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Electrophoresis in dilute polymer solutions

  • Gaojin Li (a1) and Donald L. Koch (a1)

Abstract

We analyse the electrophoresis of a weakly charged particle with a thin double layer in a dilute polymer solution. The particle velocity in polymer solutions modelled with different constitutive equations is calculated using a regular perturbation in the polymer concentration and the generalized reciprocal theorem. The analysis shows that the polymer is strongly stretched in two regions, the birefringent strand and the high-shear region inside the double layer. The electrophoretic velocity of the particle always decreases with the addition of polymers due to both increased viscosity and fluid elasticity. At a small Weissenberg number ( $Wi$ ), which is the product of the polymer relaxation time and the shear rate, the polymers inside the double layer contribute to most of the velocity reduction by increasing the fluid viscosity. With increasing $Wi$ , viscoelasticity decreases and shear thinning increases the particle velocity. Polymer elasticity alters the fluid velocity disturbance outside the double layer from that of a neutral squirmer to a puller-type squirmer. At high $Wi$ , the strong extensional stress inside the birefringent strand downstream of the particle dominates the velocity reduction. The scaling of the birefringent strand is used to estimate the particle velocity.

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Corresponding author

Email address for correspondence: dlk15@cornell.edu

References

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Afonso, A. M., Alves, M. A. & Pinho, F. T. 2009 Analytical solution of mixed electro-osmotic/pressure driven flows of viscoelastic fluids in microchannels. J. Non-Newtonian Fluid Mech. 159 (1–3), 5063.
Ardekani, A. M., Rangel, R. H. & Joseph, D. D. 2007 Motion of a sphere normal to a wall in a second-order fluid. J. Fluid Mech. 587, 163172.
Ardekani, A. M., Rangel, R. H. & Joseph, D. D. 2008 Two spheres in a free stream of a second-order fluid. Phys. Fluids 20 (6), 063101.
Arigo, M. T., Rajagopalan, D., Shapley, N. & McKinley, G. H. 1995 The sedimentation of a sphere through an elastic fluid. Part 1. Steady motion. J. Non-Newtonian Fluid Mech. 60 (2-3), 225257.
Barron, A. E., Soane, D. S. & Blanch, H. W. 1993 Capillary electrophoresis of DNA in uncross-linked polymer solutions. J. Chromatogr. A 652 (1), 316.
Becherer, P., van Saarloos, W. & Morozov, A. N. 2008 Scaling of singular structures in extensional flow of dilute polymer solutions. J. Non-Newtonian Fluid Mech. 153 (2-3), 183190.
Becherer, P., van Saarloos, W. & Morozov, A. N. 2009 Stress singularities and the formation of birefringent strands in stagnation flows of dilute polymer solutions. J. Non-Newtonian Fluid Mech. 157 (1), 126132.
Besra, L. & Liu, M. 2007 A review on fundamentals and applications of electrophoretic deposition (EPD). Prog. Mater. Sci. 52 (1), 161.
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics. Wiley.
Bird, R. B. & Wiest, J. M. 1995 Constitutive equations for polymeric liquids. Annu. Rev. Fluid Mech. 27 (1), 169193.
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46 (1), 199208.
Chang, H.-C. & Yeo, L. Y. 2010 Electrokinetically driven microfluidics and nanofluidics. Cambridge University Press.
Chilcott, M. D. & Rallison, J. M. 1988 Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newtonian Fluid Mech. 29, 381432.
Cox, R. G. 1965 The steady motion of a particle of arbitrary shape at small Reynolds numbers. J. Fluid Mech. 23 (4), 625643.
Datt, C. & Elfring, G. J. 2019 A note on higher-order perturbative corrections to squirming speed in weakly viscoelastic fluids. J. Non-Newtonian Fluid Mech. 270, 5155.
Datt, C., Natale, G., Hatzikiriakos, S. G. & Elfring, G. J. 2017 An active particle in a complex fluid. J. Fluid Mech. 823, 675688.
Debye, P. & Bueche, A. M. 1948 Intrinsic viscosity, diffusion, and sedimentation rate of polymers in solution. J. Chem. Phys. 16 (6), 573579.
Duke, T. & Viovy, J. L. 1994 Theory of DNA electrophoresis in physical gels and entangled polymer solutions. Phys. Rev. E 49 (3), 24082416.
Einarsson, J., Yang, M. & Shaqfeh, E. S. 2018 Einstein viscosity with fluid elasticity. Phys. Rev. Fluids 3 (1), 013301.
Elfring, G. J. 2017 Force moments of an active particle in a complex fluid. J. Fluid Mech. 829, R3.
Fabris, D., Muller, S. J. & Liepmann, D. 1999 Wake measurements for flow around a sphere in a viscoelastic fluid. Phys. Fluids 11 (12), 35993612.
Harlen, O. G. 1990 High-Deborah-number flow of a dilute polymer solution past a sphere falling along the axis of a cylindrical tube. J. Non-Newtonian Fluid Mech. 37 (2-3), 157173.
Harlen, O. G., Rallison, J. M. & Chilcott, M. D. 1990 High-Deborah-number flows of dilute polymer solutions. J. Non-Newtonian Fluid Mech. 34 (3), 319349.
Helgeson, M. E., Reichert, M. D., Hu, Y. T. & Wagner, N. J. 2009 Relating shear banding, structure, and phase behavior in wormlike micellar solutions. Soft Matt. 5 (20), 38583869.
Henry, D. C. 1931 The cataphoresis of suspended particles. Part I. The equation of cataphoresis. Proc. R. Soc. Lond. A 133, 106129.
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65 (2), 365400.
Ho, B. P. & Leal, L. G. 1976 Migration of rigid spheres in a two-dimensional unidirectional shear flow of a second-order fluid. J. Fluid Mech. 76 (4), 783799.
Howse, J. R., Jones, R. A., Ryan, A. J., Gough, T., Vafabakhsh, R. & Golestanian, R. 2007 Self-motile colloidal particles: from directed propulsion to random walk. Phys. Rev. Lett. 99 (4), 048102.
Hsu, J.-P., Hung, S.-H. & Yu, H.-Y. 2004 Electrophoresis of a sphere at an arbitrary position in a spherical cavity filled with Carreau fluid. J. Colloid Interface Sci. 280 (1), 256263.
Hsu, J.-P., Yeh, L.-H. & Ku, M.-H. 2006 Electrophoresis of a spherical particle along the axis of a cylindrical pore filled with a Carreau fluid. Colloid Polym. Sci. 284 (8), 886892.
Khair, A. S., Posluszny, D. E. & Walker, L. M. 2012 Coupling electrokinetics and rheology: electrophoresis in non-Newtonian fluids. Phys. Rev. E 85 (1), 016320.
Khair, A. S. & Squires, T. M. 2010 Active microrheology: a proposed technique to measure normal stress coefficients of complex fluids. Phys. Rev. Lett. 105 (15), 156001.
Koch, D. L., Lee, E. F. & Mustafa, I. 2016 Stress in a dilute suspension of spheres in a dilute polymer solution subject to simple shear flow at finite Deborah numbers. Phys. Rev. Fluids 1 (1), 013301.
Koch, D. L. & Subramanian, G. 2006 The stress in a dilute suspension of spheres suspended in a second-order fluid subject to a linear velocity field. J. Non-Newtonian Fluid Mech. 138 (2-3), 8797.
Kostal, V., Katzenmeyer, J. & Arriaga, E. A. 2008 Capillary electrophoresis in bioanalysis. Anal. Chem. 80 (12), 45334550.
Lauga, E. 2014 Locomotion in complex fluids: integral theorems. Phys. Fluids 26 (8), 081902.
Lauga, E. & Michelin, S. 2016 Stresslets induced by active swimmers. Phys. Rev. Lett. 117 (14), 148001.
Leal, L. G. 1979 The motion of small particles in non-Newtonian fluids. J. Non-Newtonian Fluid Mech. 5, 3378.
Lee, E., Chen, C.-T. & Hsu, J.-P. 2005 Electrophoresis of a rigid sphere in a Carreau fluid normal to a planar surface. J. Colloid Interface Sci. 285 (2), 857864.
Leslie, F. M. & Tanner, R. I. 1961 The slow flow of a visco-elastic liquid past a sphere. Q. J. Mech. Appl. Maths 14 (1), 3648.
Li, G., Archer, L. A. & Koch, D. L. 2019 Electroconvection in a viscoelastic electrolyte. Phys. Rev. Lett. 122 (12), 124501.
Li, G., Karimi, A. & Ardekani, A. M. 2014 Effect of solid boundaries on swimming dynamics of microorganisms in a viscoelastic fluid. Rheol. Acta 53 (12), 911926.
Lighthill, M. J. 1952 On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Maths 5 (2), 109118.
Mangelsdorf, C. S. & White, L. R. 1992 Electrophoretic mobility of a spherical colloidal particle in an oscillating electric field. J. Chem. Soc. Faraday Trans. 88 (24), 35673581.
Moore, M. N. J. & Shelley, M. J. 2012 A weak-coupling expansion for viscoelastic fluids applied to dynamic settling of a body. J. Non-Newtonian Fluid Mech. 183, 2536.
Morrison, F. A. 1970 Electrophoresis of a particle of arbitrary shape. J. Colloid Interface Sci. 34 (2), 210214.
Natale, G., Datt, C., Hatzikiriakos, S. G. & Elfring, G. J. 2017 Autophoretic locomotion in weakly viscoelastic fluids at finite Péclet number. Phys. Fluids 29 (12), 123102.
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. & Hunter, R. J. 1981 The electrophoretic mobility of large colloidal particles. Can. J. Chem. 59 (13), 18781887.
O’Brien, R. W. & White, L. R. 1978 Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc. Faraday Trans. 74, 16071626.
Ohshima, H. 2013 Electrokinetic phenomena of soft particles. Curr. Opin. Colloid Interface Sci. 18 (2), 7382.
Rallison, J. M. 2012 The stress in a dilute suspension of liquid spheres in a second-order fluid. J. Fluid Mech. 693, 500507.
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 (1), 321337.
Sawatzky, R. P. & Babchin, A. J. 1993 Hydrodynamics of electrophoretic motion in an alternating electric field. J. Fluid Mech. 246, 321334.
Schleiniger, G. & Weinacht, R. J. 1991 A remark on the Giesekus viscoelastic fluid. J. Rheol. 35 (6), 11571170.
Schnitzer, O. & Yariv, E. 2012a Macroscale description of electrokinetic flows at large zeta potentials: nonlinear surface conduction. Phys. Rev. E 86 (2), 021503.
Schnitzer, O. & Yariv, E. 2012b Strong-field electrophoresis. J. Fluid Mech. 701, 333351.
Schnitzer, O., Zeyde, R., Yavneh, I. & Yariv, E. 2013 Weakly nonlinear electrophoresis of a highly charged colloidal particle. Phys. Fluids 25 (5), 052004.
Sellier, A. 1999 Sur l’électrophorèse d’un ensemble de particules portant la même densité uniforme de charges. C.R. Acad. Sci. Paris IIB 327 (5), 443448.
Shoele, K. & Eastham, P. S. 2018 Effects of nonuniform viscosity on ciliary locomotion. Phys. Rev. Fluids 3 (4), 043101.
Smoluchowski, M. von 1903 Contribution to the theory of electro-osmosis and related phenomena. Bull. Inter. Acad. Sci. Cracovie 3, 184199.
Southern, E. M. 1975 Detection of specific sequences among DNA fragments separated by gel electrophoresis. J. Mol. Biol. 98 (3), 503517.
Vidybida, A. K. & Serikov, A. A. 1985 Electrophoresis by alternating fields in a non-Newtonian fluid. Phys. Lett. A 108 (3), 170172.
Wapperom, P. & Renardy, M. 2005 Numerical prediction of the boundary layers in the flow around a cylinder using a fixed velocity field. J. Non-Newtonian Fluid Mech. 125 (1), 3548.
Wei, S., Cheng, Z., Nath, P., Tikekar, M. D., Li, G. & Archer, L. A. 2018 Stabilizing electrochemical interfaces in viscoelastic liquid electrolytes. Sci. Adv. 4 (3), eaao6243.
Wiersema, P. H., Loeb, A. L. & Overbeek, J. T. G. 1966 Calculation of the electrophoretic mobility of a spherical colloid particle. J. Colloid Interface Sci. 22 (1), 7899.
Woolley, A. T. & Mathies, R. A. 1994 Ultra-high-speed DNA fragment separations using microfabricated capillary array electrophoresis chips. Proc. Natl Acad. Sci. USA 91 (24), 1134811352.
Yang, M. & Shaqfeh, E. S. 2018 Mechanism of shear thickening in suspensions of rigid spheres in Boger fluids. Part II. Suspensions at finite concentration. J. Rheol. 62 (6), 13791396.
Yariv, E. 2006 Force-free electrophoresis? Phys. Fluids 18 (3), 031702.
Zhao, C. & Yang, C. 2009 Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels. Appl. Math. Comput. 211 (2), 502509.
Zhao, C. & Yang, C. 2013 Electrokinetics of non-Newtonian fluids: a review. Adv. Colloid Interface Sci. 201, 94108.
Zhou, J. & Schmid, F. 2015 Computer simulations of single particles in external electric fields. Soft Matt. 11 (34), 67286739.
Zhu, L., Lauga, E. & Brandt, L. 2012 Self-propulsion in viscoelastic fluids: pushers versus pullers. Phys. Fluids 24 (5), 051902.
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Electrophoresis in dilute polymer solutions

  • Gaojin Li (a1) and Donald L. Koch (a1)

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