Advertisement

Active Brownian motion of emulsion droplets: Coarsening dynamics at the interface and rotational diffusion

  • M. SchmittEmail author
  • H. Stark
Open Access
Regular Article
First Online:
Part of the following topical collections:
  1. Nonequilibrium Collective Dynamics in Condensed and Biological Matter

Abstract.

A micron-sized droplet of bromine water immersed in a surfactant-laden oil phase can swim (S. Thutupalli, R. Seemann, S. Herminghaus, New J. Phys. 13 073021 (2011). The bromine reacts with the surfactant at the droplet interface and generates a surfactant mixture. It can spontaneously phase-separate due to solutocapillary Marangoni flow, which propels the droplet. We model the system by a diffusion-advection-reaction equation for the mixture order parameter at the interface including thermal noise and couple it to fluid flow. Going beyond previous work, we illustrate the coarsening dynamics of the surfactant mixture towards phase separation in the axisymmetric swimming state. Coarsening proceeds in two steps: an initially slow growth of domain size followed by a nearly ballistic regime. On larger time scales thermal fluctuations in the local surfactant composition initiates random changes in the swimming direction and the droplet performs a persistent random walk, as observed in experiments. Numerical solutions show that the rotational correlation time scales with the square of the inverse noise strength. We confirm this scaling by a perturbation theory for the fluctuations in the mixture order parameter and thereby identify the active emulsion droplet as an active Brownian particle.

Graphical abstract

Open image in new window

Keywords

Flowing Matter: Interfacial phenomena 
Download to read the full article text

References

  1. 1.
    A. Najafi, R. Golestanian, Phys. Rev. E 69, 062901 (2004)ADSCrossRefGoogle Scholar
  2. 2.
    R. Dreyfus, J. Baudry, M.L. Roper, M. Fermigier, H.A. Stone, J. Bibette, Nature 437, 862 (2005)ADSCrossRefGoogle Scholar
  3. 3.
    E. Gauger, H. Stark, Phys. Rev. E 74, 021907 (2006)ADSCrossRefGoogle Scholar
  4. 4.
    E. Lauga, T.R. Powers, Rep. Prog. Phys. 72, 096601 (2009)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Elgeti, R.G. Winkler, G. Gompper, Rep. Prog. Phys. 78, 056601 (2015)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    T. Fenchel, Protist 152, 329 (2001)CrossRefGoogle Scholar
  7. 7.
    T. Qiu, T.C. Lee, A.G. Mark, K.I. Morozov, R. Münster, O. Mierka, S. Turek, A.M. Leshansky, P. Fischer, Nat. Commun. 5, 5119 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    D. Alizadehrad, T. Krüger, M. Engstler, H. Stark, PLoS Comput. Biol. 11, e1003967 (2015)ADSCrossRefGoogle Scholar
  9. 9.
    C.C. Maass, C. Krüger, S. Herminghaus, C. Bahr, Annu. Rev. Condens. Matter 7, 171 (2016)ADSCrossRefGoogle Scholar
  10. 10.
    M. Enculescu, H. Stark, Phys. Rev. Lett. 107, 058301 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    P. Romanczuk, M. Bär, W. Ebeling, B. Lindner, L. Schimansky-Geier, Eur. Phys. J. ST 202, 1 (2012)CrossRefGoogle Scholar
  12. 12.
    A. Zöttl, H. Stark, Phys. Rev. Lett. 108, 218104 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    S. Michelin, E. Lauga, D. Bartolo, Phys. Fluids 25, 061701 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    J.R. Howse, R.A. Jones, A.J. Ryan, T. Gough, R. Vafabakhsh, R. Golestanian, Phys. Rev. Lett. 99, 048102 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    V. Zaburdaev, S. Uppaluri, T. Pfohl, M. Engstler, R. Friedrich, H. Stark, Phys. Rev. Lett. 106, 208103 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    M. Theves, J. Taktikos, V. Zaburdaev, H. Stark, C. Beta, Biophys. J. 105, 1915 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    F. Kümmel, B. ten Hagen, R. Wittkowski, I. Buttinoni, R. Eichhorn, G. Volpe, H. Löwen, C. Bechinger, Phys. Rev. Lett. 110, 198302 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    G. Volpe, I. Buttinoni, D. Vogt, H.J. Kummerer, C. Bechinger, Soft Matter 7, 8810 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    K. Drescher, J. Dunkel, L.H. Cisneros, S. Ganguly, R.E. Goldstein, Proc. Natl. Acad. Sci. U.S.A. 108, 10940 (2011)ADSCrossRefGoogle Scholar
  20. 20.
    T. Majmudar, E.E. Keaveny, J. Zhang, M.J. Shelley, J. R. Soc. Interface 9, 1809 (2012)CrossRefGoogle Scholar
  21. 21.
    K. Schaar, A. Zöttl, H. Stark, Phys. Rev. Lett. 115, 038101 (2015)ADSCrossRefGoogle Scholar
  22. 22.
    M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, R.A. Simha, Rev. Mod. Phys. 85, 1143 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    T. Ishikawa, T.J. Pedley, Phys. Rev. Lett. 100, 088103 (2008)ADSCrossRefGoogle Scholar
  24. 24.
    A.A. Evans, T. Ishikawa, T. Yamaguchi, E. Lauga, Phys. Fluids 23, 111702 (2011)ADSCrossRefGoogle Scholar
  25. 25.
    J. Dunkel, S. Heidenreich, K. Drescher, H.H. Wensink, M. Bär, R.E. Goldstein, Phys. Rev. Lett. 110, 228102 (2013)ADSCrossRefGoogle Scholar
  26. 26.
    F. Alarcón, I. Pagonabarraga, J. Mol. Liq. 185, 56 (2013)CrossRefGoogle Scholar
  27. 27.
    A. Zöttl, H. Stark, Phys. Rev. Lett. 112, 118101 (2014)ADSCrossRefGoogle Scholar
  28. 28.
    M. Hennes, K. Wolff, H. Stark, Phys. Rev. Lett. 112, 238104 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    O. Pohl, H. Stark, Phys. Rev. Lett. 112, 238303 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    A. Zöttl, H. Stark, J. Phys.: Condens. Matter 28, 253001 (2016)ADSGoogle Scholar
  31. 31.
    A. Walther, A.H. Müller, Soft Matter 4, 663 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    T. Bickel, A. Majee, A. Würger, Phys. Rev. E 88, 012301 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    R. Golestanian, T.B. Liverpool, A. Ajdari, Phys. Rev. Lett. 94, 220801 (2005)ADSCrossRefGoogle Scholar
  34. 34.
    W.F. Paxton, A. Sen, T.E. Mallouk, Chem. Eur. J. 11, 6462 (2005)CrossRefGoogle Scholar
  35. 35.
    J.L. Moran, J.D. Posner, J. Fluid. Mech. 680, 31 (2011)MathSciNetCrossRefGoogle Scholar
  36. 36.
    I. Buttinoni, G. Volpe, F. Kümmel, G. Volpe, C. Bechinger, J. Phys.: Condens. Matter 24, 284129 (2012)Google Scholar
  37. 37.
    M. Schmitt, H. Stark, EPL 101, 44008 (2013)ADSCrossRefGoogle Scholar
  38. 38.
    M.J. Lighthill, Commun. Pur. Appl. Math. 5, 109 (1952)MathSciNetCrossRefGoogle Scholar
  39. 39.
    J.R. Blake, J. Fluid. Mech. 46, 199 (1971)ADSCrossRefGoogle Scholar
  40. 40.
    M.T. Downton, H. Stark, J. Phys.: Condens. Matter 21, 204101 (2009)ADSGoogle Scholar
  41. 41.
    O. Pak, E. Lauga, J. Eng. Math. 88, 1 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    M. Schmitt, H. Stark, Phys. Fluids 28, 012106 (2016)ADSCrossRefGoogle Scholar
  43. 43.
    M.M. Hanczyc, T. Toyota, T. Ikegami, N. Packard, T. Sugawara, J. Am. Chem. Soc. 129, 9386 (2007)CrossRefGoogle Scholar
  44. 44.
    T. Toyota, N. Maru, M.M. Hanczyc, T. Ikegami, T. Sugawara, J. Am. Chem. Soc. 131, 5012 (2009)CrossRefGoogle Scholar
  45. 45.
    S. Thutupalli, R. Seemann, S. Herminghaus, New J. Phys. 13, 073021 (2011)ADSCrossRefGoogle Scholar
  46. 46.
    H. Kitahata, N. Yoshinaga, K.H. Nagai, Y. Sumino, Phys. Rev. E 84, 015101 (2011)ADSCrossRefGoogle Scholar
  47. 47.
    T. Banno, R. Kuroha, T. Toyota, Langmuir 28, 1190 (2012)CrossRefGoogle Scholar
  48. 48.
    T. Ban, T. Yamagami, H. Nakata, Y. Okano, Langmuir 29, 2554 (2013)CrossRefGoogle Scholar
  49. 49.
    S. Herminghaus, C.C. Maass, C. Krüger, S. Thutupalli, L. Goehring, C. Bahr, Soft Matter 10, 7008 (2014)ADSCrossRefGoogle Scholar
  50. 50.
    Z. Izri, M.N. van der Linden, S. Michelin, O. Dauchot, Phys. Rev. Lett. 113, 248302 (2014)ADSCrossRefGoogle Scholar
  51. 51.
    Y.J. Chen, Y. Nagamine, K. Yoshikawa, Phys. Rev. E 80, 016303 (2009)ADSCrossRefGoogle Scholar
  52. 52.
    O. Bliznyuk, H.P. Jansen, E.S. Kooij, H.J.W. Zandvliet, B. Poelsema, Langmuir 27, 11238 (2011)CrossRefGoogle Scholar
  53. 53.
    A.Y. Rednikov, Y.S. Ryazantsev, M.G. Velarde, J. Non-Equil. Thermodyn. 19, 95 (1994)ADSCrossRefGoogle Scholar
  54. 54.
    A.Y. Rednikov, Y.S. Ryazantsev, M.G. Velarde, Phys. Fluids 6, 451 (1994)ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    M.G. Velarde, A.Y. Rednikov, Y.S. Ryazantsev, J. Phys.: Condens. Matter 8, 9233 (1996)ADSGoogle Scholar
  56. 56.
    M.G. Velarde, Phil. Trans. R. Soc. Lond. A 356, 829 (1998)ADSCrossRefGoogle Scholar
  57. 57.
    N. Yoshinaga, K.H. Nagai, Y. Sumino, H. Kitahata, Phys. Rev. E 86, 016108 (2012)ADSCrossRefGoogle Scholar
  58. 58.
    E. Tjhung, D. Marenduzzo, M.E. Cates, Proc. Natl. Acad. Sci. U.S.A. 109, 12381 (2012)ADSCrossRefGoogle Scholar
  59. 59.
    N. Yoshinaga, Phys. Rev. E 89, 012913 (2014)ADSCrossRefGoogle Scholar
  60. 60.
    S. Yabunaka, T. Ohta, N. Yoshinaga, J. Chem. Phys. 136, 074904 (2012)ADSCrossRefGoogle Scholar
  61. 61.
    K. Furtado, C.M. Pooley, J.M. Yeomans, Phys. Rev. E 78, 046308 (2008)ADSCrossRefGoogle Scholar
  62. 62.
    A. Bray, Phil. Trans. R. Soc. Lond. A 361, 781 (2003)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    V.A. Nepomniashchii, M.G. Velarde, P. Colinet, Interfacial phenomena and convection, 1st edition (Chapman & Hall/CRC, 2002)Google Scholar
  64. 64.
    R.C. Desai, R. Kapral, Dynamics of Self-Organized and Self-Assembled Structures (Cambridge University Press, 2009)Google Scholar
  65. 65.
    S. Chandrasekhar, Hydrodynamic and hydromagnetic stability (Oxford University Press, 1961)Google Scholar
  66. 66.
    H.A. Stone, A.D. Samuel, Phys. Rev. Lett. 77, 4102 (1996)ADSCrossRefGoogle Scholar
  67. 67.
    J. Bławzdziewicz, P. Vlahovska, M. Loewenberg, Physica A 276, 50 (2000)ADSCrossRefGoogle Scholar
  68. 68.
    J.A. Hanna, P.M. Vlahovska, Phys. Fluids 22, 013102 (2010)ADSCrossRefGoogle Scholar
  69. 69.
    J.T. Schwalbe, F.R. Phelan Jr., P.M. Vlahovska, S.D. Hudson, Soft Matter 7, 7797 (2011)ADSCrossRefGoogle Scholar
  70. 70.
    O.S. Pak, J. Feng, H.A. Stone, J. Fluid. Mech. 753, 535 (2014)ADSMathSciNetCrossRefGoogle Scholar
  71. 71.
    J.H. Ferziger, M. Perić, Computational methods for fluid dynamics, Vol. 3 (Springer, Berlin, 1996)Google Scholar
  72. 72.
    R. Eymard, T. Gallouët, R. Herbin, Handb. Numer. Anal. 7, 713 (2000)Google Scholar
  73. 73.
    J.R. Baumgardner, P.O. Frederickson, SIAM J. Numer. Anal. 22, 1107 (1985)ADSMathSciNetCrossRefGoogle Scholar
  74. 74.
    J.A. Pudykiewicz, J. Comput. Phys. 213, 358 (2006)ADSMathSciNetCrossRefGoogle Scholar
  75. 75.
    N.J. Mottram, C.J. Newton, arXiv:1409.3542 (2014)
  76. 76.
    L. Longa, H.R. Trebin, Phys. Rev. A 42, 3453 (1990)ADSMathSciNetCrossRefGoogle Scholar
  77. 77.
    P. Kaiser, W. Wiese, S. Hess, J. Non-Equil. Thermody. 17, 153 (1992)ADSCrossRefGoogle Scholar
  78. 78.
    A.J. Bray, Adv. Phys. 51, 481 (2002)ADSMathSciNetCrossRefGoogle Scholar
  79. 79.
    Y. Brenier, F. Otto, C. Seis, SIAM J. Math. Anal. 43, 114 (2011)MathSciNetCrossRefGoogle Scholar
  80. 80.
    D.S. Felix Otto, Christian Seis, Commun. Math. Sci. 11, 441 (2013)MathSciNetCrossRefGoogle Scholar
  81. 81.
    V. Lobaskin, D. Lobaskin, I. Kulić, Eur. Phys. J. ST 157, 149 (2008)CrossRefGoogle Scholar
  82. 82.
    P.S. Lovely, F. Dahlquist, J. Theor. Biol. 50, 477 (1975)CrossRefGoogle Scholar
  83. 83.
    M. Doi, S. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1986)Google Scholar
  84. 84.
    S. Dukhin, G. Kretzschmar, R. Miller, Dynamics of Adsorption at Liquid Interfaces: Theory, Experiment, Application, Studies in Interface Science (Elsevier Science, 1995), ISBN 9780080530611Google Scholar
  85. 85.
    J. Dhont, An Introduction to Dynamics of Colloids (Elsevier, 1996)Google Scholar
  86. 86.
    K.A. Hawick, D.P. Playne, Int. J. Comput. Aided Engin. Technol. 2, 78 (2010)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany

Personalised recommendations