- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Chirotactic response of microswimmers in fluids with odd viscosity

Yuto Hosaka1,*, Michalis Chatzittofi1, Ramin Golestanian1,2,†, and Andrej Vilfan1,3,‡

  • *Contact author: yuto.hosaka@ds.mpg.de
  • †Contact author: ramin.golestanian@ds.mpg.de
  • ‡Contact author: andrej.vilfan@ds.mpg.de
PDF

Phys. Rev. Research 6, L032044 – Published 27 August, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L032044

Abstract

Odd viscosity is a property of chiral active fluids with broken time-reversal and parity symmetries. We show that the flow of such a fluid around a rotating axisymmetric body is exactly solvable and use this solution to determine the orientational dynamics of surface-driven microswimmers. Swimmers with a force-dipole moment exhibit precession around the axis of the odd viscosity. In addition, pushers show bimodal chirotaxis, i.e., alignment parallel or antiparallel to the axis, while pullers orbit in a plane perpendicular to it. A chiral swimmer that itself has a broken parity symmetry can exhibit unimodal chirotaxis and always align in the same direction.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (87)

  1. G. Gompper, R. G. Winkler, T. Speck, A. Solon, C. Nardini, F. Peruani, H. Löwen, R. Golestanian, U. B. Kaupp, L. Alvarez et al., The 2020 motile active matter roadmap, J. Phys.: Condens. Matter 32, 193001 (2020).
  2. M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
  3. M. E. Cates and J. Tailleur, Motility-induced phase separation, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
  4. R. Golestanian, Phoretic active matter, in Active Matter and Nonequilibrium Statistical Physics: Lecture Notes of the Les Houches Summer School: Volume 112, September 2018 (Oxford University Press, Oxford, 2022).
  5. A. Snezhko and I. S. Aranson, Magnetic manipulation of self-assembled colloidal asters, Nat. Mater. 10, 698 (2011).
  6. S. Zhou, A. Sokolov, O. D. Lavrentovich, and I. S. Aranson, Living liquid crystals, Proc. Natl. Acad. Sci. USA 111, 1265 (2014).
  7. P. Guillamat, J. Ignés-Mullol, and F. Sagués, Control of active liquid crystals with a magnetic field, Proc. Natl. Acad. Sci. USA 113, 5498 (2016).
  8. J. A. Cohen and R. Golestanian, Emergent cometlike swarming of optically driven thermally active colloids, Phys. Rev. Lett. 112, 068302 (2014).
  9. P. W. Ellis, D. J. G. Pearce, Y.-W. Chang, G. Goldsztein, L. Giomi, and A. Fernandez-Nieves, Curvature-induced defect unbinding and dynamics in active nematic toroids, Nat. Phys. 14, 85 (2018).
  10. N. Waisbord, C. T. Lefèvre, L. Bocquet, C. Ybert, and C. Cottin-Bizonne, Destabilization of a flow focused suspension of magnetotactic bacteria, Phys. Rev. Fluids 1, 053203 (2016).
  11. A. Aubret, M. Youssef, S. Sacanna, and J. Palacci, Targeted assembly and synchronization of self-spinning microgears, Nat. Phys. 14, 1114 (2018).
  12. B. Vincenti, C. Douarche, and E. Clement, Actuated rheology of magnetic micro-swimmers suspensions: Emergence of motor and brake states, Phys. Rev. Fluids 3, 033302 (2018).
  13. H. Massana-Cid, F. Meng, D. Matsunaga, R. Golestanian, and P. Tierno, Tunable self-healing of magnetically propelling colloidal carpets, Nat. Commun. 10, 2444 (2019).
  14. D. Matsunaga, J. K. Hamilton, F. Meng, N. Bukin, E. L. Martin, F. Y. Ogrin, J. M. Yeomans, and R. Golestanian, Controlling collective rotational patterns of magnetic rotors, Nat. Commun. 10, 4696 (2019).
  15. K. Han, G. Kokot, O. Tovkach, A. Glatz, I. S. Aranson, and A. Snezhko, Emergence of self-organized multivortex states in flocks of active rollers, Proc. Natl. Acad. Sci. USA 117, 9706 (2020).
  16. D. Bray, Cell Movements: From Molecules to Motility (Garland Science, New York, 2000).
  17. M. Eisenbach, A. Tamada, G. Omann, J. Segall, R. Firtel, R. Meili, D. Gutnick, M. Varon, J. W. Lengeler, and F. Murakami, Chemotaxis (World Scientific Publishing Company, Singapore, 2004).
  18. A. Bahat, I. Tur-Kaspa, A. Gakamsky, L. C. Giojalas, H. Breitbart, and M. Eisenbach, Thermotaxis of mammalian sperm cells: A potential navigation mechanism in the female genital tract, Nat. Med. 9, 149 (2003).
  19. G. Jékely, Evolution of phototaxis, Philos. Trans. R. Soc. B 364, 2795 (2009).
  20. R. R. Bennett and R. Golestanian, A steering mechanism for phototaxis in Chlamydomonas, J. R. Soc., Interface 12, 20141164 (2015).
  21. K. C. Leptos, M. Chioccioli, S. Furlan, A. I. Pesci, and R. E. Goldstein, Phototaxis of Chlamydomonas arises from a tuned adaptive photoresponse shared with multicellular Volvocine green algae, Phys. Rev. E 107, 014404 (2023).
  22. R. Blakemore, Magnetotactic bacteria, Science 190, 377 (1975).
  23. J. O. Kessler, Individual and collective fluid dynamics of swimming cells, J. Fluid Mech. 173, 191 (1986).
  24. Marcos, H. C. Fu, T. R. Powers, and R. Stocker, Bacterial rheotaxis, Proc. Natl. Acad. Sci. USA 109, 4780 (2012).
  25. B. ten Hagen, F. Kümmel, R. Wittkowski, D. Takagi, H. Löwen, and C. Bechinger, Gravitaxis of asymmetric self-propelled colloidal particles, Nat. Commun. 5, 4829 (2014).
  26. S. Das, A. Garg, A. I. Campbell, J. Howse, A. Sen, D. Velegol, R. Golestanian, and S. J. Ebbens, Boundaries can steer active Janus spheres, Nat. Commun. 6, 8999 (2015).
  27. J. Simmchen, J. Katuri, W. E. Uspal, M. N. Popescu, M. Tasinkevych, and S. Sánchez, Topographical pathways guide chemical microswimmers, Nat. Commun. 7, 10598 (2016).
  28. C. Lozano, B. ten Hagen, H. Löwen, and C. Bechinger, Phototaxis of synthetic microswimmers in optical landscapes, Nat. Commun. 7, 12828 (2016).
  29. C. Jin, C. Krüger, and C. C. Maass, Chemotaxis and autochemotaxis of self-propelling droplet swimmers, Proc. Natl. Acad. Sci. USA 114, 5089 (2017).
  30. S. Shankar, A. Souslov, M. J. Bowick, M. C. Marchetti, and V. Vitelli, Topological active matter, Nat. Rev. Phys. 4, 380 (2022).
  31. J. E. Avron, R. Seiler, and P. G. Zograf, Viscosity of quantum Hall fluids, Phys. Rev. Lett. 75, 697 (1995).
  32. J. E. Avron, Odd viscosity, J. Stat. Phys. 92, 543 (1998).
  33. D. Banerjee, A. Souslov, A. G. Abanov, and V. Vitelli, Odd viscosity in chiral active fluids, Nat. Commun. 8, 1573 (2017).
  34. Y. Hosaka and S. Komura, Nonequilibrium transport induced by biological nanomachines, Biophys. Rev. Lett. 17, 51 (2022).
  35. M. Fruchart, C. Scheibner, and V. Vitelli, Odd viscosity and odd elasticity, Annu. Rev. Condens. Matter Phys. 14, 471 (2023).
  36. A. Souslov, K. Dasbiswas, M. Fruchart, S. Vaikuntanathan, and V. Vitelli, Topological waves in fluids with odd viscosity, Phys. Rev. Lett. 122, 128001 (2019).
  37. C. Tauber, P. Delplace, and A. Venaille, A bulk-interface correspondence for equatorial waves, J. Fluid Mech. 868, R2 (2019).
  38. Y. Hosaka, S. Komura, and D. Andelman, Nonreciprocal response of a two-dimensional fluid with odd viscosity, Phys. Rev. E 103, 042610 (2021).
  39. T. Khain, C. Scheibner, M. Fruchart, and V. Vitelli, Stokes flows in three-dimensional fluids with odd and parity-violating viscosities, J. Fluid Mech. 934, A23 (2022).
  40. Y. Hosaka, D. Andelman, and S. Komura, Pair dynamics of active force dipoles in an odd-viscous fluid, Eur. Phys. J. E 46, 18 (2023).
  41. H. Yuan and M. Olvera de la Cruz, Stokesian dynamics with odd viscosity, Phys. Rev. Fluids 8, 054101 (2023).
  42. R. Lier, C. Duclut, S. Bo, J. Armas, F. Jülicher, and P. Surówka, Lift force in odd compressible fluids, Phys. Rev. E 108, L023101 (2023).
  43. Y. Hosaka, S. Komura, and D. Andelman, Hydrodynamic lift of a two-dimensional liquid domain with odd viscosity, Phys. Rev. E 104, 064613 (2021).
  44. Y. Hosaka, R. Golestanian, and A. Vilfan, Lorentz reciprocal theorem in fluids with odd viscosity, Phys. Rev. Lett. 131, 178303 (2023).
  45. M. Doi, Soft Matter Physics (Oxford University Press, Oxford, 2013).
  46. M. Doi, Onsager principle as a tool for approximation, Chin. Phys. B 24, 020505 (2015).
  47. N. Oppenheimer, D. B. Stein, and M. J. Shelley, Rotating membrane inclusions crystallize through hydrodynamic and steric interactions, Phys. Rev. Lett. 123, 148101 (2019).
  48. S. P. Thampi, A. Doostmohammadi, T. N. Shendruk, R. Golestanian, and J. M. Yeomans, Active micromachines: Microfluidics powered by mesoscale turbulence, Sci. Adv. 2, e1501854 (2016).
  49. L. Yamauchi, T. Hayata, M. Uwamichi, T. Ozawa, and K. Kawaguchi, Chirality-driven edge flow and non-Hermitian topology in active nematic cells, arXiv:2008.10852.
  50. V. Yashunsky, D. J. G. Pearce, C. Blanch-Mercader, F. Ascione, P. Silberzan, and L. Giomi, Chiral edge current in nematic cell monolayers, Phys. Rev. X 12, 041017 (2022).
  51. N. Uchida and R. Golestanian, Synchronization and collective dynamics in a carpet of microfluidic rotors, Phys. Rev. Lett. 104, 178103 (2010).
  52. N. Uchida and R. Golestanian, Synchronization in a carpet of hydrodynamically coupled rotors with random intrinsic frequency, Europhys. Lett. 89, 50011 (2010).
  53. S. Ebbens, R. A. L. Jones, A. J. Ryan, R. Golestanian, and J. R. Howse, Self-assembled autonomous runners and tumblers, Phys. Rev. E 82, 015304(R) (2010).
  54. H. Massana-Cid, D. Levis, R. J. H. Hernández, I. Pagonabarraga, and P. Tierno, Arrested phase separation in chiral fluids of colloidal spinners, Phys. Rev. Res. 3, L042021 (2021).
  55. E. S. Bililign, F. Balboa Usabiaga, Y. A. Ganan, A. Poncet, V. Soni, S. Magkiriadou, M. J. Shelley, D. Bartolo, and W. T. M. Irvine, Motile dislocations knead odd crystals into whorls, Nat. Phys. 18, 212 (2022).
  56. M. Vilfan, B. Lampret, Ž. Gregorin, L. Cmok, A. Vilfan, J. Klepp, J. Kohlbrecher, P. Hribar Boštjančič, D. Lisjak, and A. Mertelj, Spontaneous chiral symmetry breaking and lane formation in ferromagnetic ferrofluids, Small 19, 2304387 (2023).
  57. K. Drescher, K. C. Leptos, I. Tuval, T. Ishikawa, T. J. Pedley, and R. E. Goldstein, Dancing Volvox: Hydrodynamic bound states of swimming algae, Phys. Rev. Lett. 102, 168101 (2009).
  58. T. H. Tan, A. Mietke, J. Li, Y. Chen, H. Higinbotham, P. J. Foster, S. Gokhale, J. Dunkel, and N. Fakhri, Odd dynamics of living chiral crystals, Nature (London) 607, 287 (2022).
  59. V. Soni, E. S. Bililign, S. Magkiriadou, S. Sacanna, D. Bartolo, M. J. Shelley, and W. T. M. Irvine, The odd free surface flows of a colloidal chiral fluid, Nat. Phys. 15, 1188 (2019).
  60. Q. Yang, H. Zhu, P. Liu, R. Liu, Q. Shi, K. Chen, N. Zheng, F. Ye, and M. Yang, Topologically protected transport of cargo in a chiral active fluid aided by odd-viscosity-enhanced depletion interactions, Phys. Rev. Lett. 126, 198001 (2021).
  61. J. Mecke, Y. Gao, C. A. Ramírez Medina, D. G. Aarts, G. Gompper, and M. Ripoll, Simultaneous emergence of active turbulence and odd viscosity in a colloidal chiral active system, Commun. Phys. 6, 324 (2023).
  62. C. Hargus, K. Klymko, J. M. Epstein, and K. K. Mandadapu, Time reversal symmetry breaking and odd viscosity in active fluids: Green–Kubo and NEMD results, J. Chem. Phys. 152, 201102 (2020).
  63. M. Han, M. Fruchart, C. Scheibner, S. Vaikuntanathan, J. J. de Pablo, and V. Vitelli, Fluctuating hydrodynamics of chiral active fluids, Nat. Phys. 17, 1260 (2021).
  64. Z. Zhao, B. Wang, S. Komura, M. Yang, F. Ye, and R. Seto, Emergent stripes of active rotors in shear flows, Phys. Rev. Res. 3, 043229 (2021).
  65. X. Lou, Q. Yang, Y. Ding, P. Liu, K. Chen, X. Zhou, F. Ye, R. Podgornik, and M. Yang, Odd viscosity-induced Hall-like transport of an active chiral fluid, Proc. Natl. Acad. Sci. USA 119, e2201279119 (2022).
  66. M. F. Lapa and T. L. Hughes, Swimming at low Reynolds number in fluids with odd, or Hall, viscosity, Phys. Rev. E 89, 043019 (2014).
  67. Y. Hosaka, R. Golestanian, and A. Daddi-Moussa-Ider, Hydrodynamics of an odd active surfer in a chiral fluid, New J. Phys. 25, 083046 (2023).
  68. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L032044 for a movie of the angular dynamics of a chiral microswimmer in Figs. 3, the definition of the Euler angles, and the approximate expression for the critical angular velocity, which includes Ref. [69].
  69. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Pearson, Boston MA, 2001).
  70. T. Markovich and T. C. Lubensky, Odd viscosity in active matter: Microscopic origin and 3D effects, Phys. Rev. Lett. 127, 048001 (2021).
  71. S. Ganeshan and A. G. Abanov, Odd viscosity in two-dimensional incompressible fluids, Phys. Rev. Fluids 2, 094101 (2017).
  72. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Springer, Netherlands, 1983).
  73. In a classical pressureless Stokes flow (p=const and thus ∇2v=0 without ηo), the curl of the vorticity is zero, ∇×ω=0. Since the vorticity in 2D fluids has only the z-component (ω=ωzez) with ez≡e, one finds ∂xωz=0 and ∂yωz=0, hence ∇ωz=0.
  74. A. T. Chwang and T. Y.-T. Wu, Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies, J. Fluid Mech. 63, 607 (1974).
  75. J. C. Everts and B. Cichocki, Dissipative effects in odd viscous Stokes flow around a single sphere, Phys. Rev. Lett. 132, 218303 (2024).
  76. H. Masoud and H. A. Stone, The reciprocal theorem in fluid dynamics and transport phenomena, J. Fluid Mech. 879, P1 (2019).
  77. H. A. Stone and A. D. T. Samuel, Propulsion of microorganisms by surface distortions, Phys. Rev. Lett. 77, 4102 (1996).
  78. The stresslet defined as S with a prefactor 8πηe is equivalent to Eq. (64) in Ref. [79].
  79. B. Nasouri and G. J. Elfring, Higher-order force moments of active particles, Phys. Rev. Fluids 3, 044101 (2018).
  80. T. Ishikawa, M. P. Simmonds, and T. J. Pedley, Hydrodynamic interaction of two swimming model micro-organisms, J. Fluid Mech. 568, 119 (2006).
  81. E. Lauga, The Fluid Dynamics of Cell Motility (Cambridge University Press, Cambridge, 2020).
  82. V. A. Shaik and G. J. Elfring, Densitaxis: Active particle motion in density gradients, Proc. Natl. Acad. Sci. USA 121, e2405466121 (2024).
  83. J. S. Lintuvuori, A. Würger, and K. Stratford, Hydrodynamics defines the stable swimming direction of spherical squirmers in a nematic liquid crystal, Phys. Rev. Lett. 119, 068001 (2017).
  84. H. Chi, M. Potomkin, L. Zhang, L. Berlyand, and I. S. Aranson, Surface anchoring controls orientation of a microswimmer in nematic liquid crystal, Commun. Phys. 3, 162 (2020).
  85. M. Lakshmanan, The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview, Philos. Trans. R. Soc. A 369, 1280 (2011).
  86. L. Perko, Differential Equations and Dynamical Systems, 3rd ed. (Springer, New York, 2001).
  87. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed. (Westview, Boulder, CO, 2015).

Outline

Information

Phys. Rev. Research 6, L032044– Published 27 August, 2024
  • Received 2 May 2024
  • Accepted 7 August 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation