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Two-dimensional long-range uniaxial order in three-dimensional active fluids

Nature Physics volume 19, pages 733–740 (2023)Cite this article

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Abstract

Elongated active units cannot spontaneously break rotation symmetry in bulk fluids to form nematic or polar phases. This has led to the image of active suspensions as spontaneously evolving, spatiotemporally chaotic fluids. Here, in contrast, I show that bulk active fluids have stable active nematic and polar states at fluid–fluid or fluid–air interfaces. The active flow-mediated long-range interactions that destroy the ordered phase in bulk lead to long-range order at the interface. Thus, active fluids have a surface ordering transition and form states with quiescent, ordered surfaces and chaotic bulk. I further consider active units that are constrained to live at an interface to examine the minimal conditions for the existence of two-dimensional order in bulk three-dimensional fluids. In this case, immotile units do not order, but motile particles still form a long-range-ordered polar phase. This prediction of stable, uniaxial, active phases in bulk fluids may have functional consequences for active transport.

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Fig. 1: Interfacial active fluids.
Fig. 2: Stability diagram of active uniaxial phases at the z = 0 interfacial plane between two fluids or at the boundary of a bulk fluid.
Fig. 3: A log–log plot of the activity lengthscale ℓa, the lengthscale associated with the fastest-growing mode in active uniaxial fluids living at the interface between two fluids or at the boundary of a bulk medium, as a function of activity ζ.
Fig. 4: Stability region for a motile flock living either at a two-fluid interface or at the boundary of a bulk fluid in the \({{{{\mathscr{R}}}}}_{1}-{{{{\mathscr{R}}}}}_{2}\) plane for a specific value of λ = 2, as implied by equation (10).

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No datasets were generated or analysed during the current study.

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No custom code was used in the current study.

References

  1. Ramaswamy, S. Active matter. J. Stat. Mech. 2017, 054002 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  2. Dadhichi, L. P., Maitra, A. & Ramaswamy, S. Origins and diagnostics of the nonequilibrium character of active systems. J. Stat. Mech. 2018, 123201 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  3. Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1189 (2013).

    Article  ADS  Google Scholar 

  4. Prost, J., Jülicher, F. & Joanny, J.-F. Active gel physics. Nat. Phys. 11, 111–117 (2015).

    Article  Google Scholar 

  5. Jülicher, F., Grill, S. W. & Salbreux, G. Hydrodynamic theory of active matter. Rep. Prog. Phys. 81, 076601 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  6. Ramaswamy, S. The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys. 1, 323–345 (2010).

    Article  ADS  Google Scholar 

  7. Simha, R. A. & Ramaswamy, S. Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Phys. Rev. Lett. 89, 058101 (2002).

    Article  ADS  MATH  Google Scholar 

  8. Voituriez, R., Joanny, J.-F. & Prost, J. Spontaneous flow transition in active polar gels. Europhys. Lett. 70, 404 (2005).

    Article  ADS  Google Scholar 

  9. Alert, R., Joanny, J.-F. & Casademunt, J. Universal scaling of active nematic turbulence. Nat. Phys. 16, 682–688 (2020).

    Article  Google Scholar 

  10. Alert, R., Joanny, J.-F. & Casademunt, J. Active turbulence. Annu. Rev. Condens. Matter Phys. 13, 143–170 (2022).

  11. Maitra, A. et al. A nonequilibrium force can stabilize 2D active nematics. Proc. Natl Acad. Sci. USA 115, 6934–6939 (2018).

    Article  ADS  Google Scholar 

  12. Maitra, A., Srivastava, P., Marchetti, M. C., Ramaswamy, S. & Lenz, M. Swimmer suspensions on substrates: anomalous stability and long-range order. Phys. Rev. Lett. 124, 028002 (2020).

    Article  ADS  Google Scholar 

  13. Sarkar, N., Basu, A. & Toner, J. Swarming bottom feeders: flocking at solid-liquid interfaces. Phys. Rev. Lett. 127, 268004 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  14. Sarkar, N., Basu, A. & Toner, J. Hydrodynamic theory of flocking at a solid-liquid interface: long-range order and giant number fluctuations. Phys. Rev. E 104, 064611 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  15. Sluckin, T. J. & Poniewierski, A. Novel surface phase transition in nematic liquid crystals: wetting and the Kosterlitz-Thouless transition. Phys. Rev. Lett. 55, 2907–2910 (1985).

    Article  ADS  Google Scholar 

  16. Langer, S. A., Liu, A. J. & Toner, J. Hydrodynamics of two-dimensional smectics on fluid surfaces. Phys. Rev. Lett. 70, 2443–2446 (1993).

    Article  ADS  Google Scholar 

  17. Binder, K. & Hohenberg, P. C. Phase transitions and static spin correlations in Ising models with free surfaces. Phys. Rev. B 6, 3461–3487 (1972).

    Article  ADS  Google Scholar 

  18. Lubensky, T. C. & Rubin, M. H. Critical phenomena in semi-infinite systems. I. ϵ expansion for positive extrapolation length. Phys. Rev. B 11, 4533–4546 (1975).

    Article  ADS  Google Scholar 

  19. Lubensky, T. C. & Rubin, M. H. Critical phenomena in semi-infinite systems. II. Mean-field theory. Phys. Rev. B 12, 3885–3901 (1975).

    Article  ADS  Google Scholar 

  20. Poniewierski, A. & Sluckin, T. J. Statistical mechanics of a simple model of the nematic liquid crystal-wall interface. Mol. Crys. Liq. Crys. 111, 373–386 (1984).

    Article  Google Scholar 

  21. Perlekar, P., Benzi, R., Nelson, D. R. & Toschi, F. Population dynamics at high Reynolds number. Phys. Rev. Lett. 105, 144501 (2010).

    Article  ADS  Google Scholar 

  22. Lopez, D. & Lauga, E. Dynamics of swimming bacteria at complex interfaces. Phys. Fluids 26, 071902 (2014).

    Article  ADS  Google Scholar 

  23. Sanchez, T., Chen, D. T. N., DeCamp, S. J., Heymann, M. & Dogic, Z. Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–434 (2012).

    Article  ADS  Google Scholar 

  24. Martínez-Prat, B., Ignés-Mullol, J., Casademunt, J. & Sagués, F. Selection mechanism at the onset of active turbulence. Nat. Phys. 15, 362–366 (2019).

    Article  Google Scholar 

  25. Senoussi, A. et al. Tunable corrugated patterns in an active nematic sheet. Proc. Natl Acad. Sci. USA 116, 22464–22470 (2019).

    Article  ADS  Google Scholar 

  26. Strübing, T. et al. Wrinkling instability in 3D active nematics. Nano Lett. 20, 6281–6288 (2020).

    Article  ADS  Google Scholar 

  27. Gao, T., Blackwell, R., Glaser, M. A., Betterton, M. D. & Shelley, M. J. Multiscale modeling and simulation of microtubule-motor-protein assemblies. Phys. Rev. E 92, 062709 (2015).

    Article  ADS  Google Scholar 

  28. Gao, T., Blackwell, R., Glaser, M. A., Betterton, M. D. & Shelley, M. J. Multiscale polar theory of microtubule and motor-protein assemblies. Phys. Rev. Lett. 114, 048101 (2015).

    Article  ADS  Google Scholar 

  29. Subbiahdoss, G. & Reimhult, E. Biofilm formation at oil-water interfaces is not a simple function of bacterial hydrophobicity. Colloids Surf. B 194, 111163 (2020).

    Article  Google Scholar 

  30. Zutic, V., Ivosevic, N., Svetlicic, V., Long, R. A. & Azam, F. Film formation by marine bacteria at a model fluid interface. Aquat. Microb. Ecol. 17, 231–238 (1999).

    Article  Google Scholar 

  31. Vaccari, L., Molaei, M., Leheny, R. L. & Stebe, K. J. Cargo carrying bacteria at interfaces. Soft Matter 14, 5643–5653 (2018).

    Article  ADS  Google Scholar 

  32. Niepa, T. H. R. et al. Films of bacteria at interfaces (FBI): remodeling of fluid interfaces by Pseudomonas aeruginosa. Sci. Rep. 7, 17864 (2017).

    Article  ADS  Google Scholar 

  33. Chaudhuri, A., Bhattacharya, B., Gowrishankar, K., Mayor, S. & Rao, M. Spatiotemporal regulation of chemical reactions by active cytoskeletal remodeling. Proc. Natl Acad. Sci. USA 108, 14825–14830 (2011).

    Article  ADS  Google Scholar 

  34. Goswami, D. et al. Nanoclusters of GPI-anchored proteins are formed by cortical actin-driven activity. Cell 135, 1085–1097 (2008).

    Article  Google Scholar 

  35. Mayor, S. & Rao, M. Rafts: scale-dependent, active lipid organization at the cell surface. Traffic 5, 231–240 (2004).

    Article  Google Scholar 

  36. Gowrishankar, K. et al. Active remodeling of cortical actin regulates spatiotemporal organization of cell surface molecules. Cell 149, 1353–1367 (2012).

    Article  Google Scholar 

  37. Gidituri, H., Shen, Z., Würger, A. & Lintuvuori, J. S. Reorientation dynamics of microswimmers at fluid-fluid interfaces. Phys. Rev. Fluids 7, L042001 (2022).

    Article  ADS  Google Scholar 

  38. Ahmadzadegan, A., Wang, S., Vlachos, P. P. & Ardekani, A. M. Hydrodynamic attraction of bacteria to gas and liquid interfaces. Phys. Rev. E 100, 062605 (2019).

    Article  ADS  Google Scholar 

  39. Salbreux, G., Prost, J. & Joanny, J.-F. Hydrodynamics of cellular cortical flows and the formation of contractile rings. Phys. Rev. Lett. 103, 058102 (2009).

    Article  ADS  Google Scholar 

  40. Toner, J. Birth, death, and flight: a theory of Malthusian flocks. Phys. Rev. Lett. 108, 088102 (2012).

    Article  ADS  Google Scholar 

  41. Khandkar, M. D. & Barma, M. Orientational correlations and the effect of spatial gradients in the equilibrium steady state of hard rods in two dimensions: a study using deposition-evaporation kinetics. Phys. Rev. E 72, 051717 (2005).

    Article  ADS  Google Scholar 

  42. Marenduzzo, D., Orlandini, E., Cates, M. E. & Yeomans, J. M. Steady-state hydrodynamic instabilities of active liquid crystals: hybrid lattice Boltzmann simulations. Phys. Rev. E 76, 031921 (2007).

    Article  ADS  Google Scholar 

  43. Underhill, P. T., Hernandez-Ortiz, J. P. & Graham, M. D. Diffusion and spatial correlations in suspensions of swimming particles. Phys. Rev. Lett. 100, 248101 (2008).

    Article  ADS  Google Scholar 

  44. da Gama, M. M. Telo. The interfacial properties of a model of a nematic liquid crystal. I. The nematic-isotropic and the nematic-vapour interfaces. Mol. Phys. 52, 585–610 (1984).

    Article  ADS  Google Scholar 

  45. da Gama, M. M. Telo. The interfacial properties of a model of a nematic liquid crystal. II. Induced orientational order and wetting transitions at a solid–fluid interface. Mol. Phys. 52, 611–630 (1984).

    Article  ADS  Google Scholar 

  46. Nakanishi, H. & Fisher, M. E. Multicriticality of wetting, prewetting and surface transitions. Phys. Rev. Lett. 49, 1565–1568 (1982).

    Article  ADS  Google Scholar 

  47. Sheng, P. Boundary-layer phase transition in nematic liquid crystals. Phys. Rev. A 26, 1610–1617 (1982).

    Article  ADS  Google Scholar 

  48. Ramaswamy, S., Simha, R. A. & Toner, J. Active nematics on a substrate: giant number fluctuations and long-time tails. Europhys. Lett. 62, 196 (2003).

    Article  ADS  Google Scholar 

  49. Chaté, H. Dry aligning dilute active matter. Annu. Rev. Condens. Matter Phys. 11, 189–212 (2019).

    Article  ADS  Google Scholar 

  50. Shankar, S., Ramaswamy, S. & Marchetti, M. C. Low-noise phase of a two-dimensional active nematic system. Phys. Rev. E 97, 012707 (2018).

    Article  ADS  Google Scholar 

  51. Bertin, E. et al. Mesoscopic theory for fluctuating active nematics. N. J. Phys. 15, 085032 (2013).

    Article  Google Scholar 

  52. Boffetta, G., Davoudi, J., Eckhardt, B. & Schumacher, J. Lagrangian tracers on a surface flow: the role of time correlations. Phys. Rev. Lett. 93, 134501 (2004).

    Article  ADS  Google Scholar 

  53. Lee, S. H., Chadwick, R. S. & Leal, L. G. Motion of a sphere in the presence of a plane interface. Part 1. An approximate solution by generalization of the method of Lorentz. J. Fluid Mech. 93, 705–726 (1979).

    Article  ADS  MATH  Google Scholar 

  54. Dörr, A., Hardt, S., Masoud, H. & Stone, H. A. Drag and diffusion coefficients of a spherical particle attached to a fluid-fluid interface. J. Fluid Mech. 790, 607–618 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Cai, W. & Lubensky, T. C. Hydrodynamics and dynamic fluctuations of fluid membranes. Phys. Rev. E 52, 4251–4266 (1995).

    Article  ADS  Google Scholar 

  56. Martínez-Prat, B. et al. Scaling regimes of active turbulence with external dissipation. Phys. Rev. X 11, 031065 (2021).

    Google Scholar 

  57. Mahault, B. & Chaté, H. Long-range nematic order in two-dimensional active matter. Phys. Rev. Lett. 127, 048003 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  58. Toner, J. Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks. Phys. Rev. E 86, 031918 (2013).

    Article  ADS  Google Scholar 

  59. Chatterjee, R., Rana, N., Simha, R. A., Perlekar, P. & Ramaswamy, S. Inertia drives a flocking phase transition in viscous active fluids. Phys. Rev. X 11, 031063 (2021).

    Google Scholar 

  60. Lemma, L. M., DeCamp, S. J., You, Z., Giomi, L. & Dogic, Z. Statistical properties of autonomous flows in 2D active nematics. Soft Matter 15, 3264–3272 (2019).

    Article  ADS  Google Scholar 

  61. Lemma, L. M. et al. Multiscale microtubule dynamics in active nematics. Phys. Rev. Lett. 127, 148001 (2021).

    Article  ADS  Google Scholar 

  62. Guillamat, P., Ignés-Mullol, J., Shankar, S., Marchetti, M. C. & Sagués, F. Probing the shear viscosity of an active nematic film. Phys. Rev. E 94, 060602 (2016).

    Article  ADS  Google Scholar 

  63. Maitra, A., Srivastava, P., Rao, M. & Ramaswamy, S. Activating membranes. Phys. Rev. Lett. 112, 258101 (2014).

    Article  ADS  Google Scholar 

  64. Singh, R. & Cates, M. E. Hydrodynamically interrupted droplet growth in scalar active matter. Phys. Rev. Lett. 123, 148005 (2019).

    Article  ADS  Google Scholar 

  65. Zwicker, D., Seyboldt, R., Weber, C. A., Hyman, A. A. & Jülicher, F. Growth and division of active droplets provides a model for protocells. Nat. Phys. 13, 408–413 (2017).

    Article  Google Scholar 

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Acknowledgements

I thank Sriram Ramaswamy and Raphael Voituriez for insightful comments and discussions. I also thank Sriram Ramaswamy for his careful reading of the manuscript and for crucial suggestions. I acknowledge a TALENT fellowship awarded by CY Cergy Paris University.

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  1. Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université, Cergy-Pontoise, France

    Ananyo Maitra

  2. Sorbonne Université and CNRS, Laboratoire Jean Perrin, Paris, France

    Ananyo Maitra

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A.M. planned the research, performed the theoretical analysis, wrote the article and revised it.

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Maitra, A. Two-dimensional long-range uniaxial order in three-dimensional active fluids. Nat. Phys. 19, 733–740 (2023). https://doi.org/10.1038/s41567-023-01937-4

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