The Electrokinetic Cross-Coupling Coefficient: Two-Scale Homogenization Approach

ABSTRACT

By the two-scale homogenization approach we justify a two-scale model of ion transport through a layered membrane, with flows being driven by a pressure gradient and an external electrical field. By up-scaling, the electroosmotic flow equations in horizontal thin slits separated by thin solid layers are approximated by a homogenized system of macroscale equations in the form of the Poisson equation for induced vertical electrical field and Onsager's reciprocity relations between global fluxes (hydrodynamic and electric) and forces (horizontal pressure gradient and external electrical field). In addition, the two-scale approach provides macroscopic mobility coefficients in the Onsager relations. On this way, the cross-coupling kinetic coefficient is obtained in a form which does involves the ζ -potential among the data provided the surface current is negligible.

By the two-scale homogenization approach we justify a two-scale model of ion transport through a layered membrane, with flows being driven by a pressure gradient and an external electrical field. By up-scaling, the electroosmotic flow equations in horizontal thin slits separated by thin solid layers are approximated by a homogenized system of macroscale equations in the form of the Poisson equation for induced vertical electrical field and Onsager's reciprocity relations between global fluxes (hydrodynamic and electric) and forces (horizontal pressure gradient and external electrical field). In addition, the two-scale approach provides macroscopic mobility coefficients in the Onsager relations. On this way, the cross-coupling kinetic coefficient is obtained in a form which does involves the ζ -potential among the data provided the surface current is negligible.

Cite this paper

nullV. Shelukhin, I. Yeltsov and I. Paranichev, "The Electrokinetic Cross-Coupling Coefficient: Two-Scale Homogenization Approach,"*World Journal of Mechanics*, Vol. 1 No. 3, 2011, pp. 127-136. doi: 10.4236/wjm.2011.13018.

nullV. Shelukhin, I. Yeltsov and I. Paranichev, "The Electrokinetic Cross-Coupling Coefficient: Two-Scale Homogenization Approach,"

References

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[20] A. N. Chatterjee, D. M. Cannon, E. N. Gatimu, J. V. Sweedler, N. R. Aluru and P. W. Bohn, “Modelling and Simulation of Ionic Currents in Three Dimensional Microfluidic Devices with Nanofluidic Interconnects,” Journal of Nanopart. Research, Vol. 7, No. 4-5, 2005, pp. 507-516. doi:10.1007/s11051-005-5133-x

[21] Y. Amirat and V. Shelukhin, “Homogenization of Electroosmotic Flow Equations in Porous Media,” Journal of Mathematical Analysis and Applications, Vol. 342, No. 2, 2008, pp. 1227-1245. doi:10.1016/j.jmaa.2007.12.075

[22] Y. Amirat and V. Shelukhin, “Homogenization of the Poisson-Boltzmann Equation” In: A. V. Fursikov, G. P. Galdi and V. V. Pukhnachev Eds., Advances in Mathematical Fluid Mechanics, The Alexander V. Kazhikhov Memorial Volume, Birkh?user Verlag, Basel, 2010, pp. 29-41,

[23] V. V. Shelukhin and Y. Amirat, “On Electrolytes Flow in Porous Media,” Applied Mathematics and Technical Physics, Vol. 49, No. 4, 2008, 162-173.

[24] J. C. Maxwell, “Treatise on Electricity and Magnetism,” Clarendon Press, Oxford, 1881.

[25] S. R. de Groot and P. G. Mazur, “Non-Equilibrium Thermodynamics,” North-Holland, Amsterdam, 1962.

[26] J. Neev and F. R. Yeatts, “Electrokinetic Effects in Fluid Saturated Poroelastic Media,” Physical Review B, Vol. 40, No. 13, 1989, pp. 409135-409141. doi:10.1103/PhysRevB.40.9135

[27] V. V. Kormiltsev and V. V. Ratushnyak, “Theoretical and Experimental Basis for Spontaneous Polarization of Rocks in the Oil and Gas Wells,” Ekaterinburg, URO RAS (in Russian), 2007.

[28] P. Sennet and J. P. Oliver, “Colloidal Dispersion, Electrokinetic Effects, and the Concept of the Zeta Potential; Chemistry and Physics of Interfaces,” American Chemical Society, Washington, D.C., 1965.

[29] V. V. Shelukhin and S. A. Terentev, “Frequency Dispersion of Dielectric Permittivity and Electric Conductivity of Rocks via Two-Scale Homogenization of the Maxwell Equations,” Progress in Electromagnetic Research B, Vol. 14, 2009, pp. 175-202. doi:10.2528/PIERB09021804

[30] V. V. Shelukin and S. A. Terentev, “Homogenization of Maxwell Equations and the Maxwell-Wagner Dispersion,” Doklady Earth Sciences, Vol. 424, No. 3, 2009, pp. 155-159. doi:10.1134/S1028334X09010334

[31] E. D. Shchukin, A. V. Pertsev and E. A. Amelina, “Colloidal Chemistry,” Vysshaya Shkola, Moscow, 1992.

[32] B. Wurmstich and F. D. Morgan, “Modeling of Streaming Potential Responses Caused by Oil Pumping,” Geophysics, Vol. 59, No. 1, 1994, pp. 46-56. doi:10.1190/1.1443533

[1] A. Bolève, A. Crespy, A. Revil, F. Janod and J. I. Mattiuzzo, “Streaming Potentials of Granular Media: Influencies of the Dukhin and Reynolds Numbers,” Journal of Geophysical Research, Vol. 112, No. B8, 2007, pp. 1-14.

[2] A. Jardani, A. Revil, A. Bolève and J. P. Dupont, “The Three-Dimensional Inversion of Self-Potential Data Used to Constrain the Pattern of Groundwater Flow in Geothermal Fields,” Journal of Geophysical Research, Vol. 113, No. 89, 2008, pp. 1-22.

[3] G. de Marsily, “Quantative Hydrology,” Academic Press, London, 1986.

[4] B. J. Kirby and E. F. (Jr.) Hasselbrink, “Zeta Potential of Microfluidic Substrates. 1. Theory, Experimental Techni- ques and Effect on Separations,” Electrophoresis, Vol. 25, No. 2, 2004, pp. 187-202. doi:10.1002/elps.200305754

[5] Z. Zhu, M. N. Toks?z and X. Zhan, “Experimental Studies of Streaming Potential and High Frequency Seismoelectric Conversion,” Consortium Reports of MIT Earth Resources Laboratory, 2009.

[6] J. H. Saunders, M. D. Jackson and C. C. Pain, “A New Numerical Model of Electrokinetic Potential Response during Hydrocarbon Recovery,” Geophysical Reserach Letters, Vol. 33, No. 15, 2006, pp. L15316, 1-6.

[7] J. Lyklema, “Fundamentals of Colloid and Interface Science,” Academic Press, London, 1993.

[8] T. Ishido and H. Mizutani, “Experimental and Theoretical Basis for Electrikinetic Phenomena in Rock-Water Systems and Its Application to Geophysis,” Journal of Geophysical Research, Vol. 86, 1981, pp. 1763-1775. doi:10.1029/JB086iB03p01763

[9] D. B. Pengra, S. X. Li and P. Wong, “Determination of Rock Properties by Low Frequency AC Electrokinetics,” Journal of Geophysical Research, Vol. 104, No. B12, 1999, pp. 29485-29508. doi:10.1029/1999JB900277

[10] A. Revil, D. Hermitte, M. Voltz, R. Moussa, J.-G. Lacas, G. Bourrié and F. Troland, “Self-Potential Sygnals Associated with Variations of the Hydrolic Head during an Infiltration Experiment,” Geophysical Research Letters, Vol. 29, No. 7, 2002, pp. 10.1029/2001GLO14 294.

[11] R. Burridge and J. B. Keller, “Poroelastisity Equations Derived from Microstructure,” Journal of the Acoustical Society of America, Vol. 70, No. 4, 1981, pp. 1140-1146. doi:10.1121/1.386945

[12] E. Sanchez-Palencia, “Non-Homogeneous Media and Vibration Theory,” Springer, Heidelberg, 1980.

[13] S. R. Pride, A. F. Gangi and F. D. Morgan, “Deriving the Equations of Motion for Porous Isotropic Media,” Journal of the Acoustical Society of America, Vol. 92, No. 6, 1992, pp. 3278-3290. doi:10.1121/1.404178

[14] S. R. Pride and J. G. Berryman, “Linear Dymanics of Double-Porosity Dual-Permeability Materials i. Gover- ning Equations and Acoustic Attenuation,” Physical Review E, Vol. 68, No. 3, 2003, p. 036603. doi:10.1103/PhysRevE.68.036603

[15] S. R. Pride, “Governing Equations for the Coupled Electromagnetics and Acoustics of Porous Media,” Physical Review B, Vol. 50, No. 21, 1994, pp. 15678- 15696. doi:10.1103/PhysRevB.50.15678

[16] G. Allaire, “Homogenization and Two-Scale Conver- gence,” SIAM Journal on Mathematical Analysis, Vol. 23, No. 6, 1992, pp. 1482-1518. doi:10.1137/0523084

[17] A. Bensoussan, J. -L. Lions and G. Papanicolaou, “Asy- mptotic Analysis for Periodic Structures,” North-Holland, Amsterdam, 1978.

[18] L. Tartar, “An Introduction to Navier-Stokes Equation and Oceanography,” Series: Lecture Notes of the Unione Matematica Italiana, Vol. 1, 2006.

[19] J. G. Berryman, “Comparison of Two Up-Scaling Methods in Poroelasticity and Its Generalizations,” Proceedings of 17th ASCE Engineering Mechanics Conference, Newark, June 2004, pp. 13-16.

[20] A. N. Chatterjee, D. M. Cannon, E. N. Gatimu, J. V. Sweedler, N. R. Aluru and P. W. Bohn, “Modelling and Simulation of Ionic Currents in Three Dimensional Microfluidic Devices with Nanofluidic Interconnects,” Journal of Nanopart. Research, Vol. 7, No. 4-5, 2005, pp. 507-516. doi:10.1007/s11051-005-5133-x

[21] Y. Amirat and V. Shelukhin, “Homogenization of Electroosmotic Flow Equations in Porous Media,” Journal of Mathematical Analysis and Applications, Vol. 342, No. 2, 2008, pp. 1227-1245. doi:10.1016/j.jmaa.2007.12.075

[22] Y. Amirat and V. Shelukhin, “Homogenization of the Poisson-Boltzmann Equation” In: A. V. Fursikov, G. P. Galdi and V. V. Pukhnachev Eds., Advances in Mathematical Fluid Mechanics, The Alexander V. Kazhikhov Memorial Volume, Birkh?user Verlag, Basel, 2010, pp. 29-41,

[23] V. V. Shelukhin and Y. Amirat, “On Electrolytes Flow in Porous Media,” Applied Mathematics and Technical Physics, Vol. 49, No. 4, 2008, 162-173.

[24] J. C. Maxwell, “Treatise on Electricity and Magnetism,” Clarendon Press, Oxford, 1881.

[25] S. R. de Groot and P. G. Mazur, “Non-Equilibrium Thermodynamics,” North-Holland, Amsterdam, 1962.

[26] J. Neev and F. R. Yeatts, “Electrokinetic Effects in Fluid Saturated Poroelastic Media,” Physical Review B, Vol. 40, No. 13, 1989, pp. 409135-409141. doi:10.1103/PhysRevB.40.9135

[27] V. V. Kormiltsev and V. V. Ratushnyak, “Theoretical and Experimental Basis for Spontaneous Polarization of Rocks in the Oil and Gas Wells,” Ekaterinburg, URO RAS (in Russian), 2007.

[28] P. Sennet and J. P. Oliver, “Colloidal Dispersion, Electrokinetic Effects, and the Concept of the Zeta Potential; Chemistry and Physics of Interfaces,” American Chemical Society, Washington, D.C., 1965.

[29] V. V. Shelukhin and S. A. Terentev, “Frequency Dispersion of Dielectric Permittivity and Electric Conductivity of Rocks via Two-Scale Homogenization of the Maxwell Equations,” Progress in Electromagnetic Research B, Vol. 14, 2009, pp. 175-202. doi:10.2528/PIERB09021804

[30] V. V. Shelukin and S. A. Terentev, “Homogenization of Maxwell Equations and the Maxwell-Wagner Dispersion,” Doklady Earth Sciences, Vol. 424, No. 3, 2009, pp. 155-159. doi:10.1134/S1028334X09010334

[31] E. D. Shchukin, A. V. Pertsev and E. A. Amelina, “Colloidal Chemistry,” Vysshaya Shkola, Moscow, 1992.

[32] B. Wurmstich and F. D. Morgan, “Modeling of Streaming Potential Responses Caused by Oil Pumping,” Geophysics, Vol. 59, No. 1, 1994, pp. 46-56. doi:10.1190/1.1443533