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Dynamic electric-field-induced response of charged spherical colloids in uncharged hydrogels

Published online by Cambridge University Press:  02 November 2009

MU WANG  and
REGHAN J. HILL
MU WANG
Affiliation:
Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 2B2, Canada
REGHAN J. HILL*
Affiliation:
Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 2B2, Canada McGill Institute for Advanced Materials, McGill University, Montreal, Quebec H3A 2B2, Canada
*
†Email address for correspondence: reghan.hill@mcgill.ca
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Abstract

Embedding colloidal particles in polymeric hydrogels often endows the polymer skeleton with appealing characteristics for microfluidics and biosensing applications. This theoretical study provides a rigorous foundation for interpreting active electrical microrheology and electroacoustic experiments on such materials. In addition to viscoelastic properties of the composites, these techniques sense physicochemical characteristics of the particle–polymer interface. Wang & Hill (Soft Matter, vol. 4, 2008, p. 1048) studied the steady response of a rigid, impenetrable sphere in a compressible hydrogel skeleton. Here, we extend their analysis to arbitrary frequencies, showing, in general, how the frequency response depends on the particle size and charge, ionic strength of the electrolyte and elastic and hydrodynamic characteristics of the polymer skeleton. Our calculations capture the transition from quasi-steady compressible to quasi-steady incompressible dynamics as the frequency passes through the reciprocal draining time of the gel. Above the reciprocal draining time, the skeleton and fluid move in unison, so the dynamics are incompressible and, thus, given to an excellent approximation by the well-known dynamic electrophoretic mobility but with the Newtonian shear viscosity replaced by a complex, frequency-dependent value.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 357 - 400
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Ahualli, S., Delgado, A. V., Miklavcic, S. J. & White, L. R. 2006 Dynamic electrophoretic mobility of concentrated dispersions of spherical colloidal particle. On the consistent use of the cell model. Langmuir 22, 7041–7051.CrossRefGoogle ScholarPubMed
Allersma, M. W., Gittes, F., de Castro, M. J., Stewart, R. J. & Schmidt, C. F. 1998 Two-dimensional tracking of ncd motility by back focal plane interferometry. Biophys. J. 74, 1074–1085.CrossRefGoogle ScholarPubMed
Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A. & Sorensen, D. 1999 LAPACK Users' Guide, 3rd edn. Society for Industrial and Applied Mathematics.Google Scholar
Ascher, U. M., Mattheij, R. M. M. & Russell, R. D. 1988 Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice Hall.Google Scholar
Barber, J. R. 2003 Elasticity, 2nd edn. Springer.Google Scholar
Barndl, F., Sommer, F. & Goepferich, A. 2007 Rational design of hydrogels for tissue engineering: impact of physical factors on cell behaviour. Biomaterials 28 (2), 134–146.CrossRefGoogle Scholar
Berg-Sørensen, K. & Flyvbjerg, H. 2004 Power spectrum analysis for optical tweezers. Rev. Sci. Inst. 75, 594–612.CrossRefGoogle Scholar
Breuer, K. (ed.) 2005 Microscale Diagnostic Techniques. Springer.Google Scholar
Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34.CrossRefGoogle Scholar
Cash, J. R. & Mazzia, F. 2006 Hybrid mesh selection algorithms based on conditioning for two-point boundary value problems. J. Numer. Anal. Indust. Appl. Math. 1, 81–90.Google Scholar
Chaterji, S., Kwon, I. K. & Park, K. 2007 Smart polymeric gels: redefining the limits of biomedical devices. Prog. Polym. Sci. 32, 1083–1122.CrossRefGoogle ScholarPubMed
Chung, Y.-I., Ahn, K.-M., Jeon, S.-H., Lee, S.-Y., Lee, J.-H. & Tae, G. 2007 Enhanced bone regeneration with BMP-2 loaded function nanoparticle–hydrogel complex. J. Control. Release 121, 91–99.CrossRefGoogle ScholarPubMed
Cicuta, P. & Donald, A. M. 2007 Microrheology: a review of the method and applications. Soft Matt. 3, 1449–1455.CrossRefGoogle ScholarPubMed
Crocker, J. C., Valentine, M. T. & Weeks, E. R. 2000 Two-point microrheology of inhomogeneous soft materials. Phys. Rev. Lett. 85 (4), 888–891.CrossRefGoogle ScholarPubMed
Dayton, P. A. & Ferrara, K. W. 2002 Targeted imaging using ultrasound. J. Magnet. Reson. Imag. 16, 362–377.Google Scholar
DeLacey, E. H. B. & White, L. R. 1981 Dielectric response and conductivity of dilute suspensions of colloidal particles. J. Chem. Soc., Faraday Trans. 77, 2007–2039.CrossRefGoogle Scholar
Drury, J. L. & Mooney, D. J. 2003 Hydrogels for tissue engineering: scaffold design variables and applications. Biomaterials 24, 4337–4351.CrossRefGoogle ScholarPubMed
Eddington, D. T. & Beebe, D. J. 2004 Flow control with hydrogels. Adv. Drug Deliv. Rev. 56, 199–210.CrossRefGoogle ScholarPubMed
Enge, A., Pélissier, P. & Zimmermann, P. 2007 MPC: Multiple Precision Complex Library. INRIA.Google Scholar
Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P. & Zimmermann, P. 2007 MPFR: A multiple-precision binary floating-point library with correct rounding. ACM T. Math. Software 33, 13.Google Scholar
Galneder, R., Kahl, V., Arbuzova, A., Rebecchi, M., Radler, J. O. & McLaughlin, S. 2001 Microelectrophoresis of a bilayer-coated silica bead in an optical trap: application to enzymology. Biophys. J. 80, 2298–2309.CrossRefGoogle Scholar
Geissler, E. & Hecht, A. M. 1980 The Poisson ratio in polymer gels. Macromolecules 13, 1276–1280.CrossRefGoogle Scholar
Geissler, E. & Hecht, A. M. 1981 The Poisson ratio in polymer gel. 2. Macromolecules 14, 185–188.Google Scholar
Gibb, S. E. & Hunter, R. J. 2000 Dynamic mobility of colloidal particles with thick double layers. J. Colloid Interface Sci. 224, 99–111.CrossRefGoogle ScholarPubMed
Granlund, T. 2007 GNU MP: The GNU Multiple Precision Arithmetic Library, 4th edn.Google Scholar
Hill, R. J. 2006 a Electric-field-induced force on a charged spherical colloid embedded in an electrolyte-saturated Brinkman medium. Phys. Fluids 18, 043103.CrossRefGoogle Scholar
Hill, R. J. 2006 b Transport in polymer-gel composites: theoretical methodology and response to an electric field. J. Fluid Mech. 551, 405–433.Google Scholar
Hill, R. J. 2007 Electric-field-enhanced transport in polyacrylamide hydrogel nanocomposites. J. Colloid Interface Sci. 316, 635–644.CrossRefGoogle ScholarPubMed
Hill, R. J. & Ostoja-Starzewski, M. 2008 Electric-field-induced displacement of a charged spherical colloid embedded in an elastic Brinkman medium. Phys. Rev. E 77, 011404.CrossRefGoogle Scholar
Hill, R. J., Saville, D. A. & Russel, W. B. 2003 a Electrophoresis of spherical polymer-coated colloidal particles. J. Colloid Interface Sci. 258, 56–74.Google Scholar
Hill, R. J., Saville, D. A. & Russel, W. B. 2003 b High-frequency dielectric relaxation of spherical colloidal particles. Phys. Chem. Chem. Phys. 5, 911–915.CrossRefGoogle Scholar
Hunter, R. J. 1998 Recent developments in the electroacoustic characterization of colloidal suspensions and emulsions. Colloids Surf. A 141, 37–65.CrossRefGoogle Scholar
Hunter, R. J. 2001 Foundations of Colloid Science, 2nd edn. Oxford University Press.Google Scholar
Hunter, R. J. & O'Brien, R. W. 1997 Electroacoustic characterization of colloids with unusual particle properties. Colloids Surf. A 126, 123–128.CrossRefGoogle Scholar
Khademhosseini, A. & Langer, R. 2007 Microengineered hydrogels for tissue engineering. Biomaterials 28, 5087–5092.Google Scholar
Kim, J. J. & Park, K. 1998 Smart hydrogels for bioseparation. Bioseparation 7, 177–184.CrossRefGoogle ScholarPubMed
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.Google Scholar
Kimura, Y. & Mizuno, D. 2007 Microrheology of a swollen lyotropic lamellar phase. Mol. Cryst. Liq. Cryst. 478, 3–13.CrossRefGoogle Scholar
Kizilay, M. Y. & Okay, O. 2003 Effect of hydrolysis on spatial inhomogeneity in poly(acrylamide) gels of various crosslink densities. Polymer 44, 5239–5250.CrossRefGoogle Scholar
Lamb, H. 1945 Hydrodynamics, 6th edn. DoverGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Larsen, T. H. & Furst, E. M. 2008 Microrheology of the liquid–solid transition during gelation. Phys. Rev. Lett. 100 (14), 146001.CrossRefGoogle ScholarPubMed
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.Google Scholar
Levine, A. J. & Lubensky, T. C. 2000 One- and two-particle microrheology. Phys. Rev. Lett. 85, 1774–1777.CrossRefGoogle ScholarPubMed
Levine, A. J. & Lubensky, T. C. 2001 Response function of a sphere in a viscoelastic two-fluid medium. Phys. Rev. E 63, 041510.Google Scholar
Lin, C.-C. & Netters, A. T. 2006 Hydrogels in controlled release formulations: network design and mathematical modelling. Adv. Drug Deliv. Rev. 58, 1379–1408.Google Scholar
Liu, J., Levine, A. L., Mattoon, J. S., Yamaguchi, M., Lee, R. J., Pan, X. L. & Rosol, T. J. 2006 Nanoparticles as image enhancing agents for ultrasonography. Phys. Med. Biol. 51, 2179–2189.Google Scholar
Loo, C., Lowery, A., Halas, N., West, J. & Drezek, R. 2005 Immunotargeted nanoshells for integrated cancer imaging and therapy. Nano Lett. 5, 709–711.Google Scholar
Lyklema, J. 1995 Fundamentals of Interface and Colloid Science. II. Solid–Liquid Interfaces. Academic.Google Scholar
MacKintosh, F. C. & Schmidt, C. F. 1999 Microrheology. Curr. Opin. Colloid Interface Sci. 4, 300–307.CrossRefGoogle Scholar
MacRobert, T. M. 1967 Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications, 3rd edn. Pergamon.Google Scholar
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, 3567–3581.CrossRefGoogle Scholar
Markov, M. G. 2005 Propagation of longitudinal elastic waves in a fluid-saturated porous medium with spherical inclusions. Acoust. Phys. 51, S115–S121.Google Scholar
Masliyah, J. H. & Bhattacharjee, S. 2006 Electrokinetic and Colloid Transport Phenomena. Wiley Interscience.CrossRefGoogle Scholar
Mason, T. & Weitz, D. 1995 Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids. Phys. Rev. Lett. 74 (7), 1250–1253.CrossRefGoogle ScholarPubMed
Matos, M., White, L. R. & Tilton, R. D. 2008 Enhanced mixing in polyacrylamide gels containing embedded silica nanoparticles as internal electro-osmotic pumps. Colloids Surf. B 61 (2), 262–269.CrossRefGoogle Scholar
Matos, M. A., White, L. R. & Tilton, R. D. 2006 Electroosmotically enhanced mass transfer through polyacrylamide gels. J. Colloid Interface Sci. 300, 429–436.CrossRefGoogle ScholarPubMed
Mizuno, D., Head, D. A., MacKintosh, F. C. & Schmidt, C. F. 2008 Active and passive microrheology in equilibrium and nonequilibrium systems. Macromolecules 41 (19), 7194–7202.CrossRefGoogle Scholar
Mizuno, D., Kimura, Y. & Hayakawa, R. 2000 Dynamic electrophoretic mobility of colloidal particles measured by the newly developed method of quasi-elastic light scattering in a sinusoidal electric field. Langmuir 16, 9547–9554.CrossRefGoogle Scholar
Mizuno, D., Kimura, Y. & Hayakawa, R. 2001 Electrophoretic microrheology in a dilute lamellar phase of a nonionic surfactant. Phys. Rev. Lett. 87, 088104.CrossRefGoogle Scholar
Mizuno, D., Kimura, Y. & Hayakawa, R. 2004 Electrophoretic microrheology of a dilute lamellar phase: relaxation mechanisms in frequency-dependent mobility of nanometre-sized particles between soft membranes. Phys. Rev. E 70, 011509.Google Scholar
Mohammadi, A. & Hill, R. J. 2009 Steady electrical and microrheological response functions for uncharged colloidal inclusions in polyelectrolyte hydrogels. Proc. R. Soc. A, (submitted).CrossRefGoogle Scholar
Nägele, G. 2003 Viscoelasticity and diffusional properties of colloidal model dispersions. J. Phys., Condens. Matt. 15 (1), S407–S414.Google Scholar
O'Brien, R. W. 1979 A method for the calculation of the effective transport properties of suspensions of interacting particles. J. Fluid Mech. 91, 17–39.CrossRefGoogle Scholar
O'Brien, R. W. 1986 The high frequency dielectric dispersion of a colloid. J. Colloid Interface Sci. 113, 81–93.CrossRefGoogle Scholar
O'Brien, R. W. 1988 Electro-acoustic effects in a dilute suspension of spherical particles. J. Fluid Mech. 190, 71–86.Google Scholar
O'Brien, R. W. 1990 The electroacoustic equations for a colloidal suspension. J. Fluid Mech. 212, 81–93.CrossRefGoogle Scholar
O'Brien, R. W., Jones, A. & Rowlands, W. N. 2003 A new formula for the dynamic mobility in a concentrated colloid. Colloids Surf. A 218, 89–101.Google Scholar
O'Brien, R. W. & White, L. R. 1978 Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc., Faraday Trans. 74, 1607–1626.CrossRefGoogle Scholar
Oestreicher, H. L. 1951 Field and impedance of an oscillating sphere in a viscoelastic medium with an application to biophysics. J. Acoust. Soc. Am. 23, 707–714.CrossRefGoogle Scholar
O'Konski, C. T. 1960 Electric properties of macromolecules. Part V. Theory of ionic polarization in polyelectrolytes. J. Phys. Chem. 64, 605–619.CrossRefGoogle Scholar
Peppas, N. A., Bures, P., Leobandung, W. & Ichikawa, H. 2000 Hydrogels in pharmaceutical formulations. Eur. J. Pharm. Biopharm. 50, 27–46.Google Scholar
Peppas, N. A., Hilt, J. Z., Khademhosseini, A. & Langer, R. 2006 Hydrogels in biology and medicine: from molecular principles to bionanotechnology. Adv. Mater. 18, 1345–1360.Google Scholar
Pine, D. J., Weitz, D. A., Chaikin, P. M. & Herbolzheimer, E. 1988 Diffusing wave spectroscopy. Phys. Rev. Lett. 61, 1134–1137.CrossRefGoogle Scholar
Pozrikidis, C. 1996 Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.Google Scholar
Preston, M. A., Kornbrekke, R. & White, L. R. 2005 Determination of the dynamic electrophoretic mobility of a spherical colloidal particle through a novel numerical solution of the electrokinetic equations. Langmuir 21, 9832–9842.CrossRefGoogle ScholarPubMed
Qiu, Y. & Park, K. 2001 Environment-sensitive hydrogels for drug delivery. Adv. Drug Deliv. Rev. 53, 321–339.CrossRefGoogle ScholarPubMed
Rider, P. F. & O'Brien, R. W. 1993 The dynamic mobility of particles in a non-dilute suspension. J. Fluid Mech. 257, 607–636.Google Scholar
Russel, W. B., Schowalter, W. R. & Saville, D. A. 1989 Colloidal Dispersions. Cambridge University Press.CrossRefGoogle Scholar
Sato, J. & Breedveld, V. 2006 Transient rheology of solvent-responsive complex fluids by integrating microrheology and microfluidics. J. Rheology 50 (1), 1–19.Google Scholar
Schnurr, B., Gittes, F., MacKintosh, F. C. & Schmidt, C. F. 1997 Determining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations. Macromolecules 30, 7781–7792.CrossRefGoogle Scholar
Sershen, S. R., Mensing, G. A., Ng, M., Halas, N. J., Beebe, D. J. & West, J. L. 2005 Independent optical control of microfluidic valves formed from optomechanically responsive nanocomposite hydrogels. Adv. Mater. 17, 1366–1368.CrossRefGoogle ScholarPubMed
Shkel, I. A., Tsodikov, O. V. & Record, M. T. Jr 2000 Complete asymptotic solution of cylindrical and spherical Poisson–Boltzmann equations at experimental salt concentrations. J. Phys. Chem. B 104, 5161–5170.CrossRefGoogle Scholar
Speight, J. G. 2005 Lange's Handbook of Chemistry, 16th edn. McGraw-Hill.Google Scholar
Takigawa, T., Morino, Y., Urayama, K. & Masuda, T. 1996 Poisson's ratio of polyacrylamide (PAAm) gels. Polym. Gels Networks 4, 1–5.Google Scholar
Takigawa, T., Yamawaki, T., Takahashi, K. & Masuda, T. 1997 Change in Young's modulus of poly(N-isopropylacrylamide) gels by volume phase transition. Polym. Gels Networks 5, 585–589.Google Scholar
Temkin, L. & Leung, C. M. 1976 On the velocity of a rigid sphere in a sound wave. J. Sound Vib. 49, 75–92.Google Scholar
Thévenot, C., Khoukh, A., Reynaud, S., Desbrières, J. & Grassl, B. 2007 Kinetic aspects, rheological properties and mechanoelectrical effects of hydrogels composed of polyacrylamide and polystyrene nanoparticles. Soft Matt. 3, 437–447.CrossRefGoogle ScholarPubMed
Valentine, M. T., Dewalt, L. E. & Ou-Yang, H. D. 1996 Forces on a colloidal particle in a polymer solution: a study using optical tweezers. J. Phys., Condens. Matt. 8, 9477–9482.Google Scholar
Verwey, E. J. W. & Overbeek, J. T. G. 1948 Theory of Stability of Lyophobic Colloids. Elsevier.Google Scholar
Wang, K. L., Burban, J. H. & Cussler, E. L. 1993 Hydrogels as separation agents. Adv. Polym. Sci. 110, 67–79.CrossRefGoogle Scholar
Wang, M. & Hill, R. J. 2008 Electric-field-induced displacement of charged spherical colloids in compressible hydrogels. Soft Matt. 4, 1048–1058.CrossRefGoogle ScholarPubMed
Yamaguchi, N., Chae, E.-S., Zhang, L., Kiick, K. L. & Furst, E. M. 2005 Rheological characterization of polysaccharide-poly(ethylene glycol) star copolymer hydrogels. Biomacromolecules 6, 1931–1940.Google Scholar
Yao, S. H., Hertzog, D. E., Zeng, S. L., Mikkelsen, J. C. & Santiago, J. G. 2003 Porous glass electro-osmotic pumps: design and experiments. J. Colloid Interface Sci. 268, 143–153.Google Scholar
Yao, S. H. & Santiago, J. G. 2003 Porous glass electro-osmotic pumps: theory. J. Colloid Interface Sci. 268, 133–142.CrossRefGoogle Scholar
Ziemann, F., Radler, J. & Sackmann, E. 1994 Local measurement of viscoelastic moduli of entangled actin networks using an oscillating magnetic bead micro-rheometer. Biophys. J. 66, 2210–2216.CrossRefGoogle ScholarPubMed