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A Variational Model for Two-Phase Immiscible Electroosmotic Flow at Solid Surfaces

Published online by Cambridge University Press:  20 August 2015

Sihong Shao  and
Tiezheng Qian
Sihong Shao*
Affiliation:
Joint KAUST-HKUST Micro/Nanofluidics Laboratory, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong; LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
Tiezheng Qian*
Affiliation:
Department of Mathematics and Joint KAUST-HKUST Micro/Nanofluidics Laboratory, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
*
Email:shaosihong@gmail.com
Corresponding author.Email:maqian@ust.hk
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Abstract

We develop a continuum hydrodynamic model for two-phase immiscible flows that involve electroosmotic effect in an electrolyte and moving contact line at solid surfaces. The model is derived through a variational approach based on the On-sager principle of minimum energy dissipation. This approach was first presented in the derivation of a continuum hydrodynamic model for moving contact line in neutral two-phase immiscible flows (Qian, Wang, and Sheng, J. Fluid Mech. 564, 333-360 (2006)). Physically, the electroosmotic effect can be formulated by the Onsager principle as well in the linear response regime. Therefore, the same variational approach is applied here to the derivation of the continuum hydrodynamic model for charged two-phase immiscible flows where one fluid component is an electrolyte exhibiting electroosmotic effect on a charged surface. A phase field is employed to model the diffuse interface between two immiscible fluid components, one being the electrolyte and the other a nonconductive fluid, both allowed to slip at solid surfaces. Our model consists of the incompressible Navier-Stokes equation for momentum transport, the Nernst-Planck equation for ion transport, the Cahn-Hilliard phase-field equation for interface motion, and the Poisson equation for electric potential, along with all the necessary boundary conditions. In particular, all the dynamic boundary conditions at solid surfaces, including the generalized Navier boundary condition for slip, are derived together with the equations of motion in the bulk region. Numerical examples in two-dimensional space, which involve overlapped electric double layer fields, have been presented to demonstrate the validity and applicability of the model, and a few salient features of the two-phase immiscible electroosmotic flows at solid surface. The wall slip in the vicinity ofmoving contact line and the Smoluchowski slip in the electric double layer are both investigated.

Keywords

47.57.jd68.08.-p83.50.Rp83.10.FfElectroosmotic flowmoving contact lineslip boundary condition
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 3 , March 2012 , pp. 831 - 862
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Stone, H. A., Stroock, A. D., and Ajdari, A.Engineering flows in small devices: Microfluidics toward a Lab-on-a-Chip. Annu. Rev. Fluid Mech., 36: 381–411, 2004.Google Scholar
[2]Karniadakis, G., Beskok, A., and Aluru, N.Microflows and Nanoflows: Fundamentals and Simulation. Springer, New York, 2005.Google Scholar
[3]Erickson, D. and Li, D. Q.Microscale flow and transport simulation for electrokinetic and lab-on-chip applications. In Bashir, R. and Wereley, S., editors, Biomems and Biomedical Nanotechnology, Volume IV: Biomolecular Sensing, Processing and Analysis, pages 277–300. Springer, New York, 2006.CrossRefGoogle Scholar
[4]Goranović, G.Electrohydrodynamic aspects of two-fluid microfluidic systems: theory and simulation. PhD thesis, Technical University of Denmark, 2003.Google Scholar
[5]Ghosal, S.Electrokinetic flow and dispersion in capillary electrophoresis. Annu. Rev. Fluid Mech., 38:309–338, 2006.Google Scholar
[6]Neto, C., Evans, D. R., Bonaccurso, E., Butt, H. J., and Craig, V. S. J.Boundary slip in Newtonian liquids: A review of experimental studies. Rep. Prog. Phys., 68:2859–2897, 2005.Google Scholar
[7]Ajdari, A. and Bocquet, L.Giant amplification of interfacially driven transport by hydrody-namic slip: Diffusio-osmosis and beyond. Phys. Rev. Lett., 96:186102, 2006.Google Scholar
[8]Eijkel, J. C. T.Liquid slip in micro- and nanofluidics: Recent research and its possible implications. Lab Chip, 7:299–301, 2007.Google Scholar
[9]Lauga, E., Brenner, M. P., and Stone, H. A.Microfluidics: The no-slip boundary condition. In Tropea, C., Yarin, A., and Foss, J. F., editors, Handbook of Experimental Fluid Mechanics, pages 1219–1240. Springer, Berlin, 2007.Google Scholar
[10]Song, H., Chen, D. L., and Ismagilov, R. F.Reactions in droplets in microfluidic channels. Angew. Chem. Int. Ed., 45:7336–7356, 2006.Google Scholar
[11]Huebner, A., Sharma, S., Srisa-Art, M., Hollfelder, F., Edel, J. B., and deMello, A. J.Micro-droplets: A sea of applications? Lab Chip, 8:1244–1254, 2008.CrossRefGoogle Scholar
[12]Kuksenok, O., Jasnow, D., Yeomans, J., and Balazs, A. C.Periodic droplet formation in chemically patterned microchannels. Phys. Rev. Lett., 91:108303, 2003.Google Scholar
[13]Anna, S. L., Bontoux, N., and Stone, H. A.Formation of dispersions using “flow focusing” in microchannels. Appl. Phys. Lett., 82:364–366, 2003.Google Scholar
[14]Moffatt, H. K.Viscous and resistive eddies near a sharp corner. J. Fluid Mech., 18:1–18, 1964.CrossRefGoogle Scholar
[15]Huh, C. and Scriven, L. E.Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci., 35:85–101, 1971.CrossRefGoogle Scholar
[16]Dussan, E. B.V. On the spreading of liquids on solid surfaces: Static and dynamic contact lines. Annu. Rev. Fluid Mech., 11:371–400, 1979.CrossRefGoogle Scholar
[17]de Gennes, P. G.Wetting: Statics and dynamics. Rev. Mod. Phys., 57:827–863, 1985.Google Scholar
[18]Bonn, D., Eggers, J., Indekeu, J., Meunier, J., and Rolley, E.Wetting and spreading. Rev. Mod. Phys., 81:739–805, 2009.Google Scholar
[19]Qian, T. Z., Wang, X. P., and Sheng, P.Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E, 68:016306, 2003.CrossRefGoogle ScholarPubMed
[20]Qian, T. Z., Wang, X. P., and Sheng, P.Molecular hydrodynamics of the moving contact line in two-phase immiscible flows. Commun. Comput. Phys., 1:1–52, 2006.Google Scholar
[21]Qian, T. Z., Wang, X. P., and Sheng, P.A variational approach to moving contact line hydro-dynamics. J. Fluid Mech., 564:333–360, 2006.Google Scholar
[22]Onsager, L.Reciprocal relations in irreversible processes. I. Phys. Rev., 37: 405–426, 1931.CrossRefGoogle Scholar
[23]Onsager, L.Reciprocal relations in irreversible processes. II. Phys. Rev., 38: 2265–2279, 1931.Google Scholar
[24]Ren, W. and Boundary, W. E.conditions for moving contact line problem. Phys. Fluids, 19: 022101, 2007.Google Scholar
[25]Audry, M.-C., Piednoir, A., Joseph, P., and Charlaix, E.Amplification of electro-osmotic flows by wall slippage: Direct measurements on OTS-surfaces. Faraday Discuss., 146:113–124, 2010.Google Scholar
[26]Ren, Y. and Stein, D.Slip-enhanced electrokinetic energy conversion in nanofluidic channels. Nanotechnology, 19:195707, 2008.Google Scholar
[27]Mugele, F. and Baret, J. C.Electrowetting: From basics to applications. J. Phys.: Condens. Matter, 17:R705–R774, 2005.Google Scholar
[28]Eck, C., Fontelos, M., Grün, G., Klingbeil, F., and Vantzos, O.On a phase-field model for electrowetting. Interfaces Free Boundaries, 11:259–290, 2009.Google Scholar
[29]Saville, D. A.Electrohydrodynamics: The Taylor-Melcher leaky dielectric model. Annu. Rev. Fluid Mech., 29:27–64, 1997.Google Scholar
[30]Tomar, G., Gerlach, D., Biswas, G., Alleborn, N., Sharma, A., Durst, F., Welch, S. W. J., and Delgado, A.Two-phase electrohydrodynamic simulations using a volume-of-fluid approach. J. Comput. Phys., 227:1267–1285,2007.Google Scholar
[31]Mählmann, S. and Papageorgiou, D. T.Numerical study of electric field effects on the deformation of two-dimensional liquid drops in simple shear flow at arbitrary Reynolds number. J. Fluid Mech., 626:367–393, 2009.CrossRefGoogle Scholar
[32]Gao, Y. D., Wong, T. N., Yang, C., and Ooi, K. T.Two-fluid electroosmotic flow in microchannels. J. Colloid Interface Sci., 284:306–314, 2005.Google Scholar
[33]Grahame, D. C.The electrical double layer and the theory of electrocapillarity. Chem. Rev., 41:441–501, 1947.CrossRefGoogle ScholarPubMed
[34]Yang, C. and Li, D. Q.Electrokinetic effects on pressure-driven liquid flows in rectangular microchannels. J. Colloid Interface Sci., 194:95–107, 1997.Google Scholar
[35]Yang, R. J., Fu, L. M., and Lin, Y. C.Electroosmotic flow in microchannels. J. Colloid Interface Sci., 239:98–105, 2001.Google Scholar
[36]Stein, D., Kruithof, M., and Dekker, C.Surface-charge-governed ion transport in nanofluidic channels. Phys. Rev. Lett., 93:035901, 2004.Google Scholar
[37]van der Heyden, F. H. J., Stein, D., and Dekker, C.Streaming currents in a single nanofluidic channel. Phys. Rev. Lett., 95:116104, 2005.Google Scholar
[38]Qu, W. L. and Li, D. Q.A model for overlapped EDL fields. J. Colloid Interface Sci., 224:397–407, 2000.Google Scholar
[39]Kwak, H. S. and Hasselbrink, E. F. Jr.Timescales for relaxation to Boltzmann equilibrium in nanopores. J. Colloid Interface Sci., 284:753–758, 2005.Google Scholar
[40]Park, H. M., Lee, J. S., and Kim, T. W.Comparison of the Nernst-Planck model and the Poisson-Boltzmann model for electroosmotic flows in microchannels. J. Colloid Interface Sci., 315:731–739, 2007.Google Scholar
[41]Fu, L. M., Lin, J. Y., and Yang, R. J.Analysis of electroosmotic flow with step change in zeta potential. J. Colloid Interface Sci., 258:266–275, 2003.Google Scholar
[42]Hlushkou, D., Kandhai, D., and Tallarek, U.Coupled lattice-Boltzmann and finite-difference simulation of electroosmosis in microfluidic channels. Int. J. Numer. Mech. Fluids, 46:507–532, 2004.Google Scholar
[43]Daiguji, H., Yang, P. D., and Majumdar, A.Ion transport in nanofluidic channels. Nano Lett., 4:137–142, 2004.Google Scholar
[44]Daiguji, H., Yang, P. D., Szeri, A. J., and Majumdar, A.Electrochemomechanical energy conversion in nanofluidic channels. Nano Lett., 4:2315–2321, 2004.Google Scholar
[45]Daiguji, H., Adachi, T., and Tatsumi, N.Ion transport through a T-intersection of nanofluidic channels. Phys. Rev. E, 78:026301, 2008.Google Scholar
[46]Huang, K. D. and Yang, R. J.Electrokinetic behaviour of overlapped electric double layers in nanofluidic channels. Nanotechnology, 18:115701, 2007.Google Scholar
[47]Onuki, A.Ginzburg-Landau theory of solvation in polar fluids: Ion distribution around an interface. Phys. Rev. E, 73:021506, 2006.Google Scholar
[48]Kang, S. and Suh, Y. K.Numerical analysis on electroosmotic flows in a microchannel with rectangle-waved surface roughness using the Poisson-Nernst-Planck model. Microfluid. Nanofluid., 6:461–477, 2009.Google Scholar
[49]Kang, S. and Suh, Y. K.Electroosmotic flows in an electric double layer overlapped channel with rectangle-waved surface roughness. Microfluid. Nanofluid., 7:337–352, 2009.CrossRefGoogle Scholar
[50]Pennathur, S., Eijkel, J. C. T., and van den Berg, A.Energy conversion in microsystems: Is there a role for micro/nanofluidics? Lab Chip, 7:1234–1237, 2007.Google Scholar
[51]Joly, L., Ybert, C., Trizac, E., and Bocquet, L.Hydrodynamics within the electric double layer on slipping surfaces. Phys. Rev. Lett., 93:257805, 2004.Google Scholar
[52]Dufreˆche, J. F., Marry, V., Malíková, N., and Turq, P.Molecular hydrodynamics for electro-osmosis in clays: From Kubo to Smoluchowski. J. Mol. Liq., 118:145–153, 2005.Google Scholar
[53]Joly, L., Ybert, C., Trizac, E., and Bocquet, L.Liquid friction on charged surfaces: From hydro-dynamic slippage to electrokinetics. J. Chem. Phys., 125:204716, 2006.Google Scholar
[54]Anderson, D. M., McFadden, G. B., and Wheeler, A. A.Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech., 30:139–165, 1998.Google Scholar
[55]Jacqmin, D.Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech., 402:57–88, 2000.Google Scholar
[56]Yue, P., Zhou, C., and Feng, J. J.Sharp-interface limit of the Cahn-Hilliard model for moving contact lines. J. Fluid Mech., 645:279–294, 2010.Google Scholar
[57]Ryham, R., Liu, C., and Zikatanov, L.Mathematical models for the deformation of electrolyte droplets. Discrete Cont. Dyn. Syst.-B, 8:649–661, 2007.Google Scholar
[58]Johnston, H. and Liu, J. G.Finite difference schemes for incompressible flow based on local pressure boundary conditions. J. Comput. Phys., 180:120–154, 2002.Google Scholar
[59]Eyre, D. J.An unconditionally stable one-step scheme for gradient systems. preprint, 1997.Google Scholar
[60]Selberherr, S.Analysis and Simulation of Semiconductor Devices. Springer-Verlag, Wien-New York, 1984.Google Scholar
[61]Kim, H. C. and Burgess, D. J.Prediction of interfacial tension between oil mixtures and water. J. Colloid Interface Sci., 241:509–513, 2001.CrossRefGoogle Scholar
[62]Hyon, Y., Eisenberg, B. and Liu, C.A mathematical model for the hard sphere repulsion in ionic solutions. Commun. Math. Sci., 9(2):459–475, 2011.Google Scholar