Skip to main content Accessibility help
×
Home
Hostname: page-component-5c569c448b-dnb4q Total loading time: 0.241 Render date: 2022-07-01T22:49:18.526Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Electrophoresis of bubbles

Published online by Cambridge University Press:  16 July 2014

Ory Schnitzer
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
Itzchak Frankel
Affiliation:
Department of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
Ehud Yariv*
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
*
Email address for correspondence: udi@technion.ac.il

Abstract

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.

Type
Papers
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
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..Google Scholar
Baygents, J. C. & Saville, D. A. 1991 Electrophoresis of drops and bubbles. J. Chem. Soc. Faraday Trans. 87 (12), 18831898.CrossRefGoogle Scholar
Booth, F. 1951 The cataphoresis of spherical fluid droplets in electrolytes. J. Chem. Phys. 19, 13311336.CrossRefGoogle Scholar
Brandon, N. P., Kelsall, G. H., Levine, S. & Smith, A. L. 1985 Interfacial electrical properties of electrogenerated bubbles. J. Appl. Electrochem. 15 (4), 485493.CrossRefGoogle Scholar
Chang, H.-C. & Yeo, L. Y. 2010 Electrokinetically Driven Microfluidics and Nanofluidics. Cambridge University Press.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
Harper, J. F. 2010 Electrophoresis of surfactant-free bubbles. J. Colloid Interface Sci. 350 (1), 361367.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Hunter, R. J. 2000 Foundations of Colloidal Science. Oxford University Press.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Khair, A. S. 2013 Diffusiophoresis of colloidal particles in neutral solute gradients at finite Péclet number. J. Fluid Mech. 731, 6494.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.Google Scholar
Liu, H. & Dasgupta, P. K. 1997 A falling drop for sample injection in capillary zone electrophoresis. Analyt. Chem. 69 (6), 12111216.CrossRefGoogle Scholar
Lyklema, J. 1995 Fundamentals of Interface and Colloid Science, vol. II. Academic Press.Google Scholar
McTaggart, H. A. 1914 The electrification at liquid–gas surfaces. Phil. Mag. 27 (158), 297314.CrossRefGoogle Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111146.CrossRefGoogle Scholar
Morrison, F. A. 1970 Electrophoresis of a particle of arbitrary shape. J. Colloid Interface Sci. 34, 210214.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
O’Brien, R. W. & White, L. R. 1978 Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc. Faraday Trans. 74, 16071626.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Quincke, G. 1861 Ueber die fortfiihrüng Materieller theilchen durch strömende Elektricität. Ann. Phys. Chem. 115, 513598.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Russel, W. B., Saville, D. A. & Schowalter, W. R. 1989 Colloidal Dispersions. Cambridge University Press.CrossRefGoogle Scholar
Saville, D. A. 1977 Electrokinetic effects with small particles. Annu. Rev. Fluid Mech. 9, 321337.CrossRefGoogle Scholar
Schnitzer, O., Frankel, I. & Yariv, E. 2013 Electrokinetic flows about conducting drops. J. Fluid Mech. 722, 394423.CrossRefGoogle Scholar
Schnitzer, O. & Yariv, E. 2012a Macroscale description of electrokinetic flows at large zeta potentials: nonlinear surface conduction. Phys. Rev. E 86, 021503.CrossRefGoogle Scholar
Schnitzer, O. & Yariv, E. 2012b Strong-field electrophoresis. J. Fluid Mech. 701, 333351.CrossRefGoogle Scholar
Teh, S. Y., Lin, R., Hung, L. H. & Lee, A. P. 2008 Droplet microfluidics. Lab on a Chip 8 (2), 198220.CrossRefGoogle ScholarPubMed
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.Google Scholar
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.CrossRefGoogle Scholar
Yariv, E. 2006 ‘Force-free’ electrophoresis? Phys. Fluids 18, 031702.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
24
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Electrophoresis of bubbles
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Electrophoresis of bubbles
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Electrophoresis of bubbles
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *