Transport and dispersion across wiggling nanopores


The transport of fluids at the nanoscale has achieved major breakthroughs over recent years1,2,3,4; however, artificial channels still cannot match the efficiency of biological porins in terms of fluxes or selectivity. Pore shape agitation—due to thermal fluctuations or in response to external stimuli—is believed to facilitate transport in biochannels5,6,7,8,9, but its impact on transport in artificial pores remains largely unexplored. Here we introduce a general theory for transport through thermally or actively fluctuating channels, which quantifies the impact of pore fluctuations on confined diffusion in terms of the spectral statistics of the channel fluctuations. Our findings demonstrate a complex interplay between transport and surface wiggling: agitation enhances diffusion via the induced fluid flow, but spatial variations in pore geometry can induce a slowing down via entropic trapping, in full agreement with molecular dynamics simulations and existing observations from the literature. Our results elucidate the impact of pore agitation in a broad range of artificial and biological porins, but also, at larger scales, in vascular motion in fungi, intestinal contractions and microfluidic surface waves. These results open up the possibility that transport across membranes can be actively tuned by external stimuli, with potential applications to nanoscale pumping, osmosis and dynamical ultrafiltration.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Spectral mixing under interface fluctuations.
Fig. 2: Renormalized diffusion under thermally fluctuating and actively driven surfaces.
Fig. 3: Enhanced or decreased transport under pore shape wiggling versus the dimensionless Péclet-like number for various fluid transporters.


  1. 1.

    Feng, J. et al. Single-layer MoS2 nanopores as nanopower generators. Nature 536, 197–200 (2016).

    ADS  Article  Google Scholar 

  2. 2.

    Secchi, E. et al. Massive radius-dependent flow slippage in carbon nanotubes. Nature 537, 210–213 (2016).

    ADS  Article  Google Scholar 

  3. 3.

    Tunuguntla, R. H. et al. Enhanced water permeability and tunable ion selectivity in subnanometer carbon nanotube porins. Science 357, 792–796 (2017).

    Article  Google Scholar 

  4. 4.

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

    ADS  Article  Google Scholar 

  5. 5.

    Wand, A. J. Dynamic activation of protein function: a view emerging from NMR spectroscopy. Nat. Struct. Mol. Biol. 8, 926–931 (2001).

    Article  Google Scholar 

  6. 6.

    Noskov, S. Y., Berneche, S. & Roux, B. Control of ion selectivity in potassium channels by electrostatic and dynamic properties of carbonyl ligands. Nature 431, 830–834 (2004).

    ADS  Article  Google Scholar 

  7. 7.

    Bhabha, G. et al. A dynamic knockout reveals that conformational fluctuations influence the chemical step of enzyme catalysis. Science 332, 234–238 (2011).

    ADS  Article  Google Scholar 

  8. 8.

    Wei, G., Xi, W., Nussinov, R. & Ma, B. Protein ensembles: how does nature harness thermodynamic fluctuations for life? The diverse functional roles of conformational ensembles in the cell. Chem. Rev. 116, 6516–6551 (2016).

    Article  Google Scholar 

  9. 9.

    Allen, T. W., Kuyucak, S. & Chung, S.-H. Molecular dynamics study of the KcsA potassium channel. Biophys. J. 77, 2502–2516 (1999).

    Article  Google Scholar 

  10. 10.

    Moseler, M. & Landman, U. Formation, stability, and breakup of nanojets. Science 289, 1165–1169 (2000).

    ADS  Article  Google Scholar 

  11. 11.

    Davidovitch, B., Moro, E. & Stone, H. A. Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations. Phys. Rev. Lett. 95, 244505 (2005).

    ADS  Article  Google Scholar 

  12. 12.

    Fetzer, R., Rauscher, M., Seemann, R., Jacobs, K. & Mecke, K. Thermal noise influences fluid flow in thin films during spinodal dewetting. Phys. Rev. Lett. 99, 114503 (2007).

    ADS  Article  Google Scholar 

  13. 13.

    Ma, M. et al. Water transport inside carbon nanotubes mediated by phonon-induced oscillating friction. Nat. Nanotech 10, 692–695 (2015).

    ADS  Article  Google Scholar 

  14. 14.

    Ma, M., Tocci, G., Michaelides, A. & Aeppli, G. Fast diffusion of water nanodroplets on graphene. Nat. Mater. 15, 66 (2016).

    ADS  Article  Google Scholar 

  15. 15.

    Cruz-Chú, E. R. et al. On phonons and water flow enhancement in carbon nanotubes. Nat. Nanotech. 12, 1106 (2017).

    ADS  Article  Google Scholar 

  16. 16.

    Yeo, L. Y. & Friend, J. R. Surface acoustic wave microfluidics. Annu. Rev. Fluid Mech. 46, 379–406 (2014).

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Alim, K., Amselem, G., Peaudecerf, F., Brenner, M. P., & Pringle, A. Random network peristalsis in Physarum polycephalum organizes fluid flows across an individual. Proc. Natl Acad. Sci. USA 110, 13306–13311 (2013).

    ADS  Article  Google Scholar 

  18. 18.

    Cremer, J. et al. Effect of flow and peristaltic mixing on bacterial growth in a gut-like channel. Proc. Natl Acad. Sci. USA 113, 11414–11419 (2016).

    ADS  Article  Google Scholar 

  19. 19.

    Taylor, G. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. London Ser. A 219, 186–203 (1953).

    ADS  Article  Google Scholar 

  20. 20.

    Aris, R. On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235, 67–77 (1956).

    ADS  Article  Google Scholar 

  21. 21.

    Reguera, D. & Rubi, J. Kinetic equations for diffusion in the presence of entropic barriers. Phys. Rev. E 64, 061106 (2001).

    ADS  Article  Google Scholar 

  22. 22.

    Malgaretti, P., Pagonabarraga, I. & Rubi, J. M. Entropic electrokinetics: recirculation, particle separation, and negative mobility. Phys. Rev. Lett. 113, 128301 (2014).

    ADS  Article  Google Scholar 

  23. 23.

    Kuroda, S., Takagi, S., Nakagaki, T. & Ueda, T. Allometry in Physarum plasmodium during free locomotion: size versus shape, speed and rhythm. J. Exp. Biol. 218, 3729–3738 (2015).

    Article  Google Scholar 

  24. 24.

    Tlalka, M., Bebber, D., Darrah, P., Watkinson, S. & Fricker, M. Emergence of self-organised oscillatory domains in fungal mycelia. Fungal Genet. Biol. 44, 1085–1095 (2007).

    Article  Google Scholar 

  25. 25.

    Bocquet, L. & Charlaix, E. Nanofluidics, from bulk to interfaces. Chem. Soc. Rev. 39, 1073–1095 (2010).

    Article  Google Scholar 

  26. 26.

    Démery, V. & Dean, D. S. Perturbative path-integral study of active- and passive-tracer diffusion in fluctuating fields. Phys. Rev. E 84, 011148 (2011).

    ADS  Article  Google Scholar 

  27. 27.

    Grün, G., Mecke, K. & Rauscher, M. Thin-film flow influenced by thermal noise. J. Stat. Phys. 122, 1261–1291 (2006).

    ADS  MathSciNet  Article  Google Scholar 

  28. 28.

    Stubenrauch, C. & Von Klitzing, R. Disjoining pressure in thin liquid foam and emulsion films? New concepts and perspectives. J. Phys. Condens. Matter 15, R1197 (2003).

    ADS  Article  Google Scholar 

  29. 29.

    Gravelle, S., Ybert, C., Bocquet, L. & Joly, L. Anomalous capillary filling and wettability reversal in nanochannels. Phys. Rev. E 93, 033123 (2016).

    ADS  Article  Google Scholar 

  30. 30.

    Haldoupis, E., Watanabe, T., Nair, S. & Sholl, D. S. Quantifying large effects of framework flexibility on diffusion in MOFs: CH4 and CO2 in ZIF-8. ChemPhysChem 13, 3449–3452 (2012).

    Article  Google Scholar 

  31. 31.

    Thomas, J. A., Turney, J. E., Iutzi, R. M., Amon, C. H. & McGaughey, A. J. Predicting phonon dispersion relations and lifetimes from the spectral energy density. Phys. Rev. B 81, 081411 (2010).

    ADS  Article  Google Scholar 

  32. 32.

    Compoint, M., Carloni, P., Ramseyer, C. & Girardet, C. Molecular dynamics study of the KcsA channel at 2.0-Å resolution: stability and concerted motions within the pore. Biochim. Biophys. Acta Biomembr. 1661, 26–39 (2004).

    Article  Google Scholar 

  33. 33.

    Park, K. S. et al. Exceptional chemical and thermal stability of zeolitic imidazolate frameworks. Proc. Natl Acad. Sci. USA 103, 10186–10191 (2006).

    ADS  Article  Google Scholar 

  34. 34.

    Qiu, H., Shen, R. & Guo, W. Vibrating carbon nanotubes as water pumps. Nano Res. 4, 284–289 (2011).

    Article  Google Scholar 

  35. 35.

    Froehlich, J. M. et al. Small bowel motility assessment with magnetic resonance imaging. J. Magn. Reson. Imaging 21, 370–375 (2005).

    Article  Google Scholar 

  36. 36.

    Stewart, P. A. & Stewart, B. T. Protoplasmic movement in slime mold plasmodia: the diffusion drag force hypothesis. Exp. Cell Res. 17, 44 (1959).

    Article  Google Scholar 

  37. 37.

    Girard, P., Prost, J. & Bassereau, P. Passive or active fluctuations in membranes containing proteins. Phys. Rev. Lett. 94, 088102 (2005).

    ADS  Article  Google Scholar 

  38. 38.

    Sazonova, V. et al. A tunable carbon nanotube electromechanical oscillator. Nature 431, 284–287 (2004).

    ADS  Article  Google Scholar 

  39. 39.

    Herterich, K. & Hasselmann, K. The horizontal diffusion of tracers by surface waves. J. Phys. Oceanogr. 12, 704–711 (1982).

    ADS  Article  Google Scholar 

  40. 40.

    Marbach, S. & Bocquet, L. Active sieving across driven nanopores for tunable selectivity. J. Chem. Phys. 147, 154701 (2017).

    ADS  Article  Google Scholar 

  41. 41.

    Kowalczyk, S. W., Wells, D. B., Aksimentiev, A. & Dekker, C. Slowing down DNA translocation through a nanopore in lithium chloride. Nano. Lett. 12, 1038–1044 (2012).

    ADS  Article  Google Scholar 

  42. 42.

    Keyser, U. F. Controlling molecular transport through nanopores. J. R. Soc. Interface 12, 1369–1378 (2011).

    Article  Google Scholar 

  43. 43.

    Israelachvili, J. N. & Adams, G. E. Measurement of forces between two mica surfaces in aqueous electrolyte solutions in the range 0–100 nm. J. Chem. Soc. Faraday Trans. 1 74, 975–1001 (1978).

    Article  Google Scholar 

  44. 44.

    Berman, H. M. et al. in International Tables for Crystallography Vol. F (eds Rossman, M. G. & Arnold, E.) 675–684 (Springer, Dordrecht, 2006).

  45. 45.

    Siria, A. & Niguès, A. Electron beam detection of a nanotube scanning force microscope. Sci. Rep. 7, 11595 (2017).

    ADS  Article  Google Scholar 

Download references


The authors are indebted to B. Rotenberg for several fruitful discussions on molecular dynamics, and to K. Alim for bringing to the discussion biologically related examples. The authors also thank F.-X. Courdert, D.avid Lacoste and J.-F. Joanny for interesting discussions. S.M. acknowledges funding from a J.-P. Aguilar grant of the CFM foundation. D.S.D. acknowledges funding from the ANR grant FISICS. L.B. acknowledges support from ANR grant Neptune. This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir supervised by the Agence Nationale pour la Recherche.

Author information




L.B. designed the research. S.M., D.S.D. and L.B. conducted research. S.M. carried out the molecular dynamics simulations. S.M., D.S.D. and L.B. wrote the paper.

Corresponding author

Correspondence to Lydéric Bocquet.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Supplementary Information

7 Figures, 7 Tables, 42 References

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Marbach, S., Dean, D.S. & Bocquet, L. Transport and dispersion across wiggling nanopores. Nature Phys 14, 1108–1113 (2018).

Download citation

Further reading