^{a)}

^{a)}This paper was presented, in part, at NAMS’11, Las Vegas, NV, June 2011.

^{1}, Randall S. Dumont

^{2}and James M. Dickson

^{1,b)}

### Abstract

Nonequilibrium molecular dynamics (NEMD) simulations are used to investigate pressure-driven water flow passing through carbon nanotube(CNT) membranes at low pressures (5.0 MPa) typical of real nanofiltration (NF) systems. The CNT membrane is modeled as a simplified NF membrane with smooth surfaces, and uniform straight pores of typical NF pore sizes. A NEMD simulation system is constructed to study the effects of the membrane structure (pores size and membrane thickness) on the pure watertransport properties. All simulations are run under operating conditions (temperature and pressure difference) similar to a real NF processes. Simulation results are analyzed to obtain water flux, density, and velocity distributions along both the flow and radial directions. Results show that water flow through a CNT membrane under a pressure difference has the unique transport properties of very fast flow and a non-parabolic radial distribution of velocities which cannot be represented by the Hagen-Poiseuille or Navier-Stokes equations. Density distributions along radial and flow directions show that water molecules in the CNT form layers with an oscillatory density profile, and have a lower average density than in the bulk flow. The NEMD simulations provide direct access to dynamic aspects of water flow through a CNT membrane and give a view of the pressure-driven transport phenomena on a molecular scale.

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 effect of pore size

1. Flux

2. Radial-direction distributions

3. Flow-direction distributions

B. The effect of membrane thickness

IV. CONCLUSIONS

### Key Topics

- Carbon nanotubes
- 137.0
- Water reservoirs
- 43.0
- High pressure
- 18.0
- Navier Stokes equations
- 17.0
- Molecule surface interactions
- 16.0

## Figures

Orthographic snapshots of the CNT membrane model produced by the VMD package for the MD simulation: (a) Top view down through the CNT membrane model and (b) side view of the CNT membrane model. Two graphene sheets are located at the ends of the CNT and each sheet has an approximated hole in the center to match the pore size of the CNT.

Orthographic snapshots of the CNT membrane model produced by the VMD package for the MD simulation: (a) Top view down through the CNT membrane model and (b) side view of the CNT membrane model. Two graphene sheets are located at the ends of the CNT and each sheet has an approximated hole in the center to match the pore size of the CNT.

Perspective snapshots of the simulation systems produced by VMD package at the beginning status of MD simulations. Shown are the two graphene sheets which act as the movable walls, the CNT membrane model shown in Fig. 1, and the two water reservoirs connected by the membrane. Forces *f* _{ t } and *f* _{ b } act by the movable walls on the fluid. 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. Shown are the two graphene sheets which act as the movable walls, the CNT membrane model shown in Fig. 1, and the two water reservoirs connected by the membrane. Forces *f* _{ t } and *f* _{ b } act by the movable walls on the fluid. Carbon atoms in green, hydrogen atoms in white, and oxygen atoms in red.

Pressure produced in each water reservoir as a function of the simulation time (10 ns as an example shown here, real simulation time is longer and dependent on each case). Each pressure value is averaged in 10 ps after the pressure difference stabilizes in the NEMD simulation. The error about the mean value of the pressure is less than 0.2% for the high pressure on the top reservoir and is less than 10% for the low pressure on the bottom reservoir. The standard deviation for the pressure differences over all seven simulations is about 0.2 MPa.

Pressure produced in each water reservoir as a function of the simulation time (10 ns as an example shown here, real simulation time is longer and dependent on each case). Each pressure value is averaged in 10 ps after the pressure difference stabilizes in the NEMD simulation. The error about the mean value of the pressure is less than 0.2% for the high pressure on the top reservoir and is less than 10% for the low pressure on the bottom reservoir. The standard deviation for the pressure differences over all seven simulations is about 0.2 MPa.

Number flow rate of #4 CNT membranes in 30 ns simulations, shown as the slope of the linear trend in the net number of water molecules passing through the membrane as a function of the simulation time. The data shown are for membrane simulations of CNT #4 (Table I): (12, 12) CNT; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. The standard error of the slope is ±0.05 molecules/nm.

Number flow rate of #4 CNT membranes in 30 ns simulations, shown as the slope of the linear trend in the net number of water molecules passing through the membrane as a function of the simulation time. The data shown are for membrane simulations of CNT #4 (Table I): (12, 12) CNT; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. The standard error of the slope is ±0.05 molecules/nm.

Flux (rate per membrane area), shown both as molecular flux and conventional flux (in m/s), as a function of the membrane pore size as represented by the pore area. The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. H-P equation refers to the Hagen-Poiseuille equation (Eq. (1)), and solid line is the trend based on the simulation results. The statistical error for each data point is evaluated and less than 0.5% (see Fig. 4 for the case of the #4 membrane).

Flux (rate per membrane area), shown both as molecular flux and conventional flux (in m/s), as a function of the membrane pore size as represented by the pore area. The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. H-P equation refers to the Hagen-Poiseuille equation (Eq. (1)), and solid line is the trend based on the simulation results. The statistical error for each data point is evaluated and less than 0.5% (see Fig. 4 for the case of the #4 membrane).

Effect of pore size as represented by the pore radius on the density distributions along radial direction for the four membrane models: density as a function of *r*; values are averaged over a thin annular section in the pore at *r* (excluding the entrance and exit regions). The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. Solid lines are the trend curves based on the simulation results. The statistical error for each data point is less than 1%.

Effect of pore size as represented by the pore radius on the density distributions along radial direction for the four membrane models: density as a function of *r*; values are averaged over a thin annular section in the pore at *r* (excluding the entrance and exit regions). The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. Solid lines are the trend curves based on the simulation results. The statistical error for each data point is less than 1%.

Orthographic snapshots of water molecules in the different CNT membranes. Atoms are shown by balls with the van der Waals volumes: carbon atoms in green, hydrogen atoms in white, and oxygen atoms in red.

Orthographic snapshots of water molecules in the different CNT membranes. Atoms are shown by balls with the van der Waals volumes: carbon atoms in green, hydrogen atoms in white, and oxygen atoms in red.

Effect of pore size as represented by the pore radius on the distributions of *z* component velocity along the radial direction for the four membrane models: velocity as a function of *r*; values are averaged over a thin annular section in the pore at *r* (excluding the entrance and exit regions). The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. N-S equation refers to the Navier-Stokes equation (Eq. (2)). Dashed lines are the trends based on the Navier-Stokes equation, and solid lines are the trend curves based on the simulation results. The statistical errors for most data points are less than 10%.

Effect of pore size as represented by the pore radius on the distributions of *z* component velocity along the radial direction for the four membrane models: velocity as a function of *r*; values are averaged over a thin annular section in the pore at *r* (excluding the entrance and exit regions). The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. N-S equation refers to the Navier-Stokes equation (Eq. (2)). Dashed lines are the trends based on the Navier-Stokes equation, and solid lines are the trend curves based on the simulation results. The statistical errors for most data points are less than 10%.

Effect of pore size on the density distributions along the flow direction for the four membrane models: density as a function of *z*; values are averaged over a thin section at *z* (cylindrical sections in the pore and cuboid sections in the water reservoirs). The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. Dashed lines are the membrane boundaries along *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. Solid lines are the trend curves based on the simulation results. The statistical error for each data point is less than 1%.

Effect of pore size on the density distributions along the flow direction for the four membrane models: density as a function of *z*; values are averaged over a thin section at *z* (cylindrical sections in the pore and cuboid sections in the water reservoirs). The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. Dashed lines are the membrane boundaries along *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. Solid lines are the trend curves based on the simulation results. The statistical error for each data point is less than 1%.

Effect of pore size on the velocity (in *z* direction) distributions along the flow direction for the four membrane models: velocity as a function of *z*; values are averaged over a thin section at *z* (cylindrical sections in the pore and square cuboids sections in the water reservoirs). The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. Dashed lines are the membrane boundaries along *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. Solid lines are the trend curves based on the simulation results. The statistical error for each data point is less than 15%.

Effect of pore size on the velocity (in *z* direction) distributions along the flow direction for the four membrane models: velocity as a function of *z*; values are averaged over a thin section at *z* (cylindrical sections in the pore and square cuboids sections in the water reservoirs). The points are based on membrane simulations on CNT #1–4 (Table I): #1 (8, 8), #2 (10, 10), #3 (11, 11), and #4 (12, 12) CNT, respectively; *L* _{ z } = 6 nm. Δ*P* = 5.0 MPa; *T* = 300 K. Dashed lines are the membrane boundaries along *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. Solid lines are the trend curves based on the simulation results. The statistical error for each data point is less than 15%.

Flux (rate per membrane area), shown both as molecular flux and conventional flux (in m/s), as a function of the inverse membrane thickness. The points are based on membrane simulations on CNT #5, #4, #6, and #7 (Table I) corresponding to different lengths of 3 nm, 6 nm, 9 nm, and 15 nm, respectively, and *r* _{ p } = 0.643 nm. Δ*P* = 5.0 MPa; *T* = 300 K. H-P equation refers to the Hagen-Poiseuille equation (Eq. (1)), and the solid line is the trend based on the simulation results. The statistical error for each data point is less than 0.5% (see Fig. 4 for the case of #4 membrane).

Flux (rate per membrane area), shown both as molecular flux and conventional flux (in m/s), as a function of the inverse membrane thickness. The points are based on membrane simulations on CNT #5, #4, #6, and #7 (Table I) corresponding to different lengths of 3 nm, 6 nm, 9 nm, and 15 nm, respectively, and *r* _{ p } = 0.643 nm. Δ*P* = 5.0 MPa; *T* = 300 K. H-P equation refers to the Hagen-Poiseuille equation (Eq. (1)), and the solid line is the trend based on the simulation results. The statistical error for each data point is less than 0.5% (see Fig. 4 for the case of #4 membrane).

Effect of membrane thickness on the distributions along the radial direction for the four membrane models; values are averaged over a thin annular section in the pore at *r* (excluding the entrance and exit regions): (a) Density as a function of *r*; (b) Velocity, in *z* direction, as a function of *r*. The points are based on membrane simulations on CNT #5, #4, #6, and #7 (Table I) corresponding to (12, 12) CNT with different lengths of 3 nm, 6 nm, 9 nm, and 15 nm, respectively, and *r* _{ p } = 0.643 nm. Δ*P* = 5.0 MPa; *T* = 300 K. N-S equation refers to the Navier-Stokes equation (Eq. (2)). Dashed lines are the trends based on the Navier-Stokes equation, and solid lines are the trend curves based on the simulation results. The statistical error for each density data point is less than 1% and the statistical errors for most data points are less than 10%.

Effect of membrane thickness on the distributions along the radial direction for the four membrane models; values are averaged over a thin annular section in the pore at *r* (excluding the entrance and exit regions): (a) Density as a function of *r*; (b) Velocity, in *z* direction, as a function of *r*. The points are based on membrane simulations on CNT #5, #4, #6, and #7 (Table I) corresponding to (12, 12) CNT with different lengths of 3 nm, 6 nm, 9 nm, and 15 nm, respectively, and *r* _{ p } = 0.643 nm. Δ*P* = 5.0 MPa; *T* = 300 K. N-S equation refers to the Navier-Stokes equation (Eq. (2)). Dashed lines are the trends based on the Navier-Stokes equation, and solid lines are the trend curves based on the simulation results. The statistical error for each density data point is less than 1% and the statistical errors for most data points are less than 10%.

Effect of membrane thickness on the distributions along flow direction for the four membrane models; values are averaged over a thin section at *z* (cylindrical sections in the pore and square cuboids sections in the water reservoirs): (a) density distributions as a function of *z* and (b) velocity, in *z* direction, as a function of *z*. The points are based on membrane simulations on CNT #5, #4, #6, and #7 (Table I) corresponding to (12, 12) CNT with different length of 3 nm, 6 nm, 9 nm, and 15 nm, respectively, and *r* _{ p } = 0.643 nm. Δ*P* = 5.0 MPa; *T* = 300 K. 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. Solid lines are the trend curves based on the simulation results. The statistical error for each density data point and each velocity data point are less than 1%, and less than 15%, respectively.

Effect of membrane thickness on the distributions along flow direction for the four membrane models; values are averaged over a thin section at *z* (cylindrical sections in the pore and square cuboids sections in the water reservoirs): (a) density distributions as a function of *z* and (b) velocity, in *z* direction, as a function of *z*. The points are based on membrane simulations on CNT #5, #4, #6, and #7 (Table I) corresponding to (12, 12) CNT with different length of 3 nm, 6 nm, 9 nm, and 15 nm, respectively, and *r* _{ p } = 0.643 nm. Δ*P* = 5.0 MPa; *T* = 300 K. 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. Solid lines are the trend curves based on the simulation results. The statistical error for each density data point and each velocity data point are less than 1%, and less than 15%, respectively.

## Tables

Specifications of different CNT membrane physical parameters used in the simulations.

Specifications of different CNT membrane physical parameters used in the simulations.

Comparison of the enhancement factors^{a} between our NEMD simulation and the experimental data of Qin.^{45}

Comparison of the enhancement factors^{a} between our NEMD simulation and the experimental data of Qin.^{45}

Enhancement factors^{a} of the #4–7 CNT membranes at 5.0 MPa pressure difference.

Enhancement factors^{a} of the #4–7 CNT membranes at 5.0 MPa pressure difference.

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