Skip to main content
SpringerLink
Log in

Mode instabilities and dynamic patterns in a colony of self-propelled surfactant particles covering a thin liquid layer

  • Regular Article
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract.

We consider a colony of point-like self-propelled surfactant particles (swimmers) without direct interactions that cover a thin liquid layer on a solid support. The particles predominantly swim normal to the free film surface with only a small component parallel to the film surface. The coupled dynamics of the swimmer density and film height profile is captured in a long-wave model allowing for diffusive and convective transport of the swimmers (including rotational diffusion). The dynamics of the film height profile is determined by i) the upward pushing force of the swimmers onto the liquid-gas interface, ii) the solutal Marangoni force due to gradients in the swimmer concentration, and iii) the rotational diffusion of the swimmers together with the in-plane active motion. After reviewing and extending the analysis of the linear stability of the uniform state, we analyse the fully nonlinear dynamic equations and show that point-like swimmers, which only interact via long-wave deformations of the liquid film, self-organise in highly regular (standing, travelling, and modulated waves) and various irregular patterns.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323 (2010)

    Article  ADS  Google Scholar 

  2. M. Marchetti, J. Joanny, S. Ramaswamy, T. Liverpool, J. Prost, M. Rao, R. Simha, Rev. Mod. Phys. 85, 1143 (2013)

    Article  ADS  Google Scholar 

  3. I. Buttinoni, J. Bialké, F. Kümmel, H. Löwen, C. Bechinger, T. Speck, Phys. Rev. Lett. 110, 238301 (2013)

    Article  ADS  Google Scholar 

  4. I. Theurkauff, C. Cottin-Bizonne, J. Palacci, C. Ybert, L. Bocquet, Phys. Rev. Lett. 108, 268303 (2012)

    Article  ADS  Google Scholar 

  5. J. Palacci, C. Cottin-Bizonne, C. Ybert, L. Bocquet, Phys. Rev. Lett. 105, 088304 (2010)

    Article  ADS  Google Scholar 

  6. J. Palacci, S. Sacanna, A.P. Steinberg, D.J. Pine, P.M. Chaikin, Science 339, 936 (2013)

    Article  ADS  Google Scholar 

  7. S. Thutupalli, R. Seemann, S. Herminghaus, New J. Phys. 13, 073021 (2011)

    Article  ADS  Google Scholar 

  8. I.H. Riedel, K. Kruse, J. Howard, Science 309, 300 (2005)

    Article  ADS  Google Scholar 

  9. C. Dombrowski, L. Cisneros, S. Chatkaew, R.E. Goldstein, J. Kessler, Phys. Rev. Lett. 93, 098103 (2004)

    Article  ADS  Google Scholar 

  10. E. Lushia, H. Wioland, R.E. Goldstein, Proc. Natl. Acad. Sci. U.S.A. 111, 9733 (2005)

    Article  ADS  Google Scholar 

  11. A. Sokolov, I.S. Aranson, J.O. Kessler, R.E. Goldstein, Phys. Rev. Lett. 98, 158102 (2007)

    Article  ADS  Google Scholar 

  12. A. Sokolov, I.S. Aranson, Phys. Rev. Lett. 109, 248109 (2012)

    Article  ADS  Google Scholar 

  13. K.-A. Liu, L. I, Phys. Rev. E 86, 011924 (2012)

    Article  ADS  Google Scholar 

  14. A. Sokolov, R.E. Goldstein, F.I. Feldchtein, I.S. Aranson, Phys. Rev. E 80, 031903 (2009)

    Article  ADS  Google Scholar 

  15. H.H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R.E. Goldstein, H. Löwen, J.M. Yeomans, Proc. Natl. Acad. Sci. U.S.A. 109, 14308 (2012)

    Article  ADS  Google Scholar 

  16. T. Ishikawa, N. Yoshida, H. Ueno, M. Wiedeman, Y. Imai, T. Yamaguchi, Phys. Rev. Lett. 107, 028102 (2011)

    Article  ADS  Google Scholar 

  17. C. Dombrowski, L. Cisneros, S. Chatkaew, R.E. Goldstein, J.O. Kessler, Phys. Rev. Lett. 93, 098103 (2004)

    Article  ADS  Google Scholar 

  18. J. Schwarz-Linek, C. Valeriani, A. Cacciuto, M.E. Cates, D. Marenduzzo, A.N. Morozov, W.C.K. Poon, Proc. Natl. Acad. Sci. U.S.A. 109, 4052 (2012)

    Article  Google Scholar 

  19. J. Dunkel, S. Heidenreich, K. Drescher, H.H. Wensink, M. Bär, R.E. Goldstein, Phys. Rev. Lett. 110, 228102 (2013)

    Article  ADS  Google Scholar 

  20. V. Schaller, C. Weber, C. Semmrich, E. Frey, A.R. Bausch, Nature 467, 73 (2010)

    Article  ADS  Google Scholar 

  21. T. Surrey, F. Nédélec, S. Leibler, E. Karsenti, Science 292, 1167 (2001)

    Article  ADS  Google Scholar 

  22. Y. Sumino, K.H. Nagai, Y. Shitaka, D. Tanaka, K. Yoshikawa, H. Chate, K. Oiwa, Nature 483, 448 (2012)

    Article  ADS  Google Scholar 

  23. I. Derenyi, T. Vicsek, Phys. Rev. Lett. 75, 374 (1995)

    Article  ADS  Google Scholar 

  24. A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, D. Bartolo, Nature 503, 95 (2013)

    Article  ADS  Google Scholar 

  25. K.H. Nagai, Y. Sumino, R. Montagne, I.S. Aranson, H. Chaté, Phys. Rev. Lett. 114, 168001 (2015)

    Article  ADS  Google Scholar 

  26. R. Großmann, P. Romanczuk, M. Bär, L. Schimansky-Geier, Phys. Rev. Lett. 113, 258104 (2014)

    Article  ADS  Google Scholar 

  27. E. Lauga, T.R. Powers, Rep. Prog. Phys. 72, 096601 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  28. J. Bialke, H. Löwen, T. Speck, EPL 103, 30008 (2013)

    Article  ADS  Google Scholar 

  29. F. Peruani, A. Deutsch, M. Bär, Phys. Rev. E 74, 030904 (2006)

    Article  ADS  Google Scholar 

  30. A. Baskaran, M.C. Marchetti, Phys. Rev. E 77, 011920 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  31. S. van Teeffelen, H. Löwen, Phys. Rev. E 78, 020101(R) (2008)

    Article  ADS  Google Scholar 

  32. A. Zöttl, H. Stark, Phys. Rev. Lett. 112, 118101 (2014)

    Article  ADS  Google Scholar 

  33. M. Hennes, K. Wolff, H. Stark, Phys. Rev. Lett. 112, 238104 (2014)

    Article  ADS  Google Scholar 

  34. O. Pohl, H. Stark, Phys. Rev. Lett. 112, 238303 (2014)

    Article  ADS  Google Scholar 

  35. O. Pohl, H. Stark, Eur. Phys. J. E 38, 93 (2015)

    Article  Google Scholar 

  36. I.S. Aranson, A. Sokolov, J.O. Kessler, R.E. Goldstein, Phys. Rev. E 75, 040901 (2007)

    Article  ADS  Google Scholar 

  37. G. Subramanian, D.L. Koch, J. Fluid Mech. 632, 359 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  38. U. Thiele, A.J. Archer, M. Plapp, Phys. Fluids 24, 102107 (2012)

    Article  ADS  Google Scholar 

  39. S. Alonso, A. Mikhailov, Phys. Rev. E 79, 061906 (2009)

    Article  ADS  Google Scholar 

  40. A. Pototsky, U. Thiele, H. Stark, Phys. Rev. E 90, 030401(R) (2014)

    Article  ADS  Google Scholar 

  41. A. Oron, S. Davis, S. Bankoff, Rev. Mod. Phys. 69, 931 (1997)

    Article  ADS  Google Scholar 

  42. L.W. Schwartz, D.E. Weidner, R.R. Eley, Langmuir 11, 3690 (1995)

    Article  Google Scholar 

  43. V. Garbin, J.C. Crocker, K.J. Stebe, Langmuir 28, 1663 (2012)

    Article  Google Scholar 

  44. D. Alizadehrad, T. Krüger, M. Engstler, H. Stark, PLoS Comput. Biol. 11, e1003967 (2015)

    Article  ADS  Google Scholar 

  45. J.R. Howse, R.A.L. Jones, A.J. Ryan, T. Gough, R. Vafabakhsh, R. Golestanian, Phys. Rev. Lett. 99, 048102 (2007)

    Article  ADS  Google Scholar 

  46. L. Helden, R. Eichhorn, C. Bechinger, Soft Matter 11, 2379 (2015)

    Article  ADS  Google Scholar 

  47. T. Lee, M. Alarcon-Correa, C. Miksch, K. Hahn, J. Gibbs, P. Fischer, Nano Lett. 14, 2407 (2014)

    Article  ADS  Google Scholar 

  48. M. Enculescu, H. Stark, Phys. Rev. Lett. 107, 058301 (2011)

    Article  ADS  Google Scholar 

  49. K. Wolff, A. Hahn, H. Stark, Eur. Phys. J. E 36, 43 (2013)

    Article  Google Scholar 

  50. C.W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin Heidelberg, 2004)

  51. A. Pototsky, H. Stark, EPL 98, 50004 (2012)

    Article  ADS  Google Scholar 

  52. E. Stellamanns, S. Uppaluri, A. Hochstetter, N. Heddergott, M. Engstler, T. Pfohl, Sci. Rep. 4, 6515 (2014)

    Article  ADS  Google Scholar 

  53. J. Palacci, C. Cottin-Bizonne, C. Ybert, L. Bocquet, Phys. Rev. Lett. 105, 088304 (2010)

    Article  ADS  Google Scholar 

  54. B. ten Hagen, S. van Teeffelen, H. Löwen, J. Phys.: Condens. Matter 23, 194119 (2011)

    ADS  Google Scholar 

  55. R. Golestanian, Phys. Rev. Lett. 108, 038303 (2012)

    Article  ADS  Google Scholar 

  56. P. Beltrame, U. Thiele, SIAM J. Appl. Dyn. Syst. 9, 484 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  57. R. Grossmann, P. Romanczuk, M. Bar, L. Schimansky-Geier, Eur. Phys. JST 224, 1325 (2015)

    Google Scholar 

  58. F. Ziebert, Ph.D. thesis, Universität Bayreuth (2006)

  59. F. Ziebert, W. Zimmermann, Eur. Phys. J. E 18, 41 (2005)

    Article  Google Scholar 

  60. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)

    Article  ADS  Google Scholar 

  61. J. Dunkel, S. Heidenreich, M. Bär, R.E. Goldstein, New J. Phys. 15, 045016 (2013)

    Article  ADS  Google Scholar 

  62. M. Bestehorn, Hydrodynamik und Strukturbildung: Mit einer kurzen Einfuhrung in die Kontinuumsmechanik (Springer-Lehrbuch) (German Edition) (Springer, 2006), ISBN 3540337962

  63. R.A. Meyers, Encyclopedia of Complexity and Systems Science, Vol. 4 (Springer-Verlag, New York, 2009) p. 3611, ISBN 978-0-387-30440-3

  64. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed. (Springer, New York, 2010)

  65. H.A. Dijkstra, F.W. Wubs, A.K. Cliffe, E. Doedel, I.F. Dragomirescu, B. Eckhardt, A.Y. Gelfgat, A. Hazel, V. Lucarini, A.G. Salinger et al., Commun. Comput. Phys. 15, 1 (2014)

    MathSciNet  Google Scholar 

  66. K. Tunstrøm, Y. Katz, C.C. Ioannou, C. Huepe, M.J. Lutz, I.D. Couzin, PLoS Comput. Biol. 9, 1 (2013)

    Article  Google Scholar 

  67. J.V. Small, Th. Stradal, E. Vignal, K. Rottner, Trends Cell Biol. 12, 112 (2002)

    Article  Google Scholar 

  68. H. Chaté, F. Ginelli, G. Grégoire, F. Peruani, F. Raynaud, Eur. Phys. J. B 64, 451 (2008)

    Article  ADS  Google Scholar 

  69. A. Gopinath, M.F. Hagan, M.C. Marchetti, A. Baskaran, Phys. Rev. E 85, 061903 (2012)

    Article  ADS  Google Scholar 

  70. S. Mishra, A. Baskaran, M.C. Marchetti, Phys. Rev. E 81, 061916 (2010)

    Article  ADS  Google Scholar 

  71. T. Sanchez, D.T.N. Chen, S.J. DeCamp, M. Heymann, Z. Dogic, Nature 491, 431 (2012)

    Article  ADS  Google Scholar 

  72. Y. Kuznetsov, Elements of Applied Bifurcation Theory (Springer-Verlag, New York, 2004)

  73. P.C. Matthews, S.M. Cox, Nonlinearity 13, 1293 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  74. S.M. Cox, P.C. Matthews, Physica D 175, 196 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  75. D. Winterbottom, P. Matthews, S. Cox, Nonlinearity 18, 1031 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  76. U.M.B. Marconi, P. Tarazona, J. Chem. Phys. 110, 8032 (1999)

    Article  ADS  Google Scholar 

  77. A.J. Archer, R. Evans, J. Chem. Phys. 121, 4246 (2004)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science Engineering and Technology, Swinburne University of Technology, 3122, Hawthorn, Victoria, Australia

    Andrey Pototsky

  2. Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm Klemm Str. 9, 48149, Münster, Germany

    Uwe Thiele

  3. Center of Nonlinear Science (CeNoS), Westfälische Wilhelms Universität Münster, Corrensstr. 2, 48149, Münster, Germany

    Uwe Thiele

  4. Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623, Berlin, Germany

    Holger Stark

Authors
  1. Andrey Pototsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uwe Thiele
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Holger Stark
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Uwe Thiele.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pototsky, A., Thiele, U. & Stark, H. Mode instabilities and dynamic patterns in a colony of self-propelled surfactant particles covering a thin liquid layer. Eur. Phys. J. E 39, 51 (2016). https://doi.org/10.1140/epje/i2016-16051-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/i2016-16051-4

Keywords

Search

Navigation