Abstract
A theoretical framework is presented for calculating the polarization and induced-charge electrophoretic mobility of a polarizable closed (horn) toroidal micro-particle exposed to a non-uniform axial AC electric forcing. The analysis is based on employing the standard (linearized) ‘weak-field’ electroosmotic model for a symmetric electrolyte. In particular, we discuss the case of traveling-wave excitation and provide analytic expressions for the tori induced-phoretic velocity in the Stokes regime in terms of the frequency and wavelength of the ambient electric field. In addition, we consider the non-linear electroosmotic flow problem about a stationary torus, which is subject to a uniform field, and provide explicit expressions for the resulting Stokes’ stream function driven by the surface Helmholtz–Smoluchowski velocity slip. Finally, we analyze the case of asymmetric (transverse) two-component electrorotation and by calculating the Maxwell electric torque, provide analytic solution for the induced-charge angular velocity of a freely suspended conducting horn torus.
Similar content being viewed by others
References
Jones TB (1995) Electro mechanics of particles. Cambridge University Press, Cambridge
Morgan H, Green NG (2003) AC electro kinetics: colloids and nanoparticles. Research Studies Press, Baldock
Squires TM, Bazant MZ (2004) Induced-charge electro-osmosis. J. Fluid Mech. 509:217–252
Henslee EA (2020) Review: dielectrophoresis in cell characterization. Electrophoresis 41:1915–1930
Miloh T (2008) A unified theory for the dipolophoresis of nanoparticles. Phys. Fluids 20:107105
Flores-Mena JS, Garcia-Sanchez P, Ramos A (2020) Dipolophoresis and travelling-wave dipolophoresis of metal microparticles. Micromachines 11:259
Miloh T, Nagler J (2021) Travelling-wave dipolophoresis: levitation and electrorotation of Janus nanoparticles. Micromachines 12:114
Frankel I, Yossifon G, Miloh T (2012) Dipolophoresis of dielectric spheroids under asymmetric fields. Phys. Fluids 24:012004
Miloh T, Weis-Goldstein B (2015) Electrophoretic rotation and orientation of spheroidal particles in AC fields. Phys. Fluids 27:022003
Weis-Goldstein B, Miloh T (2017) 3D controlled electrorotation of conducting tri-axial ellipsoidal nanoparticles. Phys. Fluid 29:052008
Zehe A, Ramirez A, Staraostenko O (2004) Mathematical modelling of electro-rotation spectra of small particles in liquid solutions. Applications to human erythrocyte aggregates. Braz. J. Med. Bio. Res. 37:173–183
Mesarec L, Gozdz W, Iglic A, Kralji-Iglic V, Virga EG, Kralji S (2019) Normal red blood cell’s shape stabilized by membranes in plane ordering. Sci. Rep. 9:19742
Techaumnal B, Panklang N (2021) Electrochemical analysis of red blood cells under AC electric fields. IEEE Trans. Magn. 10:1109
Krokhmal PA (2002) Exact solution of the displacement boundary-value problem of elasticity for a torus. J. Eng. Math. 44:345–368
Krasnitskii S, Trafinov A, Radi E, Sevestianov I (2019) Effect of a rigid toroidal inhomogeneity on the elastic properties of a composite. Math. Mech. Solids 24(4):1129–1146
Radi E, Sevestianov I (2016) Toroidal insulting inhomogeneity in an infinite space and related problems. Proc. R. Soc. A. 472:0781
Leshanski AM, Kenneth O (2008) Surface tank treading: Propulsion of Purcell’s toroidal swimmer. Phys. Fluids 20:063104
Schmeding LC, Lauga E, Montenegro-Johnson TD (2012) Autophoretic flow on a torus. Phys. Rev. Fluids 2:034201
Thaokar RM, Schiessel H, Kulic IM (2007) Hydrodynamics of a rotating torus. Eur. Phys. J. B. 60:325–336
Scharstein RW, Wilson HB (2005) Electrostatic excitation of a conducting toroid: exact solution and the thin-wire approximation. Electrophoresis 25:1–19
Wright EB, Peterson PE (1967) Magnetization of an ideal superconducting torus in transverse field. J. Appl. Phys. 38(2):855–860
Ungphaiboon SL, Attiya D, Gomez d’Ayala G, Sansongsak P, Cellesi F, Tireli N (2010) Materials for microencapsulation: What toroidal (‘doughnuts’) can do better than spherical beads. Soft Matter 6:4070–4083
Moon F, Spencer DE (1961) Field theory handbook. Springer, Berlin
Pell WH, Payne LE (1960) On Stokes flow about a torus. Mathematica 7:78–92
Takaji H (1973) Slow viscous flow due to the motion of a closed torus. J. Phys. Soc. Jpn. 35(4):1225–1227
Dorrepaal JM, Majumdar SR, O’Neill ME, Ranger KL (1976) A closed torus in Stokes flow. Quart. J. Mech. Appl. Math. 29(4):381
Majumdar SR, O’Neill ME (1979) Asymmetric Stokes flows generated by the motion of a closed torus. J. Appl. Math. Phys. 30:967–982
Williams MMR (1987) A closed torus in Stokes flow with slip boundary conditions. Quart. Mech. Appl. Math. 40(2):235–246
Wakiya S (1971) Slow motion in shear flow of doublet of two spheres in contact. J. Phys. Soc. Jpn. 31:1581–1587
Nir A, Acrivos A (1973) On the creeping flow motion of two arbitrary-sized touching spheres in a linear shear field. J. Fluid Mech. 59:209–223
Takaji H (1974) Slow rotation of two touching spheres in viscous fluid. J. Phys. Soc. Jpn. 36:875–877
Zabarankin M (2007) Asymmetric three-dimensional Stokes flows about two fuses equal spheres. Proc. R. Soc. A. 463:2249–2329
Latta GE, Hess GB (1973) Potential flow past a sphere tangent to a plane. Phys. Fluids 16:974–976
Small RD, Weihs D (1975) Axisymmetric potential flow over two spheres in contact. ASME. J. Appl. Mech. 42:763–765
Morrison FA (1976) Irrotational flow about two touching spheres. ASME. J. Appl. Mech. 43(2):365–366
Davis AM (1977) High frequency limiting virtual-mass coefficient of heaving half-immersed spheres. J. Fluid Mech. 80(2):305–319
Bentwich M, Miloh T (1978) On the exact solution for the two-sphere problem in axisymmetric potential flow. ASME. J. Appl. Mech. 45(3):463–465
Cox SJ, Cooker MJ (2000) Potential flow past a sphere touching a tangent plane. J. Eng. Math. 38:355–370
Pitkonen M (2008) Polarizability of a pair of touching dielectric spheres. J. Appl. Phys. 103:104910
Lanzoni L, Radi E, Sevostianov I (2020) Effect of spherical pores coalescence on the overall conductivity of a material. Mech. Mat. 148:103463
Miloh T (1979) The virtual mass of a closed torus. J. Eng. Math. 13(1):1–6
Gradshteyn IS, Ryzhik IM (1965) Table of Integrals, Series, and Products. Academic Press Inc., San Diego
Hobson EW (1965) The Theory of Spherical and Ellipsoidal Harmonics. Chelsea, New York
Smythe WR (1961) Flow around a sphere in a circular tube. Phys. Fluids 4(4):756–759
Miloh T (2008) Dipolophoresis of nanoparticles. Phys. Fluids 20:083303
Murtsovkin VA (1996) Nonlinear flows near polarized disperse particles. Colloid J. 58:341
Yariv E, Miloh T (2008) Electro-convection about conducting particles. J. Fluid Mech. 595:163
Miloh T (2019) AC electrokinetics of polarizable tri-axial ellipsoidal nano-antennas and quantum dot manipulations. Micromachines 10(2):83
Happel J, Brenner H (1983) Low Reynolds Hydrodynamics. Maritinus Nijhoff, The Hague
Garcia-Sanchez P, Ramos A (2015) Electrorotation of a metal sphere immersed in an electrolyte of finite Debye length. Phys. Rev. E. 92:052313
Miloh T, Nagler J (2021) Travelling-wave dipolophoresis: levitation and electrorotation of Janus nanoparticles. Micromachines 12:114
Acknowledgements
The partial support of BSF Grant 2018168 is acknowledged. Thanks also to E. Avital from QMUL for his help in preparing Fig. 2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Miloh, T. Induced-charge electroosmosis, polarization, electrorotation, and traveling-wave electrophoresis of horn toroidal particles. J Eng Math 133, 7 (2022). https://doi.org/10.1007/s10665-021-10194-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10665-021-10194-4