Skip to main content Accessibility help
×
×
Home

Phoretic self-propulsion at finite Péclet numbers

  • Sébastien Michelin (a1) and Eric Lauga (a2)
Abstract

Phoretic self-propulsion is a unique example of force- and torque-free motion on small scales. The classical framework describing the flow field around a particle swimming by self-diffusiophoresis neglects the advection of the solute field by the flow and assumes that the chemical interaction layer is thin compared to the particle size. In this paper we quantify and characterize the effect of solute advection on the phoretic swimming of a sphere. We first rigorously derive the regime of validity of the thin-interaction-layer assumption at finite values of the Péclet number ( ${Pe}$ ). Under this assumption, we solve computationally the flow around Janus phoretic particles and examine the impact of solute advection on propulsion and the flow created by the particle. We demonstrate that although advection always leads to a decrease of the swimming speed and flow stresslet at high values of the Péclet number, an increase can be obtained at intermediate values of ${Pe}$ . This possible enhancement of swimming depends critically on the nature of the chemical interactions between the solute and the surface. We then derive an asymptotic analysis of the problem at small ${Pe}$ which allows us to rationalize our computational results. Our computational and theoretical analysis is accompanied by a parallel study of the influence of reactive effects at the surface of the particle (Damköhler number) on swimming.

Copyright
Corresponding author
Email address for correspondence: sebastien.michelin@ladhyx.polytechnique.fr
References
Hide All
Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21, 6199.
Anderson, J. L. & Prieve, D. C. 1991 Diffusiophoresis caused by gradients of strongly adsorbing solutes. Langmuir 7, 403406.
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545570.
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.
Bickel, T., Majee, A. & Würger, A. 2013 Flow pattern in the vicinity of self-propelling hot Janus particles. Phys. Rev. E 88, 012301.
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199208.
Brady, J. 2011 Particle motion driven by solute gradients with application to autonomous motion: continuum and colloidal perspectives. J. Fluid Mech. 667, 216259.
Bray, D. 2000 Cell Movements: From Molecules to Motility. Garland Science.
Brennen, C. & Winnet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339398.
Broyden, C. G. 1965 A class of methods for solving nonlinear simultaneous equations. Maths Comput. 19, 577593.
Córdova-Figueroa, U. M. & Brady, J. F. 2008 Osmotic propulsion: the osmotic motor. Phys. Rev. Lett. 100 (15), 158303 ; see also the comment on this article by F. Jülicher & J. Prost, Phys. Rev. Lett. 103, 079801.
Córdova-Figueroa, U. M., Brady, J. F. & Shklyaev, S. 2013 Osmotic propulsion of colloidal particles via constant surface flux. Soft Matt. 9, 63826390.
Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A. & Bibette, J. 2005 Microscopic artificial swimmers. Nature 437, 862865.
Ebbens, S. J. & Howse, J. R. 2011 Direct observation of the direction of motion for spherical catalytic swimmers. Langmuir 27, 1229312296.
Ebbens, S., Tu, M.-H., Howse, J. R. & Golestanian, R. 2012 Size dependence of the propulsion velocity for catalytic Janus-sphere swimmers. Phys. Rev. E 85, 020401.
Gao, W., Sattayasamitsathit, S., Manesh, K. M., Weihs, D. & Wang, J. 2010 Magnetically powered flexible metal nanowire motors. J. Am. Chem. Soc. 132, 1440314405.
Ghosh, A. & Fischer, P. 2009 Controlled propulsion of artificial magnetic nanostructured propellers. Nano Lett. 9, 22432245.
Golestanian, R., Liverpool, T. B. & Ajdari, A. 2005 Propulsion of a molecular machine by asymmetric distribution of reaction products. Phys. Rev. Lett. 94 (22), 220801.
Golestanian, R., Liverpool, T. B. & Ajdari, A. 2007 Designing phoretic micro- and nano-swimmers. New J. Phys. 9, 126.
Howse, J. R., Jones, R. A. L., Ryan, A. J., Gough, T., Vafabakhsh, R. & Golestanian, R. 2007 Self-motile colloidal particles: from directed propulsion to random walk. Phys. Rev. Lett. 99, 048102.
Jiang, H.-R., Yoshinaga, N. & Sano, M. 2010 Active motion of a Janus particle by self-thermophoresis in a defocused laser beam. Phys. Rev. Lett. 105, 268302.
Jülicher, F. & Prost, J. 2009a Comment on osmotic propulsion: the osmotic motor. Phys. Rev. Lett. 103, 079801.
Jülicher, F. & Prost, J. 2009b Generic theory of colloidal transport. Eur. Phys. J. E 29 (1), 2736.
Khair, A. S. 2013 Diffusiophoresis of colloidal particles in neutral solute gradients at finite Péclet number. J. Fluid Mech. 731, 6494.
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming micro-organisms. Rep. Prog. Phys. 72, 096601.
Magar, V., Goto, T. & Pedley, T. J. 2003 Nutrient uptake by a self-propelled steady squirmer. Q. J. Mech. Appl. Maths 56, 6591.
Michelin, S. & Lauga, E. 2011 Optimal feeding is optimal swimming for all Péclet numbers. Phys. Fluids 23 (10), 101901.
Michelin, S. & Lauga, E. 2013 Unsteady feeding and optimal strokes of model ciliates. J. Fluid Mech. 715, 131.
Michelin, S., Lauga, E. & Bartolo, D. 2013 Spontaneous autophoretic motion of isotropic particles. Phys. Fluids 25, 061701.
Nelson, B. J., Kaliakatsos, I. K. & Abbott, J. J. 2010 Microrobots for minimally invasive medicine. Annu. Rev. Biomed. Engng 12 (1), 5585.
O’Brien, R. W. 1983 The solution of the electrokinetic equations for colloidal particles with thin double layers. J. Colloid Interface Sci. 92, 204216.
Palacci, J., Sacanna, S., Steinberg, A. P., Pine, D. J. & Chaikin, P. M. 2013 Living crystals of light-activated colloidal surfers. Science 339, 936940.
Paxton, W. F., Kistler, K. C., Olmeda, C. C., Sen, A., Angelo, S. K. St., Cao, Y., Mallouk, T. E., Lammert, P. E. & Crespi, V. H. 2004 Catalytic nanomotors: autonomous movement of striped nanorods. J. Am. Chem. Soc. 126 (41), 1342413431.
Popescu, M. N., Dietrich, S., Tasinkevych, M. & Ralston, J. 2010 Phoretic motion of spheroidal particles due to self-generated solute gradients. Eur. Phys J. E 31, 351367.
Prieve, D. C., Anderson, J. L., Ebel, J. P. & Lowell, M. E. 1984 Motion of a particle generated by chemical gradients. Part 2. Electrolytes. J. Fluid Mech. 148, 247269.
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45, 311.
Sabass, B. & Seifert, U. 2012 Dynamics and efficiency of a self-propelled, diffusiophoretic swimmer. J. Chem. Phys. 136, 064508.
Schmitt, M. & Stark, H. 2013 Swimming active droplet: a theoretical analysis. Eur. Phys. Lett. 101, 44008.
Sharifi-Mood, N., Koplik, J. & Maldarelli, C. 2013 Diffusiophoretic self-propulsion of colloids driven by a surface reaction: the sub-micron particle regime for exponential and van der Waals interactions. Phys. Fluids 25, 012001.
Stone, H. A. & Samuel, A. D. T. 1996 Propulsion of microorganisms by surface distorsions. Phys. Rev. Lett. 77, 4102.
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., Seemann, R. & Herminghaus, S. 2011 Swarming behaviour of simple model squirmers. New J. Phys. 13, 073021.
Walther, A. & Müller, A. H. E. 2008 Janus particles. Soft Matt. 4, 663668.
Wang, J. 2009 Can man-made nanomachines compete with nature biomotors?. ACS Nano 3, 49.
Yariv, E. 2010 An asymptotic derivation of the thin-Debye-layer limit for electrokinetic phenomena. Chem. Engng Commun. 197, 317.
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, 016108.
Zhang, L., Peyer, K. E. & Nelson, B. J. 2010 Artificial bacterial flagella for micromanipulation. Lab on a Chip 10, 22032215.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed