients should be zero because these are all trapped by the surface and difficult to move, i.e., Dz ¼ 0 when θ ¼ 0�. Alternately, in the case of water molecules on a completely hydrophobic surface, their self-diffusion co-efficients are defined as being equal to the value of the bulk because the

Table 1 Properties of MPCOF-R multilayers.

Series Wetting at 10 MPa

Diameter (Å)

Porosity (%)

Length (Å)

Hydrophilicity (�)

-(F)4 � 8.90 – 48.13 – -(Cl)2 � 6.95 – 47.58 – -(SH)2 � 7.54 – 50.22 – -NO2 ✓ 7.30 35.76 48.78 76.52 -H ✓ 9.99 43.89 49.86 49.70 -OH ✓ 8.51 43.48 48.97 47.45 -NH2 ✓ 8.29 42.83 49.47 45.44 -(NH2)2 ✓ 8.23 40.37 48.99 41.48 -SO3H ✓ 6.91 39.15 49.03 35.57 -COOH ✓ 7.04 42.53 50.23 33.54 -(OH)2 ✓ 8.49 44.04 48.37 22.46 -(COOH)2 ✓ 7.16 42.60 51.07 18.73

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super-hydrophobic surface is expected to have no interaction with the interfacial water molecules, i.e., the absence of the surface. In this case, the Dz value of 2.4 � 10-9 m2 s-1 for the bulk TIP4P-Ew water model at 300 K [30] is applied to define the value of θ ¼ 180�. Based on this data, the relation between Dz and θ can be described as:

θ¼ 75� 109Dz (10)

The θ value of each MPCOF-R is computed using Equation (10) and listed in Table 1. Notably, the θ ranges from ~18� to ~77�, which is essentially below 90�, implying they all belong to the hydrophilic category. This is consistent with the wetting behaviors of the latter nine MPCOF-Rs in Table 1. Therefore, this range of pore hydrophilicity is suitable for further investigating the water transport through the nanopores.

3.3. Pristine flux and permeance

Following the work of Grossman et al. [47], the numbers of water molecules in the feed, internal, and permeate side over the simulation time were tracked. The flux of each MPCOF-R was calculated by the slopes of number of permeated water molecules varying with simulation times in Fig. S3. The number of permeated water molecules increases linearly with the simulation time, indicating that the simulation systems have reached a steady state. As shown in Fig. 4a, all fluxes except that of the MPCOF-(COOH)2 are proportional to ΔP, despite the varying hy-drophilicities. For the most hydrophilic MPCOF-(COOH)2, the linear fitting does not start from zero because there is no flux until the ΔPs reaches 50 MPa as shown in Fig. 4a, which will be discussed in the next section.

The permeance is then calculated using F/ΔP according to Fig. 4a and the results are shown in Fig. 4b. All permeances range from 200 to 1100 Lm-2h-1bar-1 (including the permeance of MPCOF-(COOH)2). If the factor of multilayer thickness is considered (20-nm thick in typical ex-periments, but 5-nm thick in this work), the calculated permeances are as high as 100–270 Lm-2h-1bar-1, which are significantly greater than those of the polymeric nanofiltration membranes (10–50 Lm-2h-1bar-1) with similar pore sizes in the experimental measurements. The excep-tional permeance values of the COFs can be attributed to their high porosities and orderly straight-through nanopores. More importantly, the simulated permeances well match the results (considering the ratios of membrane thickness) of experimental fabricated IISERP-COOH-COF1 [27], TpHz [48] and CTF [49] membranes, whose pore sizes are all below 1 nm. In addition, it is evident that the permeance of each MPCOF-R varies, implying that pore hydrophilicity plays an important role in determining water permeance. Accordingly, the decrease in permeance in Fig. 4b roughly follows the order of increasing pore hy-drophilicity. However, there are some exceptions because other influ-encing variables (pore size, porosity, and pore length) also affect the final permeance. Therefore, these factors should be excluded to

determine the separate effect of pore hydrophilicity on water permeance.

3.4. Variable-excluded permeance

As derived in the Theory section, the parameters in Equation (8) such as porosity, pore size, and viscosity are excluded from the pristine per-meance, such that permeance is only the function of pore hydrophilicity.

After substituting all the detailed variables in Equation (8), the variable-excluded permeance, which is the reciprocal of the left side of Equation (8), is plotted in Fig. 5a as a function of θ. The fitted curve shows a good proportional relationship, from which the following expression for water transport through MPCOF-R can be obtained: �

ΔPF�

πCμ∞

∅R

�∅R2L¼

10:252� 10� 9θ

(11a)

Fig. 3. (a) Mean square displacements and (b) self-diffusion coefficients along the z direction of in-pore water molecules for nine suitable MPCOF-Rs.

Fig. 4. (a) Pure water flux under different ΔPs and (b) the corresponding permeance of each MPCOF-R.

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ΔP¼ΔP1 þ ΔP2 ¼πCμ∞

∅RF þ

7:94� 109L∅Rθ

F (11b)

Due to the few reports on COF membranes with sub-nanometer pore sizes, it is hard to compare this relationship with their experimental data. However, we found that this trend is close to experimental results of functionalized molybdenum disulfide (MoS2) membranes, which possess similar sub-nanometer pore sizes. Ries et al. [50] fabricated three types of functionalized MoS2 membranes with varying hydrophi-licity, whose contact angles were different but all below 90�. Impor-tantly, their water permeances increased with their rising hydrophobicities, which is in excellent consistence with the trend observed in our work (Fig. 5a).

However, this relationship is applicable to all range of hydrophilic-ities and hydrophobicities only at ΔP >100 MPa owing to the fitting permeance of MPCOF-(COOH)2. For application under experimentally accessible ΔPs, two particular cases (hydrophobic and completely hy-drophilic nanopores) should be considered. For hydrophobic nanopores, a high ΔPT is observed and it rises rapidly with increased hydrophobicity as systematically shown in our previous work [16]. As shown in Table 1, three types of hydrophobic MPCOFs cannot be wetted under 10 MPa, thereby producing no flux and permeance. For completely hydrophilic nanopores, considering MPCOF-(COOH)2 as an example, its flux is investigated in a broad range of ΔPs, as shown in Fig. 5b. As observed, the water flux of MPCOF-(COOH)2 is not always proportional to ΔP within all ranges of ΔPs. When ΔP > 100 MPa, the water flux is pro-portional to ΔP. However, no evident flux and permeance are observed although the nanopores can be completely wetted at ΔP < 50 MPa due to the strong affinity exerted by the pore wall atoms on water molecules. This is consistent with the default boundary condition in the traditional HP equation as mentioned in the Introduction section, where the interfacial velocity is equal to zero. When the water is confined in sub-nanometer pores, almost all water molecules are included in the interfacial layer, and hence the interaction between the pore wall and water molecules (pore hydrophilicity) instead of that between the water molecules (viscosity) dominates the water transport. This result suggests that the hydrophilicity of MPCOF-(COOH)2 surpasses the default hy-drophilicity in the HP equation. The lowest ΔP that is needed to obtain a stable permeance for the extremely hydrophilic nanopores is called start-up pressure drop (ΔPst). The ΔPst of MPCOF-(COOH)2 is greater than the experimentally accessible ΔPs, and therefore it is difficult to generate permeance under low ΔPs. Hence, the permeances of both the hydrophobic and completely hydrophilic nanopores should be equal to zero under experimental low ΔPs.

Based on the above discussion, under low ΔPs the relationship established in Fig. 5a and Equation (11b) is applicable to a range of pore hydrophilicities that are neither hydrophobic nor completely hydro-philic. The correctional relationship between variable-excluded per-meance and θ at 10 MPa is plotted in Fig. 6, suggesting that the

relationship established in Equation (11b) is applicable to moderate hydrophilicities. Therefore, a suitable pore hydrophilicity that is neither hydrophobic nor extremely hydrophilic should be selected to confirm an observable permeance under experimental ΔPs because of the ΔPst and ΔPT values of extremely hydrophilic and hydrophobic nanopores, respectively.

The correctional between water permeance and pore hydrophilicity described in Equation (11b) and Fig. 6 is practically significant to pre-dict the water permeance of COF materials with sub-nanometer pores. Upon measurement of the water contact angle of the COF pore wall, the water permeance can be theoretically obtained, which can be useful for screening the COF materials for a variety of applications.

4. Conclusion

In summary, 12 types of covalent organic frameworks (COFs) are used as models to investigate the relation between water permeance and pore hydrophilicity at the sub-nanometer scale. These 12 COFs are constructed from a water-stable parent COF, MPCOF, and modified by 12 different functional groups (-(F)4, -(Cl)2, -(SH)2, -NO2, -H, -OH, -NH2, -(OH)2, -(NH2)2, -SO3H, -COOH, -(COOH)2) to afford different hydro-philicities but similar structures and sub-nanometer pore sizes. Through the NEMD simulations, water transport through 12 MPCOF-R multi-layers is investigated. Majority of the in-pore water molecules are included in the interfacial layer. The contact angle calculated from the self-diffusion coefficient of the interfacial water is used to characterize the pore hydrophilicity. By excluding other factors influencing water permeance using an equation deduced from the definition of interfacial friction coefficient, the water permeance is found to vary linearly with the contact angle. To guide the experimental design of the nano-materials, two exceptions should be considered under the

Fig. 5. (a) Relationship between variable-excluded permeance and pore hydrophilicity at ΔPs > 100 MPa. (b) Water flux of MPCOF-(COOH)2 within the ΔP range of 0–350 MPa.

Fig. 6. Relationship between variable-excluded permeance and pore hydro-philicity at ΔPs < 10 MPa.

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experimentally accessible ΔPs. Both the hydrophobic (MPCOF-(F)4, MPCOF-(Cl)2, and MPCOF-(SH)2) and completely hydrophilic (MPCOF- (COOH)2) COFs are not found to be suitable for water transport in practical experimental applications owing to the existence of high threshold pressure drop (ΔPT) and high start-up pressure drop (ΔPst). Based on the final corrected relationship, water transport can be quan-titatively described under the experimental ΔPs. Therefore, an appro-priate hydrophilicity of the sub-nanometer pores is required for its application in water transport. This study systematically analyzes the quantitative effect of pore hydrophilicity on water transport through sub-nanometer nanopores, which is expected to guide the experimental prediction of permeance for nanoporous materials and can thus be employed for screening the nanomaterials for various applications.

Author statement

Fang Xu: Conceptualization, Methodology, Software, Validation, Investigation Formal analysis, Writing - original draft, Visualization. Mingjie Wei: Formal analysis, Writing - review &; editing, Funding acquisition. Xin Zhang: Software, Formal analysis. Yong Wang: Formal analysis, Writing - review &; editing, Supervision, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Financial support from the National Key Research and Development Program of China (2017YFC0403902), the National Natural Science Foundation of China (21825803, 21921006), and the Jiangsu Natural Science Foundations (BK20190085). We thank the High Performance Computing Center of Nanjing Tech University for supporting the computational resources. We are also grateful to the Program of Excel-lent Innovation Teams of Jiangsu Higher Education Institutions and the Project of Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

Nomenclature

ΔPT threshold pressure drop ΔP total pressure drop P desired pressure on a piston f external force on piston atoms n atom number of a piston A area of a piston Pfeed pressure on the feed piston Ppermeate pressure on the permeate piston ΔP1 pressure drop at the pore entrance/exit Q volumetric water flux per unit time C loss coefficient of both the water entry and exit process μ∞ viscosity of bulk water R effective radius of the nanopore F water flux based on surface area Φ surface porosity fdriving driving force ffriction frictional force Ain contact area inside nanopores λ friction coefficient v water velocity ΔP2 pressure drop inside nanopore L pore length θ contact angle π Pi

Dz self-diffusion coefficient in the z direction MSDz mean squared displacements in the z direction ΔPst start-up pressure drop

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi. org/10.1016/j.memsci.2020.118297.

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