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Ballistic molecular transport through two-dimensional channels

Naturevolume 558pages420424 (2018) | Download Citation


Gas permeation through nanoscale pores is ubiquitous in nature and has an important role in many technologies1,2. Because the pore size is typically smaller than the mean free path of gas molecules, the flow of the gas molecules is conventionally described by Knudsen theory, which assumes diffuse reflection (random-angle scattering) at confining walls3,4,5,6,7. This assumption holds surprisingly well in experiments, with only a few cases of partially specular (mirror-like) reflection known5,8,9,10,11. Here we report gas transport through ångström-scale channels with atomically flat walls12,13 and show that surface scattering can be either diffuse or specular, depending on the fine details of the atomic landscape of the surface, and that quantum effects contribute to the specularity at room temperature. The channels, made from graphene or boron nitride, allow helium gas flow that is orders of magnitude faster than expected from theory. This is explained by specular surface scattering, which leads to ballistic transport and frictionless gas flow. Similar channels, but with molybdenum disulfide walls, exhibit much slower permeation that remains well described by Knudsen diffusion. We attribute the difference to the larger atomic corrugations at molybdenum disulfide surfaces, which are similar in height to the size of the atoms being transported and their de Broglie wavelength. The importance of this matter-wave contribution is corroborated by the observation of a reversed isotope effect, whereby the mass flow of hydrogen is notably higher than that of deuterium, in contrast to the relation expected for classical flows. Our results provide insights into the atomistic details of molecular permeation, which previously could be accessed only in simulations10,14, and demonstrate the possibility of studying gas transport under controlled confinement comparable in size to the quantum-mechanical size of atoms.

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  1. 1.

    Baker, R. W. Membrane Technology and Applications 2nd edn (John Wiley & Sons, Chichester, 2004).

  2. 2.

    Lu, G. Q. & Zhao, X. S. Nanoporous Materials: Science and Engineering (Imperial College Press, London, 2004).

  3. 3.

    Knudsen, M. Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren. Ann. Phys. 333, 75–130 (1909).

  4. 4.

    v. Smoluchowski, M. Zur kinetischen Theorie der Transpiration und Diffusion verdünnter Gase. Ann. Phys. 338, 1559–1570 (1910).

  5. 5.

    Steckelmacher, W. A review of the molecular flow conductance for systems of tubes and components and the measurement of pumping speed. Vacuum 16, 561–584 (1966).

  6. 6.

    Livesey, R. G. in Foundations of Vacuum Science and Technology (ed. Lafferty, J. M.) Ch. 2 (John Wiley & Sons, New York, 1998).

  7. 7.

    Lei, W., Rigozzi, M. K. & McKenzie, D. R. The physics of confined flow and its application to water leaks, water permeation and water nanoflows: a review. Rep. Prog. Phys. 79, 025901 (2016).

  8. 8.

    Holt, J. K. et al. Fast mass transport through sub-2-nanometer carbon nanotubes. Science 312, 1034–1037 (2006).

  9. 9.

    Agrawal, A. & Prabhu, S. V. Survey on measurement of tangential momentum accommodation coefficient. J. Vac. Sci. Technol. A 26, 634–645 (2008).

  10. 10.

    Bhatia, S. K., Bonilla, M. R. & Nicholson, D. Molecular transport in nanopores: a theoretical perspective. Phys. Chem. Chem. Phys. 13, 15350–15383 (2011).

  11. 11.

    Majumder, M., Chopra, N. & Hinds, B. J. Mass transport through carbon nanotube membranes in three different regimes: ionic diffusion and gas and liquid flow. ACS Nano 5, 3867–3877 (2011).

  12. 12.

    Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419–425 (2013).

  13. 13.

    Radha, B. et al. Molecular transport through capillaries made with atomic-scale precision. Nature 538, 222–225 (2016).

  14. 14.

    Zhang, W.-M., Meng, G. & Wei, X. A review on slip models for gas microflows. Microfluid. Nanofluidics 13, 845–882 (2012).

  15. 15.

    Koenig, S. P., Wang, L., Pellegrino, J. & Bunch, J. S. Selective molecular sieving through porous graphene. Nat. Nanotechnol. 7, 728–732 (2012).

  16. 16.

    Berry, V. Impermeability of graphene and its applications. Carbon 62, 1–10 (2013).

  17. 17.

    Celebi, K. et al. Ultimate permeation across atomically thin porous graphene. Science 344, 289–292 (2014).

  18. 18.

    Wang, L. et al. Molecular valves for controlling gas phase transport made from discrete ångström-sized pores in graphene. Nat. Nanotechnol. 10, 785–790 (2015).

  19. 19.

    Kim, H. W. et al. Selective gas transport through few-layered graphene and graphene oxide membranes. Science 342, 91–95 (2013).

  20. 20.

    Li, H. et al. Ultrathin, molecular-sieving graphene oxide membranes for selective hydrogen separation. Science 342, 95–98 (2013).

  21. 21.

    Liu, G., Jin, W. & Xu, N. Two-dimensional-material membranes: a new family of high-performance separation membranes. Angew. Chem. Int. Ed. 55, 13384–13397 (2016).

  22. 22.

    Wang, L. et al. Fundamental transport mechanisms, fabrication and potential applications of nanoporous atomically thin membranes. Nat. Nanotechnol. 12, 509–522 (2017).

  23. 23.

    Falk, K. et al. Molecular origin of fast water transport in carbon nanotube membranes: superlubricity versus curvature dependent friction. Nano Lett. 10, 4067–4073 (2010).

  24. 24.

    Gass, M. H. et al. Free-standing graphene at atomic resolution. Nat. Nanotechnol. 3, 676–681 (2008).

  25. 25.

    Haigh, S. J. et al. Cross-sectional imaging of individual layers and buried interfaces of graphene-based heterostructures and superlattices. Nat. Mater. 11, 764–767 (2012).

  26. 26.

    Li, Z. et al. Effect of airborne contaminants on the wettability of supported graphene and graphite. Nat. Mater. 12, 925–931 (2013).

  27. 27.

    Arya, G., Chang, H.-C. & Maginn, E. J. Knudsen diffusivity of a hard sphere in a rough slit pore. Phys. Rev. Lett. 91, 026102 (2003).

  28. 28.

    You, H. X., Brown, N. M. D. & Al-Assadi, K. F. Radio-frequency plasma etching of graphite with oxygen: a scanning tunnelling microscope study. Surf. Sci. 284, 263–272 (1993).

  29. 29.

    Addou, R., Colombo, L. & Wallace, R. M. Surface defects on natural MoS2. ACS Appl. Mater. Interfaces 7, 11921–11929 (2015).

  30. 30.

    Mehio, N., Dai, S. & Jiang, D. Quantum mechanical basis for kinetic diameters of small gaseous molecules. J. Phys. Chem. A 118, 1150–1154 (2014).

  31. 31.

    Sharvin, Y. V. A possible method for studying Fermi surfaces. Sov. Phys. JETP 21, 655–656 (1965).

  32. 32.

    Giannuzzi, L. A. & Stevie, F. A. (eds) Introduction to Focused Ion Beams: Instrumentation, Theory, Techniques and Practice (Springer, New York, 2005).

  33. 33.

    Esfandiar, A. et al. Size effect in ion transport through angstrom-scale slits. Science 358, 511–513 (2017).

  34. 34.

    Schaffer, M., Schaffer, B. & Ramasse, Q. Sample preparation for atomic-resolution STEM at low voltages by FIB. Ultramicroscopy 114, 62–71 (2012).

  35. 35.

    Bentley, J. Electron density as a descriptor of thermal molecular size. J. Phys. Chem. A 104, 9630–9635 (2000).

  36. 36.

    VandeVondele, J. et al. Quickstep: fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comput. Phys. Commun. 167, 103–128 (2005).

  37. 37.

    Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

  38. 38.

    VandeVondele, J. & Hutter, J. Gaussian basis sets for accurate calculations on mole-cular systems in gas and condensed phases. J. Chem. Phys. 127, 114105 (2007).

  39. 39.

    Goedecker, S., Teter, M. & Hutter, J. Separable dual-space Gaussian pseudopotentials. Phys. Rev. B 54, 1703–1710 (1996).

  40. 40.

    Lu, T. & Chen, F. Multiwfn: a multifunctional wavefunction analyzer. J. Comput. Chem. 33, 580–592 (2012).

  41. 41.

    Jorgensen, W. L., Maxwell, D. S. & Tirado-Rives, J. Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic liquids. J. Am. Chem. Soc. 118, 11225–11236 (1996).

  42. 42.

    Laio, A. & Parrinello, M. Escaping free-energy minima. Proc. Natl Acad. Sci. USA 99, 12562–12566 (2002).

  43. 43.

    Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1–19 (1995).

  44. 44.

    Rubinstein, M. & Colby, R. H. Polymer Physics Ch. 2, 3 (Oxford Univ. Press, Oxford, 2003).

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This work was supported by the European Research Council, Lloyd’s Register Foundation, the EU Graphene Flagship and the Royal Society. B.R. acknowledges a Leverhulme Early Career Fellowship, a L’Oréal Fellowship for Women in Science and EPSRC grant EP/R013063/1. F.C.W. acknowledges support from the National Natural Science Foundation of China (11772319 and 11572307) and the Shanghai Supercomputer Center. S.J.H. and A.P.R. were funded by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grant agreements ERC-2016-STG-EvoluTEM-715502 and DISCOVERER-2017 737183), the US Defence Threat Reduction Agency (HDTRA1-12-1-0013) and the EPSRC (EP/P009050/1 and EP/K016946/1).

Reviewer information

Nature thanks L. Bocquet and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Author information


  1. School of Physics and Astronomy, University of Manchester, Manchester, UK

    • A. Keerthi
    • , A. K. Geim
    • , A. Janardanan
    • , S. A. Dar
    • , I. V. Grigorieva
    • , F. C. Wang
    •  & B. Radha
  2. National Graphene Institute, University of Manchester, Manchester, UK

    • A. Keerthi
    • , A. K. Geim
    • , A. Esfandiar
    • , S. Hu
    • , S. A. Dar
    •  & B. Radha
  3. School of Materials, University of Manchester, Manchester, UK

    • A. P. Rooney
    •  & S. J. Haigh
  4. Department of Physics, Sharif University of Technology, Tehran, Iran

    • A. Esfandiar
  5. Chinese Academy of Sciences Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, University of Science and Technology of China, Hefei, China

    • F. C. Wang


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A.K.G. and B.R. designed and directed the project. A.K., B.R., A.J., S.H., A.E and S.A.D. fabricated the devices. A.K., B.R. and A.J. performed the measurements and their analysis. F.C.W. provided theoretical support. A.P.R. and S.J.H. carried out TEM imaging. A.K.G., B.R., F.C.W., A.K. and I.V.G. wrote the manuscript. All authors contributed to discussions.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to A. K. Geim or B. Radha.

Extended data figures and tables

  1. Extended Data Fig. 1 Fabrication procedures.

    a, Thin crystal was transferred to cover an opening in a SiN x membrane. b, The opening is extended through the bottom crystal. c, Spacer stripes were deposited on top of the bottom layer and etched from the back side. The top crystal is then transferred on top to fully cover the rectangular opening. d, Left, optical image of a single-channel capillary device made entirely from graphite; N = 5. The SiN x /Si wafer is seen in dark green; the SiN x membrane appears as a light-green square; the top graphite layer shows up in bright yellow; and the rectangular opening (lighter green) is indicated by the black arrow. Centre, atomic force micrograph near the channel entry, where the top graphite does not cover the spacer layer (the scan area is shown by the red contour). The height profile taken along the dotted white line is shown on the right, indicating h ≈ 1.7 ± 0.1 nm. The side cavities perpendicular to the 2D channel were made to prevent contamination bubbles25 across the main channel. e, A gold mask is placed on top of the trilayer assembly for final etching, to define L and to unblock the channel entry. f, Optical image of the final capillary device in the transmission mode. The SiN x membrane is fully transparent (bright). The Au mask is partially transparent, and both the top graphite and the rectangular hole in SiN x can be seen underneath the Au, as indicated by the arrows.

  2. Extended Data Fig. 2 Visualization of 2D channels.

    a, Array of 2D slits made entirely from MoS2, as imaged in bright-field STEM. For guidance, the edges of one of the channels are indicated by red marks. The vertical stripes are from the curtaining effect caused by ion milling32. b, High-magnification STEM image of a 2D channel with the top and bottom walls made from MoS2 and bilayer graphene as the spacer (right panel). The channel is white in the bright-field image, and atomic layers of MoS2 can be seen as dark lines running parallel to the channel. The left panel shows a contrast profile across the region indicated by the red rectangle. Cross-sectional images of 2D slits made entirely from graphite crystals can be found in ref. 13.

  3. Extended Data Fig. 3 Helium permeation measurements.

    a, Schematic of our experimental set-up. b, Helium flow through round apertures of various diameters as measured by our He-leak detector (symbols). Red line, expected Knudsen flow through these apertures (no fitting parameters). Inset, optical image of one of the apertures. The error bars are from measurements using different devices.

  4. Extended Data Fig. 4 Intrinsic roughness of atomically flat surfaces.

    Electron-density profiles near graphite, h-BN and MoS2 surfaces are shown. Schematics of the atomic structures are shown on top. The red curves indicate the thermal exclusion surfaces.

  5. Extended Data Fig. 5 MD simulations for a heavy polymer molecule inside ångström-scale channels.

    a, Sketch of our simulation set-up. b, c, Energy of a PMMA molecule (M = 40,000) for slits with N = 4 (b) and N = 12 (c). The axes show the position of the centre of mass with respect to the entrance edge. The origin of the axes is shown in a. The colour scales to the right show the relative free energy (potential of mean force, PMF).

  6. Extended Data Fig. 6 Height of absorbed PMMA.

    MD results are shown for the apparent (circles) and gyration (squares) heights of PMMA on graphene. The shaded areas indicate the standard errors using the data for M ≥ 40,000. Error bars show the standard error from our simulation runs lasting 10 ns.

  7. Extended Data Fig. 7 Finite-size entry effect.

    If an incident atom of diameter d hits one of the edges of the channel, it can be reflected. To avoid this, the centre trajectory should be d/(2cosθ) away from the edge, effectively reducing the entry aperture to h*(θ) = h − d/cosθ.

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