Hostname: page-component-f554764f5-fnl2l Total loading time: 0 Render date: 2025-04-16T11:03:28.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-propulsion near the onset of Marangoni instability of deformable active droplets

Published online by Cambridge University Press:  11 December 2018

Matvey Morozov Open the ORCID record for Matvey Morozov [Opens in a new window]  and
Sébastien Michelin Open the ORCID record for Sébastien Michelin [Opens in a new window]
Matvey Morozov
Affiliation:
LadHyX – Département de Mécanique, École Polytechnique – CNRS, 91128 Palaiseau CEDEX, France
Sébastien Michelin*
Affiliation:
LadHyX – Département de Mécanique, École Polytechnique – CNRS, 91128 Palaiseau CEDEX, France
*
†Email address for correspondence: sebastien.michelin@ladhyx.polytechnique.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Experimental observations indicate that chemically active droplets suspended in a surfactant-laden fluid can self-propel spontaneously. The onset of this motion is attributed to a symmetry-breaking Marangoni instability resulting from the nonlinear advective coupling of the distribution of surfactant to the hydrodynamic flow generated by Marangoni stresses at the droplet’s surface. Here, we use a weakly nonlinear analysis to characterize the self-propulsion near the instability threshold and the influence of the droplet’s deformability. We report that, in the vicinity of the threshold, deformability enhances self-propulsion of viscous droplets, but hinders propulsion of drops that are roughly less viscous than the surrounding fluid. Our asymptotics further reveals that droplet deformability may alter the type of bifurcation leading to symmetry breaking: for moderately deformable droplets, the onset of self-propulsion is transcritical and a regime of steady self-propulsion is stable; while in the case of highly deformable drops, no steady flows can be found within the asymptotic limit considered in this paper, suggesting that the bifurcation is subcritical.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Drops and Bubbles: Drops Low-Reynolds-number flows: Low-Reynolds-number flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 860 , 10 February 2019 , pp. 711 - 738
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Acrivos, A. & Taylor, T. D. 1962 Heat and mass transfer from single spheres in Stokes flow. Phys. Fluids 5, 387–394.Google Scholar
Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21, 61–99.Google Scholar
Baret, J. F. 1969 Theoretical model for an interface allowing a kinetic study of adsorption. J. Colloid Interface Sci. 30, 1–12.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199–208.Google Scholar
Caschera, F., Rasmussen, S. & Hanczyc, M. M. 2013 An oil droplet division–fusion cycle. ChemPlusChem 78, 52–54.Google Scholar
Fadda, F., Gonnella, G., Lamura, A. & Tiribocchi, A. 2017 Lattice Boltzmann study of chemically-driven self-propelled droplets. Eur. Phys. J. E 40 (12), 112.Google Scholar
Frenkel, M., Dombrovsky, L., Multanen, V., Danchuk, V., Legchenkova, I., Shoval, S., Bormashenko, Y., Binks, B. P. & Bormashenko, E. 2018 Self-propulsion of water-supported liquid marbles filled with sulfuric acid. J. Phys. Chem. B 122, 7936–7942.Google Scholar
Golovin, A. A., Gupalo, Y. P. & Ryazantsev, Y. S. 1989 Change in shape of drop moving due to the chemithermocapillary effect. J. Appl. Mech. Tech. Phys. 30, 602–609.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Mechanics of Fluids and Transport Processes). Springer.Google Scholar
Herminghaus, S., Maass, C. C., Krüger, C., Thutupalli, S., Goehring, L. & Bahr, C. 2014 Interfacial mechanisms in active emulsions. Soft Matt. 10, 7008–7022.Google Scholar
Holmes, M. H. 1995 Introduction to Perturbation Methods. Springer.Google Scholar
Ibrahim, Y., Golestanian, R. & Liverpool, T. B. 2018 Shape dependent phoretic propulsion of slender active particles. Phys. Rev. Fluids 3, 033101.Google Scholar
Izri, Z., van der Linden, M. N., Michelin, S. & Dauchot, O. 2014 Self-propulsion of pure water droplets by spontaneous Marangoni-stress-driven motion. Phys. Rev. Lett. 113, 248302.Google Scholar
Jin, C., Hokmabad, B. V., Baldwin, K. A. & Maass, C. C. 2018 Chemotactic droplet swimmers in complex geometries. J. Phys.: Condens. Matter 30, 054003.Google Scholar
Kree, R., Burada, P. S. & Zippelius, A. 2017 From active stresses and forces to self-propulsion of droplets. J. Fluid Mech. 821, 595–623.Google Scholar
Krüger, C., Bahr, C., Herminghaus, S. & Maass, C. C. 2016a Dimensionality matters in the collective behaviour of active emulsions. Eur. Phys. J. E 39, 64–72.Google Scholar
Krüger, C., Klös, G., Bahr, C. & Maass, C. C. 2016b Curling liquid crystal microswimmers: a cascade of spontaneous symmetry breaking. Phys. Rev. Lett. 117, 048003.Google Scholar
Lamb, H. 1945 Hydrodynamics, Dover Books on Physics. Dover.Google Scholar
Lauga, E. & Michelin, S. 2016 Stresslets induced by active swimmers. Phys. Rev. Lett. 117, 148001.Google Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes, Cambridge Series in Chemical Engineering. Cambridge University Press.Google Scholar
Maass, C. C., Krüger, C., Herminghaus, S. & Bahr, C. 2016 Swimming droplets. Annu. Rev. Condens. Matter Phys. 7, 171–193.Google Scholar
Matunobu, Y. 1966 Motion of a deformed drop in Stokes flow. J. Phys. Soc. Japan 21, 1596–1602.Google Scholar
Michelin, S. & Lauga, E. 2015 Autophoretic locomotion from geometric asymmetry. Eur. Phys. J. E 38, 7.Google Scholar
Michelin, S. & Lauga, E. 2017 Geometric tuning of self-propulsion for Janus catalytic particles. Sci. Rep. 7, 42264.Google Scholar
Michelin, S., Lauga, E. & Bartolo, D. 2013 Spontaneous autophoretic motion of isotropic particles. Phys. Fluids. 25, 061701.Google Scholar
Moerman, P. G., Moyses, H. W., van der Wee, E. B., Grie, D. G., van Blaaderen, A., Kegel, W. K., Groenewold, J. & Brujic, J. 2017 Solute-mediated interactions between active droplets. Phys. Rev. E 96, 032607.Google Scholar
Moran, J. L. & Posner, J. D. 2017 Phoretic self-propulsion. Annu. Rev. Fluid Mech. 49, 511–540.Google Scholar
Nagasaka, Y., Tanaka, S., Nehiraa, T. & Amimoto, T. 2017 Spontaneous emulsification and self-propulsion of oil droplets induced by the synthesis of amino acid-based surfactants. Soft Matt. 13, 6450–6457.Google Scholar
Nourhani, A. & Lammert, P. E. 2016 Geometrical performance of self-propelled colloids and microswimmers. Phys. Rev. Lett. 116, 178302.Google Scholar
Ohta, T. 2017 Dynamics of deformable active particles. J. Phys. Soc. Japan 86, 072001.Google Scholar
Rednikov, A. Ye., Ryazantsev, Y. S. & Velarde, M. G. 1994a Active drops and drop motions due to nonequilibrium phenomena. J. Non-Equilib. Thermodyn. 19, 95–113.Google Scholar
Rednikov, A. Ye., Ryazantsev, Y. S. & Velarde, M. G. 1994b Drop motion with surfactant transfer in a homogeneous surrounding. Phys. Fluids 6, 451–468.Google Scholar
Ryazantsev, Y. S., Velarde, M. G., Rubio, R. G., Ortega, F. & López, P. 2017 Thermo- and soluto-capillarity: passive and active drops. Adv. Colloid Interface Sci. 247, 52–80.Google Scholar
Schmitt, M. & Stark, H. 2013 Swimming active droplet: a theoretical analysis. Europhys. Lett. 101, 44008.Google Scholar
Schmitt, M. & Stark, H. 2016 Marangoni flow at droplet interfaces: three-dimensional solution and applications. Phys. Fluids 28, 012106.Google Scholar
Shitara, K., Hiraiwa, T. & Ohta, T. 2011 Deformable self-propelled domain in an excitable reaction–diffusion system in three dimensions. Phys. Rev. E 83, 066208.Google Scholar
Shklyaev, S., Brady, J. F. & Cordova-Figueroa, U. M. 2014 Non-spherical osmotic motor: chemical sailing. J. Fluid Mech. 748, 488–520.Google Scholar
Suga, M., Suda, S., Ichikawa, M. & Kimura, Y. 2018 Self-propelled motion switching in nematic liquid crystal droplets in aqueous surfactant solutions. Phys. Rev. E 97, 062703.Google Scholar
Thutupalli, S., Seemann, R. & Herminghaus, S. 2011 Swarming behavior of simple model squirmers. New J. Phys. 13, 073021.Google Scholar
Weber, C. A., Zwicker, D., Jülicher, F. & Lee, C. F.2018 Physics of active emulsions. arXiv:1806.09552.Google Scholar
Winklbauer, R. 2015 Cell adhesion strength from cortical tension an integration of concepts. J. Cell Sci. 128, 3687–3693.Google Scholar
Würger, A. 2014 Thermally driven Marangoni surfers. J. Fluid Mech. 752, 589–601.Google Scholar
Yabunaka, S., Ohta, T. & Yoshinaga, N. 2012 Self-propelled motion of a fluid droplet under chemical reaction. J. Chem. Phys. 136, 074904.Google Scholar
Yabunaka, S. & Yoshinaga, N. 2016 Collision between chemically driven self-propelled drops. J. Fluid Mech. 806, 205–233.Google Scholar
Yariv, E. & Kaynan, U. 2017 Phoretic drag reduction of chemically active homogeneous spheres under force fields and shear flows. Phys. Rev. Fluids 2, 012201(R).Google Scholar
Yoshinaga, N. 2014 Spontaneous motion and deformation of a self-propelled droplet. Phys. Rev. E 89, 012913.Google Scholar
Yoshinaga, N. 2017 Simple models of self-propelled colloids and liquid drops: from individual motion to collective behaviors. J. Phys. Soc. Japan 86, 101009.Google Scholar
Yoshinaga, N., Nagai, K. N., Sumino, Y. & Kitahata, H. 2012 Drift instability in the motion of a fluid droplet with a chemically reactive surface driven by Marangoni flow. Phys. Rev. E 86, 016108.Google Scholar