A growing body of work aims at designing and testing micron-scale synthetic swimmers. One method, inspired by the locomotion of flagellated bacteria, consists of applying a rotating magnetic field to a rigid, helically shaped, propeller attached to a magnetic head. When the resulting device, termed an artificial bacteria flagellum, is aligned perpendicularly to the applied field, the helix rotates and the swimmer moves forward. Experimental investigation of artificial bacteria flagella shows that at low frequency of the applied field, the axis of the helix does not align perpendicularly to the field but wobbles around the helix, with an angle decreasing as the inverse of the field frequency. Using numerical computations and asymptotic analysis, we provide a theoretical explanation for this wobbling behavior. We numerically demonstrate the wobbling-to-swimming transition as a function of the helix geometry and the dimensionless Mason number which quantifies the ratio of viscous to magnetic torques. We then employ an asymptotic expansion for near-straight helices to derive an analytical estimate for the wobbling angle allowing to rationalize our computations and past experimental results. These results can help guide future design of artificial helical swimmers.
REFERENCES
1.
C.
Brennen
and H.
Winetl
, “Fluid mechanics of propulsion by cilia and flagella
,” Annu. Rev. Fluid Mech.
9
, 339
–98
(1977
).2.
T. J.
Pedley
and J. O.
Kessler
, “Hydrodynamic phenomena in suspensions of swimming microorganisms
,” Annu. Rev. Fluid Mech.
24
, 313
–358
(1992
).3.
B. J.
Nelson
, I. K.
Kaliakatsos
, and J. J.
Abbott
, “Microrobots for minimally invasive medicine
,” Annu. Rev. Biomed. Eng.
12
, 55
–85
(2010
).4.
J. J.
Abbott
, K. E.
Peyer
, M. C.
Lagomarsino
, L.
Zhang
, L.
Dong
, I. K.
Kaliakatsos
, and B. J.
Nelson
, “How should microrobots swim?
” Int. J. Robot. Res.
28
, 1434
(2009
).5.
G.
Kosa
, P.
Jakab
, G.
Szekely
, and N.
Hata
, “MRI driven magnetic microswimmers
,” Biomed. Microdevices
14
, 165
–178
(2011
).6.
G.
Kosa
, P.
Jakab
, N.
Hata
, F.
Jolesz
, Z.
Neubach
, M.
Shoham
, and Menashe
, “Flagellar swimming for medical micro robots: Theory, experiments and application
,” in Proceedings of the 2nd Biennial IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics
(IEEE
, New York, NY
, 2008
), pp. 258
–263
.7.
E. M.
Purcell
, “Life at low Reynolds number
,” Am. J. Phys.
45
, 3
–11
(1977
).8.
E.
Lauga
and T.
Powers
, “The hydrodynamics of swimming microorganisms
,” Rep. Prog. Phys.
72
, 096601
(2009
).9.
E.
Lauga
, “Life around the scallop theorem
,” Soft Matter
7
, 3060
–3065
(2011
).10.
W. F.
Paxton
, K. C.
Kistler
, C. C.
Olmeda
, A.
Sen
, S. K. St
Angelo
, Y. Y.
Cao
, T. E.
Mallouk
, P. E.
Lammert
, and V. H.
Crespi
, “Catalytic nanomotors: Autonomous movement of striped nanorods
,” J. Am. Chem. Soc.
126
, 13424
–13431
(2004
).11.
R.
Golestanian
, T. B.
Liverpool
, and A.
Ajdari
, “Propulsion of a molecular machine by asymmetric distribution of reaction products
,” Phys. Rev. Lett.
94
, 220801
(2005
).12.
J. R.
Howse
, R. A. L.
Jones
, A. J.
Ryan
, T.
Gough
, R.
Vafabakhsh
, and R.
Golestanian
, “Self-motile colloidal particles: From directed propulsion to random walk
,” Phys. Rev. Lett.
99
, 048102
(2007
).13.
R.
Golestanian
, T. B.
Liverpool
, and A.
Ajdari
, “Designing phoretic micro- and nano-swimmers
,” New J. Phys.
9
, 126
(2007
).14.
J.
Wang
, “Can man-made nanomachines compete with nature biomotors?
,” ACS Nano
3
, 4
–9
(2009
).15.
T. E.
Mallouk
and A.
Sen
, “Powering nanorobots
,” Sci. Am.
300
, 72
–77
(2009
).16.
T.
Mirkovic
, G. D.
Scholes
, N. S.
Zacharia
, and G. A.
Ozin
, “Nanolocomotion—catalytic nanomotors and nanorotors
,” ACS Nano
4
, 1782
–1789
(2010
).17.
P.
Tierno
, O.
Guell
, and F.
Sagues
, “Controlled propulsion in viscous fluids of magnetically actuated colloidal doublets
,” Phys. Rev. E
81
, 011402
(2010
).18.
C. E.
Sing
, L.
Schmid
, M. F.
Schneider
, T.
Franke
, and A.
Alexander-Katz
, “Controlled surface-induced flows from the motion of self-assembled colloidal walkers
,” Proc. Natl. Acad. Sci. U.S.A.
107
, 535
(2010
).19.
L.
Zhang
, T.
Petit
, Y.
Lu
, B. E.
Kratochvil
, K. E.
Peyer
, R.
Pei
, J.
Lou
, and B. J.
Nelson
, “Controlled propulsion and cargo transport of rotating nickel nanowires near a patterned solid surface
,” ACS Nano
4
, 6228
(2010
).20.
21.
R.
Dreyfus
, J.
Baudry
, M. L.
Roper
, M.
Fermigier
, H. A.
Stone
, and J.
Bibette
, “Microscopic artificial swimmers
,” Nature (London)
437
, 862
–865
(2005
).22.
W.
Gao
, S.
Sattayasamitsathit
, K. M.
Manesh
, D.
Weihs
, and J.
Wang
, “Magnetically powered flexible metal nanowire motors
,” J. Am. Chem. Soc
132
, 14403
(2010
).23.
O. S.
Pak
, W.
Gao
, J.
Wang
, and E.
Lauga
, “High-speed propulsion of flexible nanowire motors: Theory and experiments
,” Soft Matter
7
, 8169
–8181
(2011
).24.
L.
Zhang
, K. E.
Peyer
, and B. J.
Nelson
, “Artificial bacteria flagella for micromanipulation
,” Lab Chip
10
, 2203
–2215
(2010
).25.
A.
Ghosh
and P.
Fischer
, “Controlled propulsion of artificial magnetic nanostructured propellers
,” Nano Lett.
9
, 2243
–2245
(2009
).26.
A.
Ghosh
, D.
Paria
, H. J.
Singh
, P. L.
Venugopalan
, and A.
Ghosh
, “Dynamical configurations and bistability of helical nanostructures under external torque
,” Phys. Rev. E
86
, 031401
(2012
).27.
K. E.
Peyer
, B. E.
Kratochvil
, L.
Zhang
, and B. J.
Nelson
, “Non-ideal swimming of artificial bacterial flagella near a surface
,” in Proceedings of the 2010 IEEE International Conference on Robotics and Automation
(IEEE
, New York, NY
, 2010
), pp. 96
–101
.28.
S.
Tottori
, L.
Zhang
, F.
Qiu
, K. K.
Krawczyk
, A.
Franco-Obregon
, and B. J.
Nelson
, “Magnetic helical micromachines: Fabrication, controlled swimming, and cargo transport
,” Adv. Mate.
24
, 811
–816
(2012
).29.
S.
Jung
, K.
Mareck
, L.
Fauci
, and M. J.
Shelley
, “Rotational dynamics of a superhelix towed in a Stokes fluid
,” Phys. Fluids
19
, 103105
(2007
).30.
R. G.
Cox
, “The motion of long slender bodies in a viscous fluid
,” J. Fluid Mech.
44
, 791
–810
(1970
).31.
G. K.
Batchelor
, “Slender-body theory for particles of arbitrary cross-section in stokes flow
,” J. Fluid Mech.
44
, 419
–440
(1970
).32.
J. B.
Keller
and S. I.
Rubinow
, “Slender body theory for slow viscous flow
,” J. Fluid Mech.
75
, 705
–714
(1976
).33.
R. E.
Johnson
, “An improved slender body theory for Stokes flow
,” J. Fluid Mech.
99
, 411
–431
(1980
).34.
O. S.
Pak
, S. E.
Spagnolie
, and E.
Lauga
, “Hydrodynamics of the double-wave structure of insect spermatozoa flagella
,” J. R. Soc., Interface
9
, 1908
–1924
(2012
).35.
S.
Jung
, K.
Mareck
, L.
Fauci
, and M. J.
Shelley
, “Rotational dynamics of a superhelix towed in a stokes fluid
,” Phys. Fluid
19
(2007
).36.
C. W.
Wolgemuth
, T. R.
Powers
, and R. E.
Goldstein
, “Twirling and whirling: Viscous dynamics of rotating elastic filaments
,” Phys. Rev. Lett.
84
, 1623
–1626
(2000
).© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.
