^{1}, J. Adler

^{1,a)}and M. Sheintuch

^{2}

### Abstract

We model and simulate gas flow through nanopores using a single-walled carbon nanotubemodel. Efficient protocols for the simulation of methane molecules in nanotubes are developed and validated for both the self-diffusivity, following a pulse perturbation, and for the transport diffusivity in an imposed concentration gradient. The former is found to be at least an order of magnitude lower than the latter, and to decline with increasing initial pressure, while the latter increases as the pressure gradient increases until it reaches an asymptotic value. Our previous analytic model, developed for single-file diffusion in narrow pores, is extended to wider pores for the case of single species transport. The model, which predicts the observed numerical results invokes four regimes of transport. The dominant transport is by ballistic motion near the wall in not too wide nanotubes when a pressure gradient or concentration is imposed; this mode is absent in the case of self-diffusion due to periodic boundary conditions. We also present results from systematic comparisons of flexible versus rigid tubes and explicit atom versus effective atomic potentials.

Animations^{49} of the molecular transport have been extended into a stereoscopic 3D version in collaboration with H. Zilken at Juelich as part of the Technion-Juelich Umbrella cooperation and were presented at International Supercomputing Conference (ISC) 2010. We thank the Umbrella project for support. We acknowledge the European inititive for High Performance Computing (HPC-Europa) project for support of the computations on the Intel EM64T cluster at HLRS in Stuttgart under Grant No. RII3-CT-2003-506079. T.M. acknowledges support from the Leonard and Diane Sherman Foundation, and thanks H. Herrman for hospitality during the HPC-EUROPA visit. M.S. acknowledges the support of the US-Israel Binational Science Foundation.

I. INTRODUCTION

II. SIMULATION METHOD AND INTERATOMIC POTENTIALS

A. Self-diffusivity simulations

B. Transport diffusivity simulations

III. THEORY

A. Narrow pores

B. Theory of transport in wider pores

C. Parameter estimation

IV. RESULTS

A. Summary of predictions

B. Self-diffusivities

C. Transport diffusivities

V. DISCUSSION OF SIMULATION RESULTS

A. Rigid vs. flexible

B. Algorithms and potentials

VI. CONCLUSIONS AND SUMMARY

### Key Topics

- Carbon nanotubes
- 64.0
- Diffusion
- 48.0
- Ballistic transport
- 21.0
- Self diffusion
- 17.0
- Trajectory models
- 15.0

##### B82B1/00

## Figures

Schematic representation of a DCV-GCMD simulation geometry. EMD simulations are performed for the entire system and GCMC simulations are preformed in between EMD steps to maintain the chemical potentials in both CVs. The hard walls bounce back approaching molecules with opposite velocity direction and no energy loss. The nanotube is of length L.

Schematic representation of a DCV-GCMD simulation geometry. EMD simulations are performed for the entire system and GCMC simulations are preformed in between EMD steps to maintain the chemical potentials in both CVs. The hard walls bounce back approaching molecules with opposite velocity direction and no energy loss. The nanotube is of length L.

X-Y section of a wide pore. Two radial zones are defined: Adsorbed layer for *r* _{ p } − σ < *r* < *r* _{ p }, and bulk layer for *r* < *r* _{ p } − σ.

X-Y section of a wide pore. Two radial zones are defined: Adsorbed layer for *r* _{ p } − σ < *r* < *r* _{ p }, and bulk layer for *r* < *r* _{ p } − σ.

Self-diffusivity, *D* _{ s }, of at 300 K in (12,12) and (10,10) SWCNTs. Sholl and Johnson's results^{11} are added for comparison. Error bars are smaller than the symbol sizes.

Self-diffusivity, *D* _{ s }, of at 300 K in (12,12) and (10,10) SWCNTs. Sholl and Johnson's results^{11} are added for comparison. Error bars are smaller than the symbol sizes.

Snapshot from EMD simulations of CH_{4} molecules diffusing in a (12,12) SWCNT under a pressure of 10 bars. The carbon atoms of the nanotubes are drawn in a smaller size, so that we can observe the molecules within the tube.

Snapshot from EMD simulations of CH_{4} molecules diffusing in a (12,12) SWCNT under a pressure of 10 bars. The carbon atoms of the nanotubes are drawn in a smaller size, so that we can observe the molecules within the tube.

Projection (along the *z* direction) from EMD simulations of CH_{4} molecules diffusing in a (12,12) SWCNT under a pressure of 10 bars. The carbon atoms of the nanotubes are drawn in a smaller size, so that we can observe the molecules within the tube.

Projection (along the *z* direction) from EMD simulations of CH_{4} molecules diffusing in a (12,12) SWCNT under a pressure of 10 bars. The carbon atoms of the nanotubes are drawn in a smaller size, so that we can observe the molecules within the tube.

Density profiles for CH_{4} adsorbed in a (12,12) nanotube as a function of the molecular pressure.

Density profiles for CH_{4} adsorbed in a (12,12) nanotube as a function of the molecular pressure.

Trajectory of a specific CH_{4} molecule in a (12,12) nanotube (**r** = 8.14 Å) at 300 K and pressures of: 50 bars (left), 30 bars (middle), and 5 bars (right).

Trajectory of a specific CH_{4} molecule in a (12,12) nanotube (**r** = 8.14 Å) at 300 K and pressures of: 50 bars (left), 30 bars (middle), and 5 bars (right).

Transport diffusivity, *D* _{ t }, of CH_{4} as a function of ΔP = P_{CV1} − P_{CV2} in flexible and rigid (10,10) SWCNT at 300 K. The values of the gas pressure in the CVs are: P_{CV1} = 7.5 bars, P_{CV2} = 9.5, 13, 17, and 30 bars.

Transport diffusivity, *D* _{ t }, of CH_{4} as a function of ΔP = P_{CV1} − P_{CV2} in flexible and rigid (10,10) SWCNT at 300 K. The values of the gas pressure in the CVs are: P_{CV1} = 7.5 bars, P_{CV2} = 9.5, 13, 17, and 30 bars.

Snapshot from DCV-GCMD simulation describing transport diffusion of CH_{4} molecules in a (12,12) SWCNT caused by applying a chemical potential gradient over the tube. The pressure of the gas in CV2 is 105 bars and in CV1 it is 7.5 bars. The carbon atoms of the nanotubes are drawn in a smaller size, so that we can observe the molecules within the tube.

Snapshot from DCV-GCMD simulation describing transport diffusion of CH_{4} molecules in a (12,12) SWCNT caused by applying a chemical potential gradient over the tube. The pressure of the gas in CV2 is 105 bars and in CV1 it is 7.5 bars. The carbon atoms of the nanotubes are drawn in a smaller size, so that we can observe the molecules within the tube.

Transport diffusivity, *D* _{ t }, of as a function of ΔP = P_{CV1} − P_{CV2}, at 300 K in (12,12) and (10,10) rigid SWCNTs. The pressure values are: P_{CV1}=7.5 bars, P_{CV2}=13, 17, 30, 40, 50, 70, 93, and 105 bars. The asymptotic values, predicted by Eq. (37), for (12,12) and (10,10) nanotubes, respectively, are marked by solid and dashed lines, respectively. Error bars on the simulation data points are smaller than the symbol sizes.

Transport diffusivity, *D* _{ t }, of as a function of ΔP = P_{CV1} − P_{CV2}, at 300 K in (12,12) and (10,10) rigid SWCNTs. The pressure values are: P_{CV1}=7.5 bars, P_{CV2}=13, 17, 30, 40, 50, 70, 93, and 105 bars. The asymptotic values, predicted by Eq. (37), for (12,12) and (10,10) nanotubes, respectively, are marked by solid and dashed lines, respectively. Error bars on the simulation data points are smaller than the symbol sizes.

Comparison of the same transport diffusivity results for (10,10) and (12,12) SWCNTs as a function of ΔP as shown in Fig 10 with points calculated from Eq. (29). The calculated points are shown by solid symbols of the same shape. Error bars on the simulations are smaller than the symbol sizes.

Comparison of the same transport diffusivity results for (10,10) and (12,12) SWCNTs as a function of ΔP as shown in Fig 10 with points calculated from Eq. (29). The calculated points are shown by solid symbols of the same shape. Error bars on the simulations are smaller than the symbol sizes.

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