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Self-propelled running droplets on solid substrates driven by chemical reactions

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Abstract.

We study chemically driven running droplets on a partially wetting solid substrate by means of coupled evolution equations for the thickness profile of the droplets and the density profile of an adsorbate layer. Two models are introduced corresponding to two qualitatively different types of experiments described in the literature. In both cases an adsorption or desorption reaction underneath the droplets induces a wettability gradient on the substrate and provides the driving force for droplet motion. The difference lies in the behavior of the substrate behind the droplet. In case I the substrate is irreversibly changed whereas in case II it recovers allowing for a periodic droplet movement (as long as the overall system stays far away from equilibrium). Both models allow for a non-saturated and a saturated regime of droplet movement depending on the ratio of the viscous and reactive time scales. In contrast to model I, model II allows for sitting drops at high reaction rate and zero diffusion along the substrate. The transition from running to sitting drops in model II occurs via a super- or subcritical drift-pitchfork bifurcation and may be strongly hysteretic implying a coexistence region of running and sitting drops.

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References

  1. I. Newton, Opticks (G. Bell & Sons LTD., London, 1931) (reprinted 4th ed. 1730) Book III, Part 1, Querie 31.

  2. F. Hauksbee, Philos. Trans. 27, 395 (1710).

    Google Scholar 

  3. M.G. Velarde, Philos. Trans. R. Soc. London Ser. A 356, 829 (1998).

    Google Scholar 

  4. F. Brochard, Langmuir 5, 432 (1989).

    Article  Google Scholar 

  5. H.P. Greenspan, J. Fluid Mech. 84, 125 (1978).

    Google Scholar 

  6. E. Raphaël, C.R. Acad. Sci., Ser. II 306, 751 (1988).

    Google Scholar 

  7. M.K. Chaudhury, G.M. Whitesides, Science 256, 1539 (1992).

    Google Scholar 

  8. K. Ichimura, S.K. Oh, M. Nakagawa, Science 288, 1624 (2000).

    Article  PubMed  Google Scholar 

  9. J.F. Joanny, F. Jülicher, J. Prost, Phys. Rev. Lett. 90, 168102 (2003).

    PubMed  Google Scholar 

  10. Venturi, Ann. Chimie XXI, 262 (1799).

  11. C. Tomlinson, Philos. Mag. Ser. 4 46, 409 (1869).

    Google Scholar 

  12. L. Rayleigh, Proc. R. Soc. London 47, 364 (1890).

    Google Scholar 

  13. Y. Hayashima, M. Nagayama, Y. Doi, S. Nakata, M. Kimura, M. Iida, Phys. Chem. Chem. Phys. 4, 1386 (2002).

    Google Scholar 

  14. R.L. Cottington, C.M. Murphy, C.R. Singleterry, Adv. Chem. Ser. 43, 341 (1964).

    Google Scholar 

  15. J. Bico, D. Quéré, Europhys. Lett. 51, 546 (2000).

    Article  Google Scholar 

  16. C.G. Marangoni, Ann. Phys. (Poggendorf) 143, 337 (1871) Observation 16.

    Google Scholar 

  17. A.Y. Rednikov, Y.S. Ryazantsev, M.G. Velarde, Phys. Fluids 6, 451 (1994).

    Article  Google Scholar 

  18. H. Riegler, personal communication (2003).

  19. A. Yochelis, L.M. Pismen, Phys. Rev. E 72, 025301(R) (2005).

    Article  Google Scholar 

  20. C.D. Bain, G.D. Burnetthall, R.R. Montgomerie, Nature 372, 414 (1994).

    Article  Google Scholar 

  21. F. Domingues Dos Santos, T. Ondarçuhu, Phys. Rev. Lett. 75, 2972 (1995).

    Article  PubMed  Google Scholar 

  22. S.W. Lee, P.E. Laibinis, J. Am. Chem. Soc. 122, 5395 (2000).

    Article  Google Scholar 

  23. S.W. Lee, D.Y. Kwok, P.E. Laibinis, Phys. Rev. E 65, 051602 (2002).

    Article  Google Scholar 

  24. Y. Sumino, N. Magome, T. Hamada, K. Yoshikawa, Phys. Rev. Lett. 94, 068301 (2005).

    Article  PubMed  Google Scholar 

  25. Y. Sumino, H. Kitahata, K. Yoshikawa, M. Nagayama, S.M. Nomura, N. Magome, Y. Mori, Phys. Rev. E 72, 041603 (2005).

    Article  Google Scholar 

  26. R. Magerle, personal communication (2003).

  27. A.K. Schmid, N.C. Bartelt, R.Q. Hwang, Science 290, 1561 (2000).

    PubMed  Google Scholar 

  28. K. Landry, N. Eustathopoulos, Acta Mater. 44, 3923 (1996).

    Article  Google Scholar 

  29. F.G. Yost, Scr. Mater. 38, 1225 (1998).

    Article  Google Scholar 

  30. J.A. Warren, W.J. Boettinger, A.R. Roosen, Acta Mater. 46, 3247 (1998).

    Article  Google Scholar 

  31. R. Voitovitch, A. Mortensen, F. Hodaj, N. Eustathopoulos, Acta Mater. 47, 1117 (1999).

    Article  Google Scholar 

  32. E. Saiz, R.M. Cannon, A.P. Tomsia, Acta Mater. 48, 4449 (2000).

    Article  Google Scholar 

  33. W.B. Webb, G.S. Grest, Scr. Mater. 47, 393 (2002).

    Article  Google Scholar 

  34. D.W. Zheng, W. Wen, K.N. Tu, Phys. Rev. E 57, R3719 (1998).

  35. S. Kalogeropoulou, C. Rado, N. Eustathopoulos, Scr. Mater. 41, 723 (1999).

    Google Scholar 

  36. F. Brochard-Wyart, P.-G. de Gennes, C.R. Acad. Sci., Ser. II 321, 285 (1995).

    Google Scholar 

  37. P.-G. de Gennes, Physica A 249, 196 (1998).

    Google Scholar 

  38. A. Mikhailov, D. Meinköhn, in Lect. Notes Phys., Vol. 484 (Springer, 1997) pp. 334-345.

  39. P.-G. de Gennes, C. R. Acad. Sci., Ser. II 327, 147 (1999).

    Google Scholar 

  40. P.G. de Gennes, Europhys. Lett. 39, 407 (1997).

    Google Scholar 

  41. U. Thiele, K. John, M. Bär, Phys. Rev. Lett. 93, 027802 (2004).

    PubMed  Google Scholar 

  42. A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys. 69, 931 (1997).

    Article  Google Scholar 

  43. P.-G. de Gennes, Rev. Mod. Phys. 57, 827 (1985).

    Article  Google Scholar 

  44. R.J. Hunter, Foundation of Colloid Science, Vol. 1 (Clarendon Press, Oxford, 1992).

  45. J.N. Israelachvili, Intermolecular and Surface Forces (Academic Press, London, 1992).

  46. U. Thiele, K. Neuffer, Y. Pomeau, M.G. Velarde, Colloid Surf. A 206, 135 (2002).

    Article  Google Scholar 

  47. R.F. Probstein, Physicochemical Hydrodynamics, 2nd ed. (Wiley, New York, 1994).

  48. Note that the disjoining pressure used in reference TJB04 was $\Pi(h)=\frac{2 S_a d_0^2}{h^3} + \frac{S_p}{l}\,\left(1+\frac{\phi}{g}\right)\exp\left[\frac{d_0-h}{l}\right]$, where for $\phi=0$, $S_a$ and $S_p$ are the apolar and polar components of the total spreading coefficient $S=S_a+S_p$, respectively, and $l$ is a correlation length Shar93. One usually describes the choice $S_a>0$ and $S_p<0$ as a combination of a stabilizing long-range van der Waals and a destabilizing short-range polar interaction. The apparent contradiction of qualitative similar results for model I for different verbal descriptions and combinations of signs used here and in reference TJB04 results from a subtle feature of the combination of exponential and power law. Combining a term $\sim 1/h^3$ and one $\sim \exp (-h)$ leads for a proper choice of parameters to a dominance of $1/h^3$ for large and very small $h$. The exponential only dominates for intermediate thicknesses (see U. Thiele, M.G. Velarde, K. Neuffer, Phys. Rev. Lett. 87, 016104 (2001) for a related phase diagram). This implies that the above verbal description only covers part of the feature of the disjoining pressure. On the contrary, the combination of two power laws used here clearly attributes the long-range and short-range forces to the terms $h^{-3}$ and $h^{-6}$, respectively. We therefore believe, that the chosen disjoining pressure more accurately represents the physical situation.

    Article  PubMed  Google Scholar 

  49. A. Sharma, Langmuir 9, 861 (1993).

    Article  Google Scholar 

  50. E. Doedel, H.B. Keller, J.P. Kernevez, Int. J. Bif. Chaos 1, 493 (1991).

    Google Scholar 

  51. E. Doedel, H.B. Keller, J.P. Kernevez, Int. J. Bif. Chaos 1, 745 (1991).

    Article  Google Scholar 

  52. E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X.J. Wang, AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (Concordia University, Montreal, 1997).

  53. U. Thiele, K. Neuffer, M. Bestehorn, Y. Pomeau, M.G. Velarde, Colloid Surf. A 206, 87 (2002).

    Article  Google Scholar 

  54. M. Kness, L.S. Tuckerman, D. Barkley, Phys. Rev. A 46, 5054 (1992).

    Article  PubMed  Google Scholar 

  55. K. Krischer, A. Mikhailov, Phys. Rev. Lett. 73, 3165 (1994).

    Article  PubMed  Google Scholar 

  56. A. Hagberg, E. Meron, Chaos 4, 477 (Sept. 1994).

  57. M. Or-Guil, M. Bode, C.P. Schenk, H.-G. Purwins, Phys. Rev. E 57, 6432 (1998).

    Article  Google Scholar 

  58. H.U. Bödeker, M.C. Röttger, A.W. Liehr, T.D. Frank, R. Friedrich, H.-G. Purwins, Phys. Rev. E 67, 056220 (2003).

    Article  Google Scholar 

  59. M.R.E. Proctor, C.A. Jones, J. Fluid Mech. 188, 301 (1988).

    Google Scholar 

  60. P. Coullet, R.E. Goldstein, G.H. Gunaratne, Phys. Rev. Lett. 63, 1954 (1989).

    Article  PubMed  Google Scholar 

  61. E. Knobloch, D.R. Moore, Phys. Rev. A 42, 4693 (1990).

    Article  PubMed  Google Scholar 

  62. H. Riecke, H.G. Paap, Phys. Rev. A 45, 8605 (1992).

    Article  PubMed  Google Scholar 

  63. U. Thiele, E. Knobloch, Physica D 190, 213 (2004).

    Article  Google Scholar 

  64. U. Thiele, E. Knobloch, Phys. Fluids 15, 892 (2003).

    Article  Google Scholar 

  65. D. Merkt, A. Pototsky, M. Bestehorn, U. Thiele, Phys. Fluids 17, 064104 (2005).

    Article  Google Scholar 

  66. NAG C library, Mark 6 (2000), www.nag.co.uk.

  67. L.M. Pismen, Y. Pomeau, Phys. Fluids 16, 2604 (2004).

    Article  Google Scholar 

  68. A. Pototsky, M. Bestehorn, D. Merkt, U. Thiele, Phys. Rev. E 70, 025201(R) (2004).

    Article  Google Scholar 

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Authors and Affiliations

  1. Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187, Dresden, Germany

    K. John & U. Thiele

  2. Physikalisch-Technische Bundesanstalt, Abbestr. 2-12, D-10587, Berlin, Germany

    M. Bär

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  1. K. John
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  3. U. Thiele
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Correspondence to K. John.

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John, K., Bär, M. & Thiele, U. Self-propelled running droplets on solid substrates driven by chemical reactions. Eur. Phys. J. E 18, 183–199 (2005). https://doi.org/10.1140/epje/i2005-10039-1

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