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Computational fluid dynamics model of rhythmic motion of charged droplets between parallel electrodes

Published online by Cambridge University Press:  06 June 2017

Rudolf Flittner Open the ORCID record for Rudolf Flittner [Opens in a new window]  and
Michal Přibyl Open the ORCID record for Michal Přibyl [Opens in a new window]
Rudolf Flittner
Affiliation:
Department of Chemical Engineering, University of Chemistry and Technology, Prague, 166 28 Prague 6, Czech Republic
Michal Přibyl*
Affiliation:
Department of Chemical Engineering, University of Chemistry and Technology, Prague, 166 28 Prague 6, Czech Republic
*
†Email address for correspondence: [email protected]
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Abstract

A mathematical model of rhythmic motion of a charged droplet between two parallel electrodes is developed in this study. The work is motivated by recent experimental findings that report oscillatory behaviour of water in oil droplets under a direct current electric field. The model considers the presence of a charged droplet placed in a dielectric medium. The droplet is immediately attracted to the electrode with the opposite polarity. When approaching the electrode, the electric charge is electrochemically reversed within the droplet, which is then repelled to the other electrode. The entire process can periodically repeat. The model is able to track a deformable liquid–liquid interface, the dynamics of the wetting process at the electrodes and the dynamics of electrochemical charge transfer between the droplet and the electrodes. The dependences of the oscillation frequency, charge acquired by the droplet and charging time on several model parameters (surface charge density on electrodes, kinetic parameter of charging, droplet–electrode contact angle, droplet size, liquid permittivity) are examined. Qualitative agreement of the model predictions with available experimental data is obtained, e.g. the oscillation frequency increases with growing electric field strength or droplet size. Our model represents the first successful attempt to predict oscillatory motion of aqueous droplets by a pseudo-three-dimensional two-phase approach. Our model also strongly supports the theory that the oscillatory motion relies on the combination of electrochemical charge injection at the electrodes and electrostatic attraction/repulsion processes.

JFM classification

Drops and Bubbles: Drops MHD and Electrohydrodynamics: MHD and Electrohydrodynamics Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , pp. 31 - 53
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Copyright
© 2017 Cambridge University Press 

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References

Ahn, B., Lee, K., Louge, R. & Oh, K. W. 2009 Concurrent droplet charging and sorting by electrostatic actuation. Biomicrofluidics 3 (4), 044102.CrossRefGoogle ScholarPubMed
Ahn, B., Lee, K., Panchapakesan, R. & Oh, K. W. 2011 On-demand electrostatic droplet charging and sorting. Biomicrofluidics 5 (2), 024113.Google Scholar
Ahn, M. M., Im, D. J., Yoo, B. S. & Kang, I. S. 2015 Characterization of electrode alignment for optimal droplet charging and actuation in droplet-based microfluidic system. Electrophoresis 36 (17), 2086–2093.Google Scholar
Amestoy, P. R., Duff, I. S., Koster, J. & L’Excellent, J.-Y. 2001 A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Applics 23 (1), 15–41.Google Scholar
Arndt, D., Thoming, J. & Baumer, M. 2013 Improving the quality of nanoparticle production by using a new biphasic synthesis in a slug flow microreactor. Chem. Engng J. 228, 1083–1091.Google Scholar
Bard, A. J. & Faulkner, L. R. 2001 Electrochemical Methods: Fundamentals and Applications. Wiley.Google Scholar
Bartlett, C. T., Genero, G. A. & Bird, J. C. 2015 Coalescence and break-up of nearly inviscid conical droplets. J. Fluid Mech. 763, 369–385.Google Scholar
Bazant, M. Z. 2015 Electrokinetics meets electrohydrodynamics. J. Fluid Mech. 782, 1–4.Google Scholar
Beranek, P., Flittner, R., Hrobar, V., Ethgen, P. & Pribyl, M. 2014 Oscillatory motion of water droplets in kerosene above co-planar electrodes in microfluidic chips. AIP Adv. 4 (6), 067103.CrossRefGoogle Scholar
Berry, J. D., Davidson, M. R. & Harvie, D. J. E. 2013 A multiphase electrokinetic flow model for electrolytes with liquid/liquid interfaces. J. Comput. Phys. 251, 209–222.Google Scholar
Cahill, B. P. 2014 Introduction. In Micro-Segmented Flow, Applications in Chemistry and Biology (ed. Köhler, M. J. & Cahill, B. P.), pp. 1–6. Springer.Google Scholar
Cao, J. & Köhler, M. 2015 Droplet-based microfluidics for microtoxicological studies. Engng Life Sci. 15 (3), 306–317.Google Scholar
Cartier, C. A., Drews, A. M. & Bishop, K. J. M. 2014 Microfluidic mixing of nonpolar liquids by contact charge electrophoresis. Lab on a Chip 14, 4230–4236.Google Scholar
Cech, J., Pribyl, M. & Snita, D. 2013 Three-phase slug flow in microchips can provide beneficial reaction conditions for enzyme liquid–liquid reactions. Biomicrofluidics 7 (5), 054103.Google Scholar
Cech, J., Schrott, W., Slouka, Z., Pribyl, M., Broz, M., Kuncova, G. & Snita, D. 2012 Enzyme hydrolysis of soybean oil in a slug flow microsystem. Biochem. Engng J. 67, 194–202.CrossRefGoogle Scholar
Cervenka, P., Hrdlicka, J., Pribyl, M. & Snita, D. 2012 Kinetic mechanism for modeling of electrochemical reactions. Phys. Rev. E 85 (4,1), 041505.Google Scholar
Collins, R. T., Sambath, K., Harris, M. T. & Basaran, O. A. 2013 Universal scaling laws for the disintegration of electrified drops. Proc. Natl Acad. Sci. USA 110 (13), 4905–4910.CrossRefGoogle ScholarPubMed
Comsol2013a Comsol Multiphysics, Reference manual, version 4.4.Google Scholar
Comsol2013b Microfluidics module, User’s guide, version 4.4.Google Scholar
Dessimoz, A. L., Cavin, L., Renken, A. & Kiwi-Minsker, L. 2008 Liquid–liquid two-phase flow patterns and mass transfer characteristics in rectangular glass microreactors. Chem. Engng Sci. 63 (16), 4035–4044.CrossRefGoogle Scholar
Ghaini, A., Kashid, M. N. & Agar, D. W. 2010 Effective interfacial area for mass transfer in the liquid–liquid slug flow capillary microreactors. Chem. Engng Process. 49 (4), 358–366.CrossRefGoogle Scholar
Grodzinsky, A. J. 2011 Fields, Forces, and Flows in Biological Systems. Garland Science.Google Scholar
Gunther, A. & Jensen, K. F. 2006 Multiphase microfluidics: from flow characteristics to chemical and materials synthesis. Lab on a Chip 6 (12), 1487–1503.Google Scholar
Hase, M., Watanabe, S. N. & Yoshikawa, K. 2006 Rhythmic motion of a droplet under a dc electric field. Phys. Rev. E 74 (4,2), 046301.Google Scholar
Hindmarsh, A. C., Brown, P. N., Grant, K. E., Lee, S. L., Serban, R., Shumaker, D. E. & Woodward, C. S. 2005 SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31 (3), 363–396.Google Scholar
Hrdlicka, J., Cervenka, P., Jindra, T., Pribyl, M. & Snita, D. 2013 Mathematical modeling of traveling wave micropumps: analysis of energy transformation. IEEE Trans. Ind. Applics 49 (2), 685–690; Annual Meeting of the IEEE Industry Applications Society (IAS), Orlando, FL, October 09–13, 2011.CrossRefGoogle Scholar
Hrdlicka, J., Cervenka, P., Pribyl, M. & Snita, D. 2010 Mathematical modeling of AC electroosmosis in microfluidic and nanofluidic chips using equilibrium and non-equilibrium approaches. J. Appl. Electrochem. 40 (5,SI), 967–980.CrossRefGoogle Scholar
Im, D. J., Ahn, M. M., Yoo, B. S., Moon, D., Lee, D. W. & Kang, I. S. 2012 Discrete electrostatic charge transfer by the electrophoresis of a charged droplet in a dielectric liquid. Langmuir 28 (32), 11656–11661.Google Scholar
Im, D. J., Noh, J., Moon, D. & Kang, I. S. 2011 Electrophoresis of a charged droplet in a dielectric liquid for droplet actuation. Analyt. Chem. 83 (13), 5168–5174.Google Scholar
Iverson, B. D. & Garimella, S. V. 2008 Recent advances in microscale pumping technologies: a review and evaluation. Microfluid. Nanofluid. 5 (2), 145–174.Google Scholar
Jalaal, M., Khorshidi, B. & Esmaeilzadeh, E. 2010 An experimental study on the motion, deformation and electrical charging of water drops falling in oil in the presence of high voltage DC electric field. Exp. Therm. Fluid Sci. 34 (8), 1498–1506.Google Scholar
Jovanovic, J., Rebrov, E. V., Nijhuis, T. A., Hessel, V. & Schouten, J. C. 2010 Phase-transfer catalysis in segmented flow in a microchannel: fluidic control of selectivity and productivity. Ind. Engng Chem. Res. 49 (6), 2681–2687.Google Scholar
Jovanovic, J., Zhou, W., Rebrov, E. V., Nijhuis, T. A., Hessel, V. & Schouten, J. C. 2011 Liquid–liquid slug flow: hydrodynamics and pressure drop. Chem. Engng Sci. 66 (1), 42–54.Google Scholar
Jung, Y. M., Oh, H. C. & Kang, I. S. 2008 Electrical charging of a conducting water droplet in a dielectric fluid on the electrode surface. J. Colloid Interface Sci. 322 (2), 617–623.Google Scholar
Kashid, M. N. & Agar, D. W. 2007 Hydrodynamics of liquid–liquid slug flow capillary microreactor: flow regimes, slug size and pressure drop. Chem. Engng J. 131 (13), 1–13.CrossRefGoogle Scholar
Kashid, M. N., Gerlach, I., Goetz, S., Franzke, J., Acker, J. F., Platte, F., Agar, D. W. & Turek, S. 2005 Internal circulation within the liquid slugs of a liquid–liquid slug-flow capillary microreactor. Ind. Engng Chem. Res. 44 (14), 5003–5010; arXiv:org/10.1021/ie0490536.Google Scholar
Kurimura, T. & Ichikawa, M. 2016 Noise-supported actuator: coherent resonance in the oscillations of a micrometer-sized object under a direct current-voltage. Appl. Phys. Lett. 108 (14), 144101.Google Scholar
Kurimura, T., Ichikawa, M., Takinoue, M. & Yoshikawa, K. 2013 Back-and-forth micromotion of aqueous droplets in a dc electric field. Phys. Rev. E 88 (4), 042918.CrossRefGoogle Scholar
Ladosz, A., Rigger, E. & von Rohr, P. R. 2016 Pressure drop of three-phase liquid–liquid–gas slug flow in round microchannels. Microfluid. Nanofluid. 20 (3), 1–14.Google Scholar
Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S. & Zanetti, G. 1994 Modelling merging and fragmentation in multiphase flows with surfer. J. Comput. Phys. 113 (1), 134–147.Google Scholar
Li, Y., Yamane, D. G., Li, S., Biswas, S., Reddy, R. K., Goettert, J. S., Nandakumar, K. & Kumar, C. S. S. R. 2013 Geometric optimization of liquid–liquid slug flow in a flow-focusing millifluidic device for synthesis of nanomaterials. Chem. Engng J. 217, 447–459.Google Scholar
Lin, Yuan, Skjetne, Paal & Carlson, Andreas 2012 A phase field model for multiphase electro-hydrodynamic flow. Intl J. Multiphase Flow 45, 1–11.CrossRefGoogle Scholar
Link, D. R., Grasland-Mongrain, E., Duri, A., Sarrazin, F., Cheng, Z. D., Cristobal, G., Marquez, M. & Weitz, D. A. 2006 Electric control of droplets in microfluidic devices. Angew. Chem. Intl Ed. Engl. 45 (16), 2556–2560.CrossRefGoogle ScholarPubMed
Martien, P., Pope, S. C., Scott, P. L. & Shaw, R. S. 1985 The chaotic behavior of the leaky faucet. Phys. Lett. A 110 (7–8), 399–404.Google Scholar
Meyer, C., Hoffmann, M. & Schluter, M. 2014 Micro-PIV analysis of gas–liquid Taylor flow in a vertical oriented square shaped fluidic channel. Intl J. Multiphase Flow 67, 140–148.Google Scholar
Mhatre, S. & Thaokar, E. M. 2013 Drop motion, deformation, and cyclic motion in a non-uniform electric field in the viscous limit. Phys. Fluids 25 (7), 072105.Google Scholar
Miller, C. A. & Scriven, L. E. 1968 Oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32 (3), 417–435.Google Scholar
Olsson, E. & Kreiss, G. 2005 A conservative level set method for two phase flow. J. Comput. Phys. 210 (1), 225–246.CrossRefGoogle Scholar
Parlitz, U. & Lauterborn, W. 1987 Period-doubling cascades and devil’s staircases of the driven van der Pol oscillator. Phys. Rev. A 36 (3), 1428–1434.Google Scholar
Pillai, R., Berry, J. D., Harvie, D. J. E. & Davidson, M. R. 2016 Electrokinetics of isolated electrified drops. Soft Matt. 12 (14), 3310–3325.Google Scholar
van der Pol, B. 1926 LXXXVIII. On relaxation-oscillations. Lond. Edin. Dublin Phil. Mag. J. Sci. 2 (11), 978–992.Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor-Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29 (1), 27–64.Google Scholar
Schnitzer, O. & Yariv, E. 2015 The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description. J. Fluid Mech. 773, 1–33.Google Scholar
Slouka, Z., Pribyl, M., Snita, D. & Postler, T. 2007 Transient behavior of an electrolytic diode. Phys. Chem. Chem. Phys. 9 (39), 5374–5381.Google Scholar
Testino, A., Pilger, F., Lucchini, M. A., Quinsaat, J. E. Q., Staehli, C. & Bowen, P. 2015 Continuous polyol synthesis of metal and metal oxide nanoparticles using a segmented flow tubular reactor (SFTR). Molecules 20 (6), 10566–10581.Google Scholar
Wallau, W., Schlawitschek, C. & Arellano-Garcia, H. 2016 Electric field driven separation of oil-water mixtures: model development and experimental verification. Ind. Engng Chem. Res. 55 (16), 4585–4598.Google Scholar
Wang, X., Cheng, C., Wang, S. & Liu, S. 2009 Electroosmotic pumps and their applications in microfluidic systems. Microfluid. Nanofluid. 6 (2), 145–162.CrossRefGoogle ScholarPubMed
Wehking, J. D. & Kumar, R. 2015 Droplet actuation in an electrified microfluidic network. Lab on a Chip 15, 793–801.Google Scholar
Zeng, J. & Korsmeyer, T. 2004 Principles of droplet electrohydrodynamics for lab-on-a-chip. Lab on a Chip 4, 265–277.Google Scholar
Zhao, C. X. 2013 Multiphase flow microfluidics for the production of single or multiple emulsions for drug delivery. Adv. Drug Deliv. Rev. 65 (11–12), 1420–1446.Google Scholar

Flittner and Přibyl supplementary movie

Aqueous droplet dispersed in a dielectric fluid oscillates between electrodes under DC electric field.

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