Skip to main content Accessibility help

Electrophoresis of bubbles

  • Ory Schnitzer (a1), Itzchak Frankel (a2) and Ehud Yariv (a1)


Smoluchowski’s celebrated electrophoresis formula is inapplicable to field-driven motion of drops and bubbles with mobile interfaces. We here analyse bubble electrophoresis in the thin-double-layer limit. To this end, we employ a systematic asymptotic procedure starting from the standard electrokinetic equations and a simple physicochemical interface model. This furnishes a coarse-grained macroscale description where the Debye-layer physics is embodied in effective boundary conditions. These conditions, in turn, represent a non-conventional driving mechanism for electrokinetic flows, where bulk concentration polarization, engendered by the interaction of the electric field and the Debye layer, results in a Marangoni-like shear stress. Remarkably, the electro-osmotic velocity jump at the macroscale level does not affect the electrophoretic velocity. Regular approximations are obtained in the respective cases of small zeta potentials, small ions, and weak applied fields. The nonlinear small-zeta-potential approximation rationalizes the paradoxical zero mobility predicted by the linearized scheme of Booth (J. Chem. Phys., vol. 19, 1951, pp. 1331–1336). For large (millimetre-size) bubbles the pertinent limit is actually that of strong fields. We have carried out a matched-asymptotic-expansion analysis of this singular limit, where salt polarization is confined to a narrow diffusive layer. This analysis establishes that the bubble velocity scales as the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2/3$ -power of the applied-field magnitude and yields its explicit functional dependence upon a specific combination of the zeta potential and the ionic drag coefficient. The latter is provided to within an $O(1)$ numerical pre-factor which, in turn, is calculated via the solution of a universal (parameter-free) nonlinear flow problem. It is demonstrated that, with increasing field magnitude, all numerical solutions of the macroscale model indeed collapse on the analytic approximation thus obtained. Existing measurements of clean-bubble electrophoresis agree neither with present theory nor with previous models; we discuss this ongoing discrepancy.


Corresponding author

Email address for correspondence:


Hide All
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Baygents, J. C. & Saville, D. A. 1989 The circulation produced in a drop by an electric field: a high field strength electrokinetic model. In Drops & Bubbles, Third International Colloquium, Monterey 1988 (ed. Wang, T.), AIP Conference Proceedings, vol. 7, pp. 717. Am. Inst. Phys..
Baygents, J. C. & Saville, D. A. 1991 Electrophoresis of drops and bubbles. J. Chem. Soc. Faraday Trans. 87 (12), 18831898.
Booth, F. 1951 The cataphoresis of spherical fluid droplets in electrolytes. J. Chem. Phys. 19, 13311336.
Brandon, N. P., Kelsall, G. H., Levine, S. & Smith, A. L. 1985 Interfacial electrical properties of electrogenerated bubbles. J. Appl. Electrochem. 15 (4), 485493.
Chang, H.-C. & Yeo, L. Y. 2010 Electrokinetically Driven Microfluidics and Nanofluidics. Cambridge University Press.
Choi, K., Kim, S. J., Jin, Y. G., Jang, Y. O., Kim, J.-S. & Chung, D. S. 2008 Single drop microextraction using commercial capillary electrophoresis instruments. Anal. Chem. 81 (1), 225230.
Davis, J. A., James, R. O. & Leckie, J. O. 1978 Surface ionization and complexation at the oxide/water interface: I. Computation of electrical double layer properties in simple electrolytes. J. Colloid Interface Sci. 63 (3), 480499.
Graciaa, A., Morel, G., Saulner, P., Lachaise, J. & Schechter, R. S. 1995 The $\zeta $ -potential of gas bubbles. J. Colloid Interface Sci. 172 (1), 131136.
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Harper, J. F. 2010 Electrophoresis of surfactant-free bubbles. J. Colloid Interface Sci. 350 (1), 361367.
Hinch, E. J., Sherwood, J. D., Chew, W. C. & Sen, P. N. 1984 Dielectric response of a dilute suspension of spheres with thin double layers in an asymmetric electrolyte. J. Chem. Soc. Faraday Trans. 2 80 (5), 535551.
Huebner, A., Sharma, S., Srisa-Art, M., Hollfelder, F., Edel, J. B. & Demello, A. J. 2008 Microdroplets: a sea of applications? Lab on a Chip 8 (8), 12441254.
Hunter, R. J. 2000 Foundations of Colloidal Science. Oxford University Press.
Kelsall, G. H., Tang, S., Smith, A. L. & Yurdakul, S. 1996a Measurement of rise and electrophoretic velocities of gas bubbles. J. Chem. Soc. Faraday Trans. 92, 38793885.
Kelsall, G. H., Tang, S., Yurdakul, S. & Smith, A. L. 1996b Electrophoretic behaviour of bubbles in aqueous electrolytes. J. Chem. Soc. Faraday Trans. 92, 38873893.
Khair, A. S. 2013 Diffusiophoresis of colloidal particles in neutral solute gradients at finite Péclet number. J. Fluid Mech. 731, 6494.
Kumar, A., Elele, E., Yeksel, M., Khusid, B., Qiu, Z. & Acrivos, A. 2006 Measurements of the fluid and particle mobilities in strong electric fields. Phys. Fluids 18, 123301.
Leroy, P., Jougnot, D., Revil, A., Lassin, A. & Azaroual, M. 2012 A double layer model of the gas bubble/water interface. J. Colloid Interface Sci. 388, 243256.
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.
Liu, H. & Dasgupta, P. K. 1997 A falling drop for sample injection in capillary zone electrophoresis. Analyt. Chem. 69 (6), 12111216.
Lyklema, J. 1995 Fundamentals of Interface and Colloid Science, vol. II. Academic Press.
McTaggart, H. A. 1914 The electrification at liquid–gas surfaces. Phil. Mag. 27 (158), 297314.
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111146.
Morrison, F. A. 1970 Electrophoresis of a particle of arbitrary shape. J. Colloid Interface Sci. 34, 210214.
O’Brien, R. W. 1983 The solution of the electrokinetic equations for colloidal particles with thin double layers. J. Colloid Interface Sci. 92 (1), 204216.
O’Brien, R. W. & White, L. R. 1978 Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc. Faraday Trans. 74, 16071626.
Ohshima, H., Healy, T. W. & White, L. R. 1984 Electrokinetic phenomena in a dilute suspension of charged mercury drops. J. Chem. Soc. Faraday Trans. 2 80 (12), 16431667.
Quincke, G. 1861 Ueber die fortfiihrüng Materieller theilchen durch strömende Elektricität. Ann. Phys. Chem. 115, 513598.
Rivette, N. J. & Baygents, J. C. 1996 A note on the electrostatic force and torque acting on an isolated body in an electric field. Chem. Engng Sci. 51 (23), 52055211.
Russel, W. B., Saville, D. A. & Schowalter, W. R. 1989 Colloidal Dispersions. Cambridge University Press.
Saville, D. A. 1977 Electrokinetic effects with small particles. Annu. Rev. Fluid Mech. 9, 321337.
Schnitzer, O., Frankel, I. & Yariv, E. 2013 Electrokinetic flows about conducting drops. J. Fluid Mech. 722, 394423.
Schnitzer, O. & Yariv, E. 2012a Macroscale description of electrokinetic flows at large zeta potentials: nonlinear surface conduction. Phys. Rev. E 86, 021503.
Schnitzer, O. & Yariv, E. 2012b Strong-field electrophoresis. J. Fluid Mech. 701, 333351.
Teh, S. Y., Lin, R., Hung, L. H. & Lee, A. P. 2008 Droplet microfluidics. Lab on a Chip 8 (2), 198220.
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.
Yang, C., Dabros, T., Li, D., Czarnecki, J. & Masliyah, J. H. 2001 Measurement of the zeta potential of gas bubbles in aqueous solutions by microelectrophoresis method. J. Colloid Interface Sci. 243 (1), 128135.
Yariv, E. 2006 ‘Force-free’ electrophoresis? Phys. Fluids 18, 031702.
Yariv, E., Schnitzer, O. & Frankel, I. 2011 Streaming-potential phenomena in the thin-Debye-layer limit. Part 1. General theory. J. Fluid Mech. 685, 306334.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Electrophoresis of bubbles

  • Ory Schnitzer (a1), Itzchak Frankel (a2) and Ehud Yariv (a1)


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