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Large-deformation electrohydrodynamics of an elastic capsule in a DC electric field

  • Sudip Das (a1) and Rochish M. Thaokar (a1)


The dynamics of a spherical elastic capsule, containing a Newtonian fluid bounded by an elastic membrane and immersed in another Newtonian fluid, in a uniform DC electric field is investigated. Discontinuity of electrical properties, such as the conductivities of the internal and external fluid media as well as the capacitance and conductance of the membrane, leads to a net interfacial Maxwell stress which can cause the deformation of such an elastic capsule. We investigate this problem considering well-established membrane laws for a thin elastic membrane, with fully resolved hydrodynamics in the Stokes flow limit, and describe the electrostatics using the capacitor model. In the limit of small deformation, the analytical theory predicts the dynamics fairly satisfactorily. Large deformations at high capillary number, though, necessitate a numerical approach (axisymmetric boundary element method in the present case) to solve this highly nonlinear problem. Akin to vesicles, at intermediate times, highly nonlinear biconcave shapes along with squaring and hexagon-like shapes are observed when the outer medium is more conducting. The study identifies the essentiality of parameters such as high membrane capacitance, low membrane conductance, low hydrodynamic time scales and high capillary number (the ratio of the destabilizing electric force to the stabilizing elastic force) for observation of these shape transitions. The transition is due to large compressive Maxwell stress at the poles at intermediate times. Thus such shape transition can be seen in spherical globules admitting electrical capacitance, possibly irrespective of the nature of the interfacial restoring force.


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Barthès-Biesel, D. 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100 (4), 831853.
Barthès-Biesel, D. 1991 Role of interfacial properties on the motion and deformation of capsules in shear flow. Physica 172 (1), 103124.
Barthès-Biesel, D. & Chhim, V. 1981 The constitutive equation of a dilute suspension of spherical microcapsules. Intl J. Multiphase Flow 7 (5), 493505.
Barthès-Biesel, D., Diaz, A. & Dhenin, E. 2002 Effect of constitutive laws for two-dimensional membranes on flow-induced capsule deformation. J. Fluid Mech. 460, 211222.
Barthès-Biesel, D. & Rallison, J. M. 1981 The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251267.
Chang, K. S. & Olbricht, W. L. 1993 Experimental studies of the deformation of a synthetic capsule in extensional flow. J. Fluid Mech. 250, 587608.
Crowley, J. M. 1973 Electrical breakdown of bimolecular lipid membranes as an electromechanical instability. Biophys. J. 13 (7), 711724.
Das, S., Mayya, Y. S. & Thaokar, R. 2015 Dynamics of a charged drop in a quadrupole electric field. Europhys. Lett. 111 (2), 24006.
DeBruin, K. A. & Krassowska, W. 1999 Modeling electroporation in a single cell. I. Effects of field strength and rest potential. Biophys. J. 77 (3), 12131224.
Deshmukh, S. D. & Thaokar, R. M. 2012 Deformation, breakup and motion of a perfect dielectric drop in a quadrupole electric field. Phys. Fluids 24 (3), 032105.
Deshmukh, S. D. & Thaokar, R. M. 2013 Deformation and breakup of a leaky dielectric drop in a quadrupole electric field. J. Fluid Mech. 731, 713733.
Diaz, A., Pelekasis, N. & Barthès-Biesel, D. 2000 Transient response of a capsule subjected to varying flow conditions: effect of internal fluid viscosity and membrane elasticity. Phys. Fluids 12 (5), 948957.
Dimova, R., Bezlyepkina, N., Jordo, M. D., Knorr, R. L., Riske, K. A., Staykova, M., Vlahovska, P. M., Yamamoto, T., Yang, P. & Lipowsky, R. 2009 Vesicles in electric fields: some novel aspects of membrane behavior. Soft Matt. 5, 32013212.
Dimova, R., Riske, K. A., Aranda, S., Bezlyepkina, N., Knorr, R. L. & Lipowsky, R. 2007 Giant vesicles in electric fields. Soft Matt. 3, 817827.
Dodson, W. R. & Dimitrakopoulos, P. 2009 Dynamics of strain-hardening and strain-softening capsules in strong planar extensional flows via an interfacial spectral boundary element algorithm for elastic membranes. J. Fluid Mech. 641, 263296.
Dubreuil, F., Elsner, N. & Fery, A. 2003 Elastic properties of polyelectrolyte capsules studied by atomic-force microscopy and RICM. Eur. Phys. J. E 12 (2), 215221.
Dupont, C., Salsac, A.-V., Barthès-Biesel, D., Vidrascu, M. & Tallec, P. L. 2015 Influence of bending resistance on the dynamics of a spherical capsule in shear flow. Phys. Fluids 27 (5), 051902.
Farutin, A. & Misbah, C. 2012 Squaring, parity breaking, and S tumbling of vesicles under shear flow. Phys. Rev. Lett. 109, 248106.
Fery, A. & Weinkamer, R. 2007 Mechanical properties of micro- and nanocapsules: single-capsule measurements. Polymer 48 (25), 72217235.
Finken, R., Kessler, S. & Seifert, U. 2011 Micro-capsules in shear flow. J. Phys.: Condens. Matter 23 (18), 184113.
Finken, R. & Seifert, U. 2006 Wrinkling of microcapsules in shear flow. J. Phys.: Condens. Matter 18 (15), L185L191.
Gao, C., Leporatti, S., Moya, S., Donath, E. & Möhwald, H. 2001 Stability and mechanical properties of polyelectrolyte capsules obtained by stepwise assembly of poly(styrenesulfonate sodium salt) and poly(diallyldimethyl ammonium) chloride onto melamine resin particles. Langmuir 17 (11), 34913495.
Green, A. E. & Adkins, J. E. 1970 Large Elastic Deformation, 2nd edn. Oxford University Press.
Grosse, C. & Schwan, H. P. 1992 Cellular membrane potentials induced by alternating fields. Biophys. J. 63 (6), 16321642.
Ha, J.-W. & Yang, S.-M. 2000 Electrohydrodynamic effects on the deformation and orientation of a liquid capsule in a linear flow. Phys. Fluids 12 (7), 16711684.
Hu, W.-F., Kim, Y. & Lai, M.-C. 2014 An immersed boundary method for simulating the dynamics of three-dimensional axisymmetric vesicles in Navier–Stokes flows. J. Comput. Phys. A 257, 670686.
Hu, W.-F., Lai, M.-C., Seol, Y. & Young, Y.-N. 2016 Vesicle electrohydrodynamic simulations by coupling immersed boundary and immersed interface method. J. Comput. Phys. 317, 6681.
Hyman, W. A. & Skalak, R. 1972 Non-Newtonian behavior of a suspension of liquid drops in tube flow. AIChE J. 18 (1), 149154.
Kang, Y., Li, D., Kalams, S. A. & Eid, J. E. 2008 DC-dielectrophoretic separation of biological cells by size. Biomedical Microdevices 10 (2), 243249.
Karyappa, R. B., Deshmukh, S. D. & Thaokar, R. M. 2014a Breakup of a conducting drop in a uniform electric field. J. Fluid Mech. 754, 550589.
Karyappa, R. B., Deshmukh, S. D. & Thaokar, R. M. 2014b Deformation of an elastic capsule in a uniform electric field. Phys. Fluids 26 (12), 122108.
Kataoka, K., Harada, A. & Nagasaki, Y. 2001 Block copolymer micelles for drug delivery: design, characterization and biological significance. Adv. Drug Deliv. Rev. 47 (1), 113131; Special Issue: Nanoparticulate systems for improved drug delivery.
Keller, M. W. & Sottos, N. R. 2006 Mechanical properties of microcapsules used in a self-healing polymer. Exp. Mech. 46 (6), 725733.
Kessler, S., Finken, R. & Seifert, U. 2008 Swinging and tumbling of elastic capsules in shear flow. J. Fluid Mech. 605, 207226.
Kessler, S., Finken, R. & Seifert, U. 2009 Elastic capsules in shear flow: analytical solutions for constant and time-dependent shear rates. Eur. Phys. J. E 29 (4), 399413.
Kolahdouz, E. M. & Salac, D. 2015a Dynamics of three-dimensional vesicles in dc electric fields. Phys. Rev. E 92, 012302.
Kolahdouz, E. M. & Salac, D. 2015b Electrohydrodynamics of three-dimensional vesicles: a numerical approach. SIAM J. Sci. Comput. 37 (3), B473B494.
Kwak, S. & Pozrikidis, C. 2001 Effect of membrane bending stiffness on the axisymmetric deformation of capsules in uniaxial extensional flow. Phys. Fluids 13 (5), 12341242.
Lac, E. & Barthès-Biesel, D. 2005 Deformation of a capsule in simple shear flow: effect of membrane prestress. Phys. Fluids 17 (7), 072105.
Lac, E., Barthès-Biesel, D., Pelekasis, N. A. & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling. J. Fluid Mech. 516, 303334.
Le, D. V. 2010 Effect of bending stiffness on the deformation of liquid capsules enclosed by thin shells in shear flow. Phys. Rev. E 82, 016318.
Lighthill, M. J. 1968 Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. J. Fluid Mech. 34 (1), 113143.
de Loubens, C., Deschamps, J., Boedec, G. & Leonetti, M. 2015 Stretching of capsules in an elongation flow, a route to constitutive law. J. Fluid Mech. 767, R3.
McConnell, L. C., Miksis, M. J. & Vlahovska, P. M. 2013 Vesicle electrohydrodynamics in dc electric fields. IMA J. Appl. Maths 78 (4), 797817.
McConnell, L. C., Miksis, M. J. & Vlahovska, P. M. 2015a Continuum modeling of the electric-field-induced tension in deforming lipid vesicles. J. Chem. Phys. 143 (24), 243132.
McConnell, L. C., Vlahovska, P. M. & Miksis, M. J. 2015b Vesicle dynamics in uniform electric fields: squaring and breathing. Soft Matt. 11, 48404846.
Navot, Y. 1998 Elastic membranes in viscous shear flow. Phys. Fluids 10 (8), 18191833.
Neumann, E., Sowers, A. E. & Jordan, C. A. 1989 Electroporation and Electrofusion in Cell Biology, 1st edn. Plenum Press.
Ouriemi, M. & Vlahovska, P. M. 2015 Electrohydrodynamic deformation and rotation of a particle-coated drop. Langmuir 31 (23), 62986305.
Pethig, R. 1996 Dielectrophoresis: using inhomogeneous ac electrical fields to separate and manipulate cells. Crit. Rev. Biotechnol. 16 (4), 331348.
Pozrikidis, C. 1990 The axisymmetric deformation of a red blood cell in uniaxial straining Stokes flow. J. Fluid Mech. 216, 231254.
Pozrikidis, C. 1995 Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow. J. Fluid Mech. 297, 123152.
Pozrikidis, C. 2001 Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J. Fluid Mech. 440, 269291.
Rallison, J. M. & Acrivos, A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89, 191200.
Ramanujan, S. & Pozrikidis, C. 1998 Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities. J. Fluid Mech. 361, 117143.
Rao, P. R., Zahalak, G. I. & Sutera, S. P. 1994 Large deformations of elastic cylindrical capsules in shear flows. J. Fluid Mech. 270, 7390.
Riske, K. A. & Dimova, R. 2006 Electric pulses induce cylindrical deformations on giant vesicles in salt solutions. Biophys. J. 91 (5), 17781786.
Schwalbe, J. T., Vlahovska, P. M. & Miksis, M. J. 2011 Vesicle electrohydrodynamics. Phys. Rev. E 83, 046309.
Skalak, R., Tozeren, A., Zarda, R. P. & Chien, S. 1973 Strain energy function of red blood cell membranes. Biophys. J. 13 (3), 245264.
Skotheim, J. M. & Secomb, T. W. 2007 Red blood cells and other nonspherical capsules in shear flow: oscillatory dynamics and the tank-treading-to-tumbling transition. Phys. Rev. Lett. 98, 078301.
Sui, Y., Chen, X. B., Chew, Y. T., Roy, P. & Low, H. T. 2010 Numerical simulation of capsule deformation in simple shear flow. Comput. Fluids 39 (2), 242250.
Sui, Y., Chew, Y. T., Roy, P. & Low, H. T. 2007 Effect of membrane bending stiffness on the deformation of elastic capsules in extensional flow: a lattice Boltzmann study. Intl J. Mod. Phys. C 18 (08), 12771291.
Sui, Y., Chew, Y. T., Roy, P. & Low, H. T. 2008a A hybrid method to study flow-induced deformation of three-dimensional capsules. J. Comput. Phys. 227 (12), 63516371.
Sui, Y., Low, H. T., Chew, Y. T. & Roy, P. 2008b Tank-treading, swinging, and tumbling of liquid-filled elastic capsules in shear flow. Phys. Rev. E 77, 016310.
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146 (858), 501523.
Thaokar, R. M. 2016 Time-dependent electrohydrodynamics of a compressible viscoelastic capsule in the small-deformation limit. Phys. Rev. E 94, 042607.
Thaokar, R. M. & Deshmukh, S. D. 2010 Rayleigh instability of charged drops and vesicles in the presence of counterions. Phys. Fluids 22 (3), 034107.
Tieleman, D. P. 2004 The molecular basis of electroporation. BMC Biochem. 5 (1), 10.
Trefethen, L. N.1996 Finite difference and spectral methods for ordinary and partial differential equations. Unpublished text
Veerapaneni, S. 2016 Integral equation methods for vesicle electrohydrodynamics in three dimensions. J. Comput. Phys. 326, 278289.
Vlahovska, P. M., Young, Y.-N., Danker, G. & Misbah, C. 2011 Dynamics of a non-spherical microcapsule with incompressible interface in shear flow. J. Fluid Mech. 678, 221247.
Yamamoto, T., Aranda-Espinoza, S., Dimova, R. & Lipowsky, R. 2010 Stability of spherical vesicles in electric fields. Langmuir 26 (14), 1239012407.
Zhou, H. & Pozrikidis, C. 1995 Deformation of liquid capsules with incompressible interfaces in simple shear flow. J. Fluid Mech. 283, 175200.
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Large-deformation electrohydrodynamics of an elastic capsule in a DC electric field

  • Sudip Das (a1) and Rochish M. Thaokar (a1)


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