^{1}, Randall S. Dumont

^{2}and James M. Dickson

^{1,a)}

### Abstract

Nonequilibrium molecular dynamics (NEMD) simulations are presented to investigate the effect of water-membrane interactions on the transport properties of pressure-driven water flow passing through carbon nanotube (CNT) membranes. The CNT membrane is modified with different physical properties to alter the van der Waals interactions or the electrostatic interactions between water molecules and the CNT membranes. The unmodified and modified CNT membranes are models of simplified nanofiltration (NF) membranes at operating conditions consistent with real NF systems. All NEMD simulations are run with constant pressure difference (8.0 MPa) temperature (300 K), constant pore size (0.643 nm radius for CNT (12, 12)), and membrane thickness (6.0 nm). The water flow rate, density, and velocity (in flow direction) distributions are obtained by analyzing the NEMD simulation results to compare transport through the modified and unmodified CNT membranes. The pressure-driven water flow through CNT membranes is from 11 to 21 times faster than predicted by the Navier-Stokes equations. For water passing through the modified membrane with stronger van der Waals or electrostatic interactions, the fast flow is reduced giving lower flow rates and velocities. These investigations show the effect of water-CNT membrane interactions on water transport under NF operating conditions. This work can help provide and improve the understanding of how these membrane characteristics affect membrane performance for real NF processes.

The Natural Sciences and Engineering Research Council Canada (NSERC) and McMaster University are thanked for their financial support. The Shared Hierarchical Academic Research Computing Network (SHARCNET, http://www.sharcnet.ca) is gratefully acknowledged for providing high performance computing technology for this work.

I. INTRODUCTION

II. SIMULATION METHODS

III. RESULTS AND DISCUSSION

A. The modified Lennard-Jones parameters of the CNT membranes

B. The polarized CNT membranes

IV. CONCLUSIONS

### Key Topics

- Carbon nanotubes
- 110.0
- Carbon
- 13.0
- Water reservoirs
- 13.0
- Electrostatics
- 12.0
- Molecular dynamics
- 12.0

## Figures

Perspective snapshots of the simulation systems produced by VMD package at the beginning status of MD simulations (adapted from the figure in Ref. ^{ 12 } ). Shown is the beginning empty CNT membrane model connecting two liquid filled reservoirs, and the two graphene sheets acted as the movable walls where the force *f* _{ t } or *f* _{ b } is applied on each carbon atom of the top or bottom wall. Carbon atoms in green, hydrogen atoms in white, and oxygen atoms in red.

Perspective snapshots of the simulation systems produced by VMD package at the beginning status of MD simulations (adapted from the figure in Ref. ^{ 12 } ). Shown is the beginning empty CNT membrane model connecting two liquid filled reservoirs, and the two graphene sheets acted as the movable walls where the force *f* _{ t } or *f* _{ b } is applied on each carbon atom of the top or bottom wall. Carbon atoms in green, hydrogen atoms in white, and oxygen atoms in red.

Bar chart of simulation and theoretical flow rates (molecules/ns) of unmodified and modified CNT membranes. The results are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each flow rate data is less than 0.5%. H-P equation refers to the Hagen-Poiseuille equation (Eq. (5) ), and the values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I .

Bar chart of simulation and theoretical flow rates (molecules/ns) of unmodified and modified CNT membranes. The results are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each flow rate data is less than 0.5%. H-P equation refers to the Hagen-Poiseuille equation (Eq. (5) ), and the values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I .

Simulation flow rate (molecules/ns) as a function of the square root of the well-depth parameter. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 0.5%. The solid line is the linear trend based on the simulation results.

Simulation flow rate (molecules/ns) as a function of the square root of the well-depth parameter. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 0.5%. The solid line is the linear trend based on the simulation results.

Effect of van der Waals interactions between water and membrane, represented by the well-depth parameter of the Lennard-Jones potential (interaction increasing from red ♦, to black ■, to green ▲), on the density distributions along the radial direction: density values are averaged over thin annular sections in the pore at the indicated *r* values over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 1%. The values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I . The solid lines are the trend curves based on the simulation results, and the dotted lines are the pore surface represented by the effective pore radius (0.643 nm).

Effect of van der Waals interactions between water and membrane, represented by the well-depth parameter of the Lennard-Jones potential (interaction increasing from red ♦, to black ■, to green ▲), on the density distributions along the radial direction: density values are averaged over thin annular sections in the pore at the indicated *r* values over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 1%. The values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I . The solid lines are the trend curves based on the simulation results, and the dotted lines are the pore surface represented by the effective pore radius (0.643 nm).

Effect of van der Waals interactions between water and membrane, represented by the well-depth parameter of the Lennard-Jones potential (interaction increasing from red ♦, to black ■, to green ▲), on the density distributions along the flow direction: density values are averaged over thin sections about the indicated *z* values (cylindrical sections in the pore and cuboid sections in the water reservoirs) over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 1%. The values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I . The solid lines are the trend curves based on the simulation results. The dashed lines are the membrane boundaries along *z* direction separating the system in three parts from left to right: the water reservoir on the high pressure side, the pore of the CNT membrane, and the water reservoir on the low pressure side.

Effect of van der Waals interactions between water and membrane, represented by the well-depth parameter of the Lennard-Jones potential (interaction increasing from red ♦, to black ■, to green ▲), on the density distributions along the flow direction: density values are averaged over thin sections about the indicated *z* values (cylindrical sections in the pore and cuboid sections in the water reservoirs) over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 1%. The values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I . The solid lines are the trend curves based on the simulation results. The dashed lines are the membrane boundaries along *z* direction separating the system in three parts from left to right: the water reservoir on the high pressure side, the pore of the CNT membrane, and the water reservoir on the low pressure side.

Effect of van der Waals interactions between water and membrane, represented by the well-depth parameter of the Lennard-Jones potential (interaction increasing from red ♦, to black ■, to green ▲), on the velocity distributions (in *z* direction) as a function of radial position: velocity (in *z* direction) values are averaged over thin annular sections in the pore at the indicated *r* values over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical errors for most data points are less than 10%. N-S equation refers to the Navier-Stokes equation (Eq. (6) ), and the values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I . The solid lines are the trend curves based on the simulation results, the dashed lines are the trends based on the Navier-Stokes equation, and the dotted lines are the pore surface represented by the effective pore radius (0.643 nm).

Effect of van der Waals interactions between water and membrane, represented by the well-depth parameter of the Lennard-Jones potential (interaction increasing from red ♦, to black ■, to green ▲), on the velocity distributions (in *z* direction) as a function of radial position: velocity (in *z* direction) values are averaged over thin annular sections in the pore at the indicated *r* values over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical errors for most data points are less than 10%. N-S equation refers to the Navier-Stokes equation (Eq. (6) ), and the values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I . The solid lines are the trend curves based on the simulation results, the dashed lines are the trends based on the Navier-Stokes equation, and the dotted lines are the pore surface represented by the effective pore radius (0.643 nm).

Effect of van der Waals interactions between water and membrane, represented by the well-depth parameter of the Lennard-Jones potential (interaction increasing from red ♦, to black ■, to green ▲), on the velocity distributions (in *z* direction) along the flow direction: velocity (in *z* direction) values are averaged over thin sections about the indicated *z* values (cylindrical sections in the pore and cuboid sections in the water reservoirs) over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 10%. The values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I . The solid lines are the trend curves based on the simulation results. The dashed lines are the membrane boundaries along the *z* direction separating the system in three parts from left to right: the water reservoir at high pressure side, the pore of the CNT membrane, and the water reservoir at low pressure side.

Effect of van der Waals interactions between water and membrane, represented by the well-depth parameter of the Lennard-Jones potential (interaction increasing from red ♦, to black ■, to green ▲), on the velocity distributions (in *z* direction) along the flow direction: velocity (in *z* direction) values are averaged over thin sections about the indicated *z* values (cylindrical sections in the pore and cuboid sections in the water reservoirs) over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 10%. The values of the well-depth parameters (*ɛ* _{(CC)}, *ɛ* _{(HH)}, and *ɛ* _{(OO)}) are shown in Table I . The solid lines are the trend curves based on the simulation results. The dashed lines are the membrane boundaries along the *z* direction separating the system in three parts from left to right: the water reservoir at high pressure side, the pore of the CNT membrane, and the water reservoir at low pressure side.

Charge patterns of the unmodified and modified CNTs: (a) unmodified CNT, (b) unwrapped CNT of the ring polarized model, and (c) unwrapped CNT of the band polarized model. This figure illustrates the relative locations of the neutral atoms in green, the positive atoms in orange, and the negative atoms in blue. Transport is from left to right as indicated by the arrows on the *z*-axes.

Charge patterns of the unmodified and modified CNTs: (a) unmodified CNT, (b) unwrapped CNT of the ring polarized model, and (c) unwrapped CNT of the band polarized model. This figure illustrates the relative locations of the neutral atoms in green, the positive atoms in orange, and the negative atoms in blue. Transport is from left to right as indicated by the arrows on the *z*-axes.

Bar chart of simulation and theoretical flow rates (molecules/ns) of unmodified and modified polarized CNT membranes. The results are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each flow rate data is less than 0.5%. H-P equation refers to the Hagen-Poiseuille equation (Eq. (5) ), and the charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) .

Bar chart of simulation and theoretical flow rates (molecules/ns) of unmodified and modified polarized CNT membranes. The results are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each flow rate data is less than 0.5%. H-P equation refers to the Hagen-Poiseuille equation (Eq. (5) ), and the charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) .

Effect of electrostatic interactions between water and membrane, caused by the charge patterns within the CNT, on the density distributions along the radial direction: density values are averaged over thin annular sections in the pore at the indicated *r* values over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 1%. The charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) . The solid lines are the trend curves based on the simulation results, and the dotted lines are the pore surface represented by the effective pore radius (0.643 nm).

Effect of electrostatic interactions between water and membrane, caused by the charge patterns within the CNT, on the density distributions along the radial direction: density values are averaged over thin annular sections in the pore at the indicated *r* values over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 1%. The charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) . The solid lines are the trend curves based on the simulation results, and the dotted lines are the pore surface represented by the effective pore radius (0.643 nm).

Effect of electrostatic interactions between water and membrane, caused by the charge patterns within the CNT, on the density distributions along the flow direction: density values are averaged over thin sections about the indicated *z* values (cylindrical sections in the pore and cuboid sections in the water reservoirs) over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 1%. The charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) . The solid lines are the trend curves based on the simulation results. The dashed lines are the membrane boundaries along the *z* direction separating the system in three parts from left to right: the water reservoir at high pressure side, the pore of the CNT membrane, and the water reservoir at low pressure side.

Effect of electrostatic interactions between water and membrane, caused by the charge patterns within the CNT, on the density distributions along the flow direction: density values are averaged over thin sections about the indicated *z* values (cylindrical sections in the pore and cuboid sections in the water reservoirs) over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 1%. The charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) . The solid lines are the trend curves based on the simulation results. The dashed lines are the membrane boundaries along the *z* direction separating the system in three parts from left to right: the water reservoir at high pressure side, the pore of the CNT membrane, and the water reservoir at low pressure side.

Effect of electrostatic interactions between water and membrane, caused by the charge patterns of the CNT, on the velocity (in *z* direction) distributions along the radial direction: velocity (in *z* direction) values are averaged over thin annular sections in the pore at the indicated *r* values over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical errors for most data points are less than 10%. N-S equation refers to the Navier-Stokes equation (Eq. (6) ), and the charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) . The solid lines are the trend curves based on the simulation results, the dashed lines are the trends based on the Navier-Stokes equation, and the dotted lines are the pore surface represented by the effective pore radius (0.643 nm).

Effect of electrostatic interactions between water and membrane, caused by the charge patterns of the CNT, on the velocity (in *z* direction) distributions along the radial direction: velocity (in *z* direction) values are averaged over thin annular sections in the pore at the indicated *r* values over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical errors for most data points are less than 10%. N-S equation refers to the Navier-Stokes equation (Eq. (6) ), and the charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) . The solid lines are the trend curves based on the simulation results, the dashed lines are the trends based on the Navier-Stokes equation, and the dotted lines are the pore surface represented by the effective pore radius (0.643 nm).

Effect of electrostatic interactions between water and membrane, caused by the charge patterns of the CNT, on the velocity (in *z* direction) distributions along the flow direction: velocity (in *z* direction) values are averaged over thin sections about the indicated *z* values (cylindrical sections in the pore and cuboid sections in the water reservoirs) over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 10%. The charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) . The solid lines are the trend curves based on the simulation results. The dashed lines are the membrane boundaries along the *z* direction separating the system in three parts from left to right: the water reservoir at high pressure side, the pore of the CNT membrane, and the water reservoir at low pressure side.

Effect of electrostatic interactions between water and membrane, caused by the charge patterns of the CNT, on the velocity (in *z* direction) distributions along the flow direction: velocity (in *z* direction) values are averaged over thin sections about the indicated *z* values (cylindrical sections in the pore and cuboid sections in the water reservoirs) over the NEMD simulation at steady state water flow. The data points are based on membrane simulations using the (12, 12) CNT, *L* _{ z }: 6.0 nm; Δ*P*: 8.0 MPa; *T*: 300 K. The statistical error for each data point is less than 10%. The charge patterns of the modified polarized CNTs are shown in Figs. 8(b) and 8(c) . The solid lines are the trend curves based on the simulation results. The dashed lines are the membrane boundaries along the *z* direction separating the system in three parts from left to right: the water reservoir at high pressure side, the pore of the CNT membrane, and the water reservoir at low pressure side.

Orthographic snapshots of water in the (a) unmodified CNT, (b) ring polarized CNT, and (c) band polarized CNT. Each snapshot is a cross-sectional view in the center section (facing *z* = −1.0 nm, from *z* = −1.0 nm to 1.0 nm; see Fig. 8 for the coordinates). Neutral carbon atoms in green, positively charged carbon atoms in orange, negatively charged carbon atoms in blue, hydrogen atoms in white, and oxygen atoms in red. For the unmodified (neutral) wall in (a) the water orientation near the pore surface is random. However, for the polarized cases (b) and (c) there is preferential orientation of water in the CNTs.

Orthographic snapshots of water in the (a) unmodified CNT, (b) ring polarized CNT, and (c) band polarized CNT. Each snapshot is a cross-sectional view in the center section (facing *z* = −1.0 nm, from *z* = −1.0 nm to 1.0 nm; see Fig. 8 for the coordinates). Neutral carbon atoms in green, positively charged carbon atoms in orange, negatively charged carbon atoms in blue, hydrogen atoms in white, and oxygen atoms in red. For the unmodified (neutral) wall in (a) the water orientation near the pore surface is random. However, for the polarized cases (b) and (c) there is preferential orientation of water in the CNTs.

## Tables

The Lennard-Jones parameters of the unmodified and modified CNT membranes.

The Lennard-Jones parameters of the unmodified and modified CNT membranes.

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