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Collisions and rebounds of chemically active droplets

  • K. Lippera (a1), M. Morozov (a1), M. Benzaquen (a1) and S. Michelin (a1)


Active droplets swim as a result of the nonlinear advective coupling of the distribution of chemical species they consume or release with the Marangoni flows created by their non-uniform surface distribution. Most existing models focus on the self-propulsion of a single droplet in an unbounded fluid, which arises when diffusion is slow enough (i.e. beyond a critical Péclet number, $Pe_{c}$ ). Despite its experimental relevance, the coupled dynamics of multiple droplets and/or collision with a wall remains mostly unexplored. Using a novel approach based on a moving fitted bi-spherical grid, the fully coupled nonlinear dynamics of the chemical solute and flow fields is solved here to characterise in detail the axisymmetric collision of an active droplet with a rigid wall (or with a second droplet). The dynamics is strikingly different depending on the convective-to-diffusive transport ratio, $Pe$ : near the self-propulsion threshold (moderate $Pe$ ), the rebound dynamics is set by chemical interactions and is well captured by asymptotic analysis; in contrast, for larger $Pe$ , a complex and nonlinear combination of hydrodynamic and chemical effects set the detailed dynamics, including a closer approach to the wall and a velocity plateau shortly after the rebound of the droplet. The rebound characteristics, i.e. minimum distance and duration, are finally fully characterised in terms of $Pe$ .


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Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.
Acrivos, A. & Taylor, T. D. 1962 Heat and mass transfer form single spheres in Stokes flow. Phys. Fluids 5, 387394.
Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21 (1), 6199.
Bechinger, C., Leonardo, R. D., Löwen, H., Reichhardt, C., Volpe, G. & Volpe, G. 2016 Active particles in complex and crowded environments. Rev. Mod. Phys. 88, 045006.
Berg, H. C. 1993 Random Walks in Biology. Princeton University Press.
Brennen, C. & Winnet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339398.
Crowdy, D. G. 2013 Wall effects on self-diffusiophoretic janus particles: a theoretical study. J. Fluid Mech. 735, 473498.
Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A. & Bibette, J. 2005 Microscopic artificial swimmers. Nature 437, 862865.
Duan, W., Wang, W., Das, S., Yadav, V., Mallouk, T. E. & Sen, A. 2015 Synthetic nano- and micromachines in analytical chemistry: sensing, migration, capture, delivery and separation. Annu. Rev. Anal. Chem. 8, 311333.
Ebbens, S. J. 2016 Active colloids: progress and challenges towards realising autonomous applications. Curr. Opin. Colloid Interface Sci. 21, 1423.
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.
Ghosh, A. & Fischer, P. 2009 Controlled propulsion of artificial magnetic nanostructured propellers. Nano Lett. 9, 22432245.
Guasto, J. S., Rusconi, R. & Stocker, R. 2012 Fluid mechanics of planktonic microorganisms. Annu. Rev. Fluid Mech. 44, 373400.
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media (Mechanics of Fluids and Transport Processes). Springer.
Herminghaus, S., Maass, C. C., Krüger, C., Thutupalli, S., Goehring, L. & Bahr, C. 2014 Interfacial mechanisms in active emulsions. Soft Matt. 10, 70087022.
Ibrahim, Y. & Liverpool, T. B. 2016 How walls affect the dynamics of self-phoretic microswimmers. Eur. Phys. J. 225 (8–9), 18431874.
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. Let. 113, 248302.
Kanso, E. & Michelin, S. 2019 Phoretic and hydrodynamic interactions of weakly confined autophoretic particles. J. Chem. Phys. 150, 044902.
Kim, S. & Karrila, S. J. 2013 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.
Kirchman, D. L. 2008 Microbial Ecology of the Oceans. Wiley.
Krüger, C., Bahr, C., Herminghaus, S. & Maass, C. C. 2016a Dimensionality matters in the collective behaviour of active emulsions. Eur. Phys. J. E 39 (6), 64.
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.
Lamb, H. 1945 Hydrodynamics. Dover Books on Physics.
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming micro-organisms. Rep. Prog. Phys. 72, 096601.
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.
Liebchen, B. & Löwen, H. 2019 Which interactions dominate in active colloids? J. Chem. Phys. 150 (6), 061102.
Maass, C. C., Krger, C., Herminghaus, S. & Bahr, C. 2016 Swimming droplets. Annu. Rev. Condens. Matter Phys. 7, 171193.
Marchetti, M. C., Joanny, J. F., Ramaswamy, S., Liverpool, T. B., Prost, J., Rao, M. & Simha, R. A. 2013 Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 11431189.
Masoud, H. & Stone, H. A. 2019 The reciprocal theorem in fluid dynamics and transport phenomena. J. Fluid Mech. 879, P1.
Medina-Sánchez, M., Schwarz, L., Meyer, A. K., Hebenstreit, F. & Schmidt, O. G. 2015 Cellular cargo delivery: toward assisted fertilization by sperm-carrying micromotors. Nano Lett. 16 (1), 555561.
Michelin, S., Gallino, G., Gallaire, F. & Lauga, E. 2019 Viscous growth and rebound or a bubble near a rigid surface. J. Fluid Mech. 860, 172199.
Michelin, S., Guérin, E. & Lauga, E. 2018 Collective dissolution of microbubbles. Phys. Rev. Fluids 3, 043601.
Michelin, S. & Lauga, E. 2014 Phoretic self-propulsion at finite Péclet numbers. J. Fluid Mech. 747, 572604.
Michelin, S. & Lauga, E. 2015 Autophoretic locomotion from geometric asymmetry. Eur. Phys. J. E 38 (2), 7.
Michelin, S., Lauga, E. & Bartolo, D. 2013 Spontaneous autophoretic motion of isotropic particles. Phys. Fluids 25, 061701.
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.
Moran, J. L. & Posner, J. D. 2017 Phoretic self-propulsion. Annu. Rev. Fluid Mech. 49, 511.
Morozov, M. & Michelin, S. 2019a Nonlinear dynamics of a chemically-active drop: from steady to chaotic self-propulsion. J. Chem. Phys. 150, 044110.
Morozov, M. & Michelin, S. 2019b Orientational instability and spontaneous rotation of active nematic droplets. Soft Matt. 15, 78147822.
Morozov, M. & Michelin, S. 2019c Self-propulsion near the onset of Marangoni instability of deformable active droplets. J. Fluid Mech. 860, 711738.
Pak, O. S., Feng, J. & Stone, H. A. 2014 Viscous Marangoni migration of a drop in a Poiseuille flow at low surface Péclet numbers. J. Fluid Mech. 753, 535552.
Palacci, J., Sacanna, S., Steinberg, A. P., Pine, D. J. & Chaikin, P. M. 2013 Living crystals of light-activated colloidal surfers. Science 339, 936940.
Park, B.-W., Zhuang, J., Yasa, O. & Sitti, M. 2017 Multifunctional bacteria-driven microswimmers for targeted active drug delivery. ACS Nano 11 (9), 89108923.
Pohl, O. & Stark, H. 2014 Dynamic clustering and chemotactic collapse of self-phoretic active particles. Phys. Rev. Lett. 112 (23), 238303.
Popescu, M. N., Tasinkevych, M. & Dietrich, S. 2011 Pulling and pushing a cargo with a catalytically active carrier. Eur. Phys. Lett. 95, 28004.
Rednikov, A. Y., Kurdyumov, V. N., Ryazantsev, Y. S. & Velarde, M. G. 1995 The role of time-varying gravity on the motion of a drop induced by Marangoni instability. Phys. Fluids 7 (11), 26702678.
Rednikov, A. Y., Ryazantsev, Y. S. & Velarde, M. G. 1994 Drop motion with surfactant transfer in a homogeneous surrounding. Phys. Fluids 6, 451.
Reigh, S. Y. & Kapral, R. 2015 Catalytic dimer nanomotors: continuum theory and microscopic dynamics. Soft Matt. 11, 31493158.
Ryazantsev, Y. S., Velarde, M. G., Rubio, R. G., Guzman, E., Ortega, F. & Lopez, P. 2017 Thermo-and soluto-capillarity: passive and active drops. Adv. Colloid Interface Sci. 247, 5280.
Saha, S., Golestanian, R. & Ramaswamy, S. 2014 Clusters, asters, and collective oscillations in chemotactic colloids. Phys. Rev. E 89 (6), 062316.
Schmitt, M. & Stark, H. 2013 Swimming active droplet: a theoretical analysis. Eur. Phys. Lett. 101 (4), 44008.
Singh, A. V., Hosseinidoust, Z., Park, B.-W., Yasa, O. & Sitti, M. 2017 Microemulsion-based soft bacteria-driven microswimmers for active cargo delivery. ACS Nano 11 (10), 97599769.
Soto, R. & Golestanian, R. 2014 Self-assembly of catalytically active colloidal molecules: tailoring activity through surface chemistry. Phys. Rev. Lett. 112, 068301.
Soto, R. & Golestanian, R. 2015 Self-assembly of active colloidal molecules with dynamic function. Phys. Rev. E 91 (5), 052304.
Stimson, M. & Jeffery, G. B. 1926 The motion of two spheres in a viscous fluid. Proc. R. Soc. Lond. 111 (757), 110116.
Suarez, S. S. & Pacey, A. A. 2006 Sperm transport in the female reproductive tract. Human Reprod. Update 12, 2337.
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.
Theurkauff, I., Cottin-Bizonne, C., Palacci, J., Ybert, C. & Bocquet, L. 2012 Dynamic clustering in active colloidal suspensions with chemical signaling. Phys. Rev. Lett. 108, 268303.
Thutupalli, S., Geyer, D., Singh, R., Adhikari, R. & Stone, H. A. 2018 Flow-induced phase separation of active particles is controlled by boundary conditions. Proc. Natl Acad. Sci. USA 115 (21), 54035408.
Thutupalli, S., Seemann, R. & Herminghaus, S. 2011 Swarming behavior of simple model squirmers. New J. Phys. 13 (7), 073021.
Toyota, T., Tsuha, H., Yamada, K., Takakura, K., Ikegami, T. & Sugawara, T. 2006 Listeria-like motion of oil droplets. Chem. Lett. 35 (7), 708709.
Uspal, W. E., Popescu, M. N., Dietrich, S. & Tasinkevych, M. 2015 Self-propulsion of a catalytically active particle near a planar wall: from reflection to sliding and hovering. Soft Matt. 11 (3), 434438.
Varma, A. & Michelin, S. 2019 Modeling chemo-hydrodynamic interactions of phoretic particles: a unified framework. Phys. Rev. Fluids 4, 124204.
Varma, A., Montenegro-Johnson, T. D. & Michelin, S. 2018 Clustering-induced self-propulsion of isotropic autophoretic particles. Soft Matt. 14 (35), 71557173.
Yabunaka, S. & Yoshinaga, N. 2016 Collision between chemically driven self-propelled drops. J. Fluid Mech. 806, 205233.
Yariv, E. 2016 Wall-induced self-diffusiophoresis of active isotropic colloids. Phys. Rev. Fluids 1 (3), 032101.
Yoshinaga, N., Nagai, K. H., 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 (1), 016108.
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Lippera et al. supplementary movie 1
Collision dynamics of a droplet with a rigid wall with Pe=6. The evolution of the concentration field is also shown (colours)

 Video (4.2 MB)
4.2 MB

Lippera et al. supplementary movie 2
Collision dynamics of a droplet with a rigid wall with Pe=20. The evolution of the concentration field is also shown (colours)

 Video (3.1 MB)
3.1 MB

Collisions and rebounds of chemically active droplets

  • K. Lippera (a1), M. Morozov (a1), M. Benzaquen (a1) and S. Michelin (a1)


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