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Squirming inside a liquid droplet with surface viscosities

Herve Nganguia and Adedoyin Adegbuyi

Matthew Uffenheimer and On Shun Pak*

  • *Contact author: opak@scu.edu
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Phys. Rev. Fluids 10, 033104 – Published 28 March, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.033104

Abstract

The possibility of encapsulating therapeutic substances with active particles inside droplets, thereby propelling their motion from within, presents exciting opportunities for biomedical applications such as targeted drug delivery. In realistic biological and environmental settings, droplet interfaces often exhibit complex interfacial rheological behaviors due to different molecules or particles laden on the interface. Motivated by this complexity, we investigate the effects of interfacial rheology on the motion of an active droplet consisting of a liquid droplet enclosing an active particle described by the squirmer model. Specifically, we examine theoretically how surface shear and dilatational viscosities impact the propulsion of both the enclosed squirmer and the enclosing droplet. Our results indicate that while surface shear viscosity has no impact on the propulsion speeds, both the droplet and the squirmer swim slower with increased surface dilatational viscosity. The independence of propulsion speeds from surface shear viscosity is a feature shared with the classical problem of a translating droplet with surface viscosities. However, we add a cautionary remark on the subtlety in interpreting the impact of surface viscosities in these two problems. We also examine how the presence of surface viscosities affects the flow field and energetic cost in this active droplet system. These findings represent a first step towards understanding how complexities arising from realistic biological or environmental settings influence the behavior of microswimmer-driven droplets, paving the way for their potential applications in these complex environments.

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References (47)

  1. A. D. Griffiths and D. S. Tawfik, Miniaturising the laboratory in emulsion droplets, Trends Biotechnol. 24, 395 (2006).
  2. B. T. Kelly, J.-C. Baret, V. Taly, and A. D. Griffiths, Miniaturizing chemistry and biology in microdroplets, Chem. Commun. 2007, 1773 (2007).
  3. M. He, J. S. Edgar, G. D. M. Jeffries, R. M. Lorenz, J. P. Shelby, and D. T. Chiu, Selective encapsulation of single cells and subcellular organelles into picoliter- and femtoliter-volume droplets, Anal. Chem. 77, 1539 (2005).
  4. M. Chabert and J.-L. Viovy, Microfluidic high-throughput encapsulation and hydrodynamic self-sorting of single cells, Proc. Natl. Acad. Sci. USA 105, 3191 (2008).
  5. J. Clausell-Tormos, D. Lieber, J.-C. Baret, A. El-Harrak, O. J. Miller, L. Frenz, J. Blouwolff, K. J. Humphry, S. Köster, H. Duan, C. Holtze, D. A. Weitz, A. D. Griffiths, and C. A. Merten, Droplet-based microfluidic platforms for the encapsulation and screening of mammalian cells and multicellular organisms, Chem. Bio. 15, 427 (2008).
  6. H. Wen, Y. Yu, G. Zhu, L. Jang, and J. Qin, A droplet microchip with substance exchange capability for the developmental study of C. elegans, Lab Chip 15, 1905 (2015).
  7. W. Gao and J. Wang, The environmental impact of micro/nanomachines: A review, ACS Nano 8, 3170 (2014).
  8. J. Li, B. E.-F. de Ávila, W. Gao, L. Zhang, and J. Wang, Micro/nanorobots for biomedicine: Delivery, surgery, sensing, and detoxification, Sci. Robot. 2, eaam6431 (2017).
  9. C. K. Schmidt, M. Medina-Sánchez, R. J. Edmondson, and O. G. Schmidt, Engineering microrobots for targeted cancer therapies from a medical perspective, Nat. Commun. 11, 5618 (2020).
  10. Z. Wu, Y. Chen, D. Mukasa, O. S. Pak, and W. Gao, Medical micro/nanorobots in complex media, Chem. Soc. Rev. 49, 8088 (2020).
  11. B. J. Nelson and S. Pané, Delivering drugs with microrobots, Science 382, 1120 (2023).
  12. R. Nauber, S. R. Goudu, M. Goeckenjan, M. Bornhäuser, C. Ribeiro, and M. Medina-Sánchez, Medical microrobots in reproductive medicine from the bench to the clinic, Nat. Commun. 14, 728 (2023).
  13. Y. Ding, F. Qiu, X. C. Solvas, F. W. Y. Chiu, B. J. Nelson, and A. DeMello, Microfluidic-based droplet and cell manipulations using artificial bacterial flagella, Micromachines 7, 25 (2016).
  14. S. Y. Reigh, L. Zhu, F. Gallaire, and E. Lauga, Swimming with a cage: low-reynolds-number locomotion inside a droplet, Soft Matter 13, 3161 (2017).
  15. R. Kree, L. Rückert, and A. Zippelius, Dynamics of a droplet driven by an internal active device, Phys. Rev. Fluids 6, 034201 (2021).
  16. K. V. S. Chaithanya and P. T. Sumesh, Deformation dynamics of an active compound particle in an imposed shear flow—a theoretical study, J. Phys. D: Appl. Phys. 53, 314001 (2020).
  17. G. G. Fuller and J. Vermant, Complex fluid-fluid interfaces: Rheology and structure, Annu. Rev. Chem. Biomol. Eng. 3, 519 (2012).
  18. J. R. Samaniuk and J. Vermant, Micro and macrorheology at fluid–fluid interfaces, Soft Matter 10, 7023 (2014).
  19. N. O. Jaensson, P. D. Anderson, and J. Vermant, Computational interfacial rheology, J. Non-Newtonian Fluid Mech. 290, 104507 (2021).
  20. V. A. Shaik, V. Vasani, and A. M. Ardekani, Locomotion inside a surfactant-laden drop at low surface Péclet numbers, J. Fluid Mech. 851, 187 (2018).
  21. A. R. Sprenger, V. A. Shaik, A. M. Ardekani, M. Lisicki, A. J. T. M. Mathijssen, F. Guzmán-Lastra, H. Löwen, A. M. Menzel, and A. Daddi-Moussa-Ider, Towards an analytical description of active microswimmers in clean and in surfactant-covered drops, Eur. Phys. J. E 43, 58 (2020).
  22. V. A. Shaik and A. M. Ardekani, Point force singularities outside a drop covered with an incompressible surfactant: Image systems and their applications, Phys. Rev. Fluids 2, 113606 (2017).
  23. M. J. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers, Commun. Pure Appl. Math. 5, 109 (1952).
  24. J. R. Blake, A spherical envelope approach to ciliary propulsion, J. Fluid Mech. 46, 199 (1971).
  25. T. J. Pedley, Spherical squirmers: Models for swimming micro-organisms, IMA J. Appl. Math. 81, 488 (2016).
  26. T. Ishikawa, Fluid dynamics of squirmers and ciliated microorganisms, Annu. Rev. Fluid Mech. 56, 119 (2024).
  27. S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications (Dover, New York, 1991).
  28. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Noordhoff International Publishing, Leyden, 1973).
  29. L. E. Scriven, Dynamics of a fluid interface, Chem. Eng. Sci. 12, 98 (1960).
  30. L. G. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes (Cambridge University Press, Cambridge, 2007).
  31. R. S. Subramanian and R. Balasubramaniam, The Motion of Bubbles and Drops in Reduced Gravity (Cambridge University Press, Cambridge, 2001).
  32. H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, 1932).
  33. O. S. Pak and E. Lauga, Generalized squirming motion of a sphere, J. Eng. Math. 88, 1 (2014).
  34. S. Y. Reigh and E. Lauga, Two-fluid model for locomotion under self-confinement, Phys. Rev. Fluids 2, 093101 (2017).
  35. H. Nganguia, L. Zhu, D. Palaniappan, and O. S. Pak, Squirming in a viscous fluid enclosed by a Brinkman medium, Phys. Rev. E 101, 063105 (2020).
  36. S. K. Agrawal and D. T. Wasan, The effect of interfacial viscosities on the motion of drops and bubbles, J. Chem. Eng. 18, 215 (1979).
  37. M. D. Levan, Motion of a droplet with a Newtonian interface, J. Colloid Interface Sci. 83, 11 (1981).
  38. V. Narsimhan, Letter: The effect of surface viscosity on the translational speed of droplets, Phys. Fluids 30, 081703 (2018).
  39. L. M. C. Sagis, Dynamic properties of interfaces in soft matter: Experiments and theory, Rev. Mod. Phys. 83, 1367 (2011).
  40. A. Andersen, J. Oertegren, P. Koelsch, D. Wantke, and H. Motschmann, Oscillating bubble SHG on surface elastic and surface viscoelastic systems: New insights in the dynamics of adsorption layers, J. Phys. Chem. B 110, 18466 (2006).
  41. J. Maldonado-Valderrama, V. B. Fainerman, M. J. Gálvez-Ruiz, A. Martín-Rodriguez, M. A. Cabrerizo-Vílchez, and R. Miller, Dilatational rheology of β-casein adsorbed layers at liquid-fluid interfaces, J. Phys. Chem. B 109, 17608 (2005).
  42. M. Lakshmanan, A. Dhathathreyan, and R. Miller, Synergy between hofmeister effect and coupled water in proteins: Unusual dilational moduli of BSA at air/solution interface, Colloids Surf. A 324, 194 (2008).
  43. F. Monroy, H. M. Hilles, F. Ortega, and R. G. Rubio, Relaxation dynamics of langmuir polymer films: A power-law analysis, Phys. Rev. Lett. 91, 268302 (2003).
  44. H. Hilles, F. Monroy, L. J. Bonales, F. Ortega, and R. G. Rubio, Fourier-transform rheology of polymer langmuir monolayers: Analysis of the non-linear and plastic behaviors, Adv. Colloid Interface Sci. 122, 67 (2006).
  45. M. Vrânceanu, K. Winkler, H. Nirschl, and G. Leneweit, Surface rheology and phase transitions of monolayers of phospholipid/cholesterol mixtures, Biophys. J. 94, 3924 (2008).
  46. L. R. Arriaga, I. López-Montero, J. Ignés-Mullol, and F. Monroy, Domain-growth kinetic origin of nonhorizontal phase coexistence plateaux in Langmuir monolayers: Compression rigidity of a raft-like lipid distribution, J. Phys. Chem. B 114, 4509 (2010).
  47. R. Dandekar and A. M. Ardekani, Effect of interfacial viscosities on droplet migration at low surfactant concentrations, J. Fluid Mech. 902, A2 (2020).

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Phys. Rev. Fluids 10, 033104– Published 28 March, 2025
  • Received 29 August 2024
  • Accepted 9 January 2025
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