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Deformation of an encapsulated bubble in steady and oscillatory electric fields

Published online by Cambridge University Press:  06 April 2018

Yunqiao Liu Open the ORCID record for Yunqiao Liu [Opens in a new window] ,
Dongdong He ,
Xiaobo Gong Open the ORCID record for Xiaobo Gong [Opens in a new window]  and
Huaxiong Huang
Yunqiao Liu
Affiliation:
Key Laboratory of Hydrodynamics (Ministry of Education), Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China
Dongdong He
Affiliation:
School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China
Xiaobo Gong*
Affiliation:
Key Laboratory of Hydrodynamics (Ministry of Education), Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China
Huaxiong Huang
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada The Fields Institute for Research in Mathematical Sciences, Toronto, Ontario M5T 3J1, Canada
*
†Email address for correspondence: x.gong@sjtu.edu.cn
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Abstract

In this paper, we investigate the dynamics of an encapsulated bubble in steady and oscillatory electric fields theoretically, based on a leaky dielectric model. On the bubble surface, an applied electric field generates a Maxwell stress, in addition to hydrodynamic traction and membrane mechanical stress. Our model also includes the effect of interfacial charge due to the jump of the current and the stretching of the interface. We focus on the axisymmetric deformation of the encapsulated bubble induced by the electric field and carry out our analysis using Legendre polynomials. In our first example, the encapsulating membrane is modelled as a nearly incompressible interface with bending rigidity. Under a steady uniform electric field, the encapsulated bubble resumes an elongated equilibrium shape, dominated by the second- and fourth-order shape modes. The deformed shape agrees well with experimental observations reported in the literature. Our model reveals that the interfacial charge distribution is determined by the magnitude of the shape modes, as well as the permittivity and conductivity of the external and internal fluids. The effects of the electric field on the natural frequency of the oscillating bubble are also shown. For our second example, we considered a bubble encapsulated with a hyperelastic membrane with bending rigidity, subject to an oscillatory electric field. We show that the bubble can modulate its oscillating frequency and reach a stable shape oscillation at an appreciable amplitude.

JFM classification

Drops and Bubbles: Bubble dynamics Drops and Bubbles: Electrohydrodynamic effects Biological Fluid Dynamics: Membranes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 844 , 10 June 2018 , pp. 567 - 596
Copyright
© 2018 Cambridge University Press 

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References

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, 251–267.10.1017/S0022112081003480Google Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics, chap. 3.1, Cambridge University Press.10.1017/CBO9780511800955Google Scholar
Brenner, M. P., Hilgenfeldt, S. & Lohse, D. 2002 Single-bubble sonoluminescence. Rev. Mod. Phys. 74 (2), 425–484.10.1103/RevModPhys.74.425Google Scholar
Chang, K. & Olbricht, W. L. 1993a Experimental studies of the deformation and breakup of a synthetic capsule in steady and unsteady simple shear flow. J. Fluid Mech. 250, 609–633.10.1017/S0022112093001582Google Scholar
Chang, K. & Olbricht, W. L. 1993b Experimental studies of the deformation of a synthetic capsule in extensional flow. J. Fluid Mech. 250, 587–608.10.1017/S0022112093001570Google Scholar
Chatterjee, D. & Sarkar, K. 2003 A Newtonian rheological model for the interface of microbubble contrast agents. Ultrasound Med. Biol. 29 (12), 1749–1757.10.1016/S0301-5629(03)01051-2Google Scholar
Church, C. C. 1995 The effects of an elastic solid surface layer on the radial pulsations of gas bubbles. J. Acoust. Soc. Am. 97 (3), 1510–1521.10.1121/1.412091Google Scholar
De Jong, N., Emmer, M., Van Wamel, A. & Versluis, M. 2009 Ultrasonic characterization of ultrasound contrast agents. Med. Biol. Engng Comput. 47 (8), 861–873.10.1007/s11517-009-0497-1Google Scholar
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, R3.10.1017/jfm.2015.69Google Scholar
De Loubens, C., Deschamps, J., Edwards-Levy, F. & Leonetti, M. 2016 Tank-treading of microcapsules in shear flow. J. Fluid Mech. 789, 750–767.10.1017/jfm.2015.758Google Scholar
Dong, W., Li, R. Y., Yu, H. L. & Yan, Y. Y. 2006 An investigation of behaviours of a single bubble in a uniform electric field. Exp. Therm. Fluid Sci. 30 (6), 579–586.10.1016/j.expthermflusci.2005.12.003Google Scholar
Dubash, N. & Mestel, A. J. 2007 Behaviour of a conducting drop in a highly viscous fluid subject to an electric field. J. Fluid Mech. 581, 469–493.10.1017/S0022112007006040Google Scholar
Feng, Z. C. & Leal, L. G. 1997 Nonlinear bubble dynamics. Annu. Rev. Fluid Mech. 29 (1), 201–243.10.1146/annurev.fluid.29.1.201Google Scholar
Ferrara, K., Pollard, R. & Borden, M. 2007 Ultrasound microbubble contrast agents: fundamentals and application to gene and drug delivery. Annu. Rev. Biomed. Engng 9, 415–447.10.1146/annurev.bioeng.8.061505.095852Google Scholar
Gracià, R. S., Bezlyepkina, N., Knorr, R. L., Lipowsky, R. & Dimova, R. 2010 Effect of cholesterol on the rigidity of saturated and unsaturated membranes: fluctuation and electrodeformation analysis of giant vesicles. Soft Matt. 6 (7), 1472–1482.10.1039/b920629aGoogle Scholar
Green, A. E. & Adkins, J. E. 1960 Large Elastic Deformations and Non-Linear Continuum Mechanics. Clarendon Press.Google Scholar
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. C 28 (11–12), 693–703.10.1515/znc-1973-11-1209Google Scholar
Kang, I. S. 1993 Dynamics of a conducting drop in a time-periodic electric field. J. Fluid Mech. 257, 229–264.10.1017/S0022112093003064Google Scholar
Knoche, S., Vella, D., Aumaitre, E., Degen, P., Rehage, H., Cicuta, P. & Kierfeld, J. 2013 Elastometry of deflated capsules: elastic moduli from shape and wrinkle analysis. Langmuir 29 (40), 12463–12471.10.1021/la402322gGoogle Scholar
Kummrow, M. & Helfrich, W. 1991 Deformation of giant lipid vesicles by electric fields. Phys. Rev. A 44 (12), 8356.10.1103/PhysRevA.44.8356Google Scholar
Kwan, J. J. & Borden, M. A. 2012 Lipid monolayer collapse and microbubble stability. Adv. Colloid. Interface Sci. 183, 82–99.10.1016/j.cis.2012.08.005Google Scholar
Lacour, T., Guédra, M., Valier-Brasier, T. & Coulouvrat, F. 2018 A model for acoustic vaporization dynamics of a bubble/droplet system encapsulated within a hyperelastic shell. J. Acoust. Soc. Am. 143 (1), 23–37.10.1121/1.5019467Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Landau, L. D., Lifshitz, E. M. & Pitaevskii, L. P. 1984 Electrodynamics of Continuous Media, 2nd edn. Elsevier Butterworth-Heinemann.Google Scholar
Lee, S. M. & Kang, I. S. 1999 Three-dimensional analysis of the steady-state shape and small-amplitude oscillation of a bubble in uniform and non-uniform electric fields. J. Fluid Mech. 384, 59–91.10.1017/S0022112098004133Google Scholar
Lefebvre, Y., Leclerc, E., Barthès-Biesel, D., Walter, J. & Edwards-Lévy, F. 2008 Flow of artificial microcapsules in microfluidic channels: a method for determining the elastic properties of the membrane. Phys. Fluids 20 (12), 123102.10.1063/1.3054128Google Scholar
Lindner, J. R. 2004 Microbubbles in medical imaging: current applications and future directions. Nat. Rev. Drug Discov. 3 (6), 527–533.10.1038/nrd1417Google Scholar
Liu, Y., Sugiyama, K. & Takagi, S. 2016 On the interaction of two encapsulated bubbles in an ultrasound field. J. Fluid Mech. 804, 58–89.10.1017/jfm.2016.525Google Scholar
Liu, Y., Sugiyama, K., Takagi, S. & Matsumoto, Y. 2011 Numerical study on the shape oscillation of an encapsulated microbubble in ultrasound field. Phys. Fluids 23 (4), 041904.10.1063/1.3578493Google Scholar
Liu, Y., Sugiyama, K., Takagi, S. & Matsumoto, Y. 2012 Surface instability of an encapsulated bubble induced by an ultrasonic pressure wave. J. Fluid Mech. 691, 315–340.10.1017/jfm.2011.477Google Scholar
Love, A. E. H. 1888 The small free vibrations and deformation of a thin elastic shell. Phil. Trans. R. Soc. Lond. A 179, 491–546.Google Scholar
Marmottant, P., van der Meer, S., Emmer, M., Versluis, M., de Jong, N., Hilgenfeldt, S. & Lohse, D. 2005 A model for large amplitude oscillations of coated bubbles accounting for buckling and rupture. J. Acoust. Soc. Am. 118 (6), 3499–3505.10.1121/1.2109427Google Scholar
van der Meer, S. M., Dollet, B., Voormolen, M. M., Chin, C. T., Bouakaz, A., de Jong, N., Versluis, M. & Lohse, D. 2007 Microbubble spectroscopy of ultrasound contrast agents. J. Acoust. Soc. Am. 121 (1), 648–656.10.1121/1.2390673Google Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1 (1), 111–146.10.1146/annurev.fl.01.010169.000551Google Scholar
Mooney, M. 1940 A theory of large elastic deformation. J. Appl. Phys. 11, 582–592.10.1063/1.1712836Google Scholar
Pereira, A., Trevelyan, P. M. J., Thiele, U. & Kalliadasis, S. 2007 Dynamics of a horizontal thin liquid film in the presence of reactive surfactants. Phys. Fluids 19 (11), 112102.10.1063/1.2775938Google Scholar
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9 (1), 145–185.10.1146/annurev.fl.09.010177.001045Google Scholar
Pozrikidis, C. 2001 Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J. Fluid Mech. 440, 269–291.10.1017/S0022112001004657Google Scholar
Prosperetti, A. 1977 Viscous effects on perturbed spherical flows. Q. Appl. Maths 34 (4), 339–352.10.1090/qam/99652Google Scholar
Prosperetti, A. 1991 The thermal behaviour of oscillating gas bubbles. J. Fluid Mech. 222, 587–616.10.1017/S0022112091001234Google Scholar
Pu, G., Borden, M. A. & Longo, M. L. 2006 Collapse and shedding transitions in binary lipid monolayers coating microbubbles. Langmuir 22 (7), 2993–2999.10.1021/la0530337Google Scholar
Salipante, P. F. & Vlahovska, P. M. 2014 Vesicle deformation in dc electric pulses. Soft Matt. 10 (19), 3386–3393.10.1039/C3SM52870GGoogle Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29 (1), 27–64.10.1146/annurev.fluid.29.1.27Google Scholar
Schnitzer, O. & Yariv, E. 2015 The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description. J. Fluid Mech. 773, 1–33.10.1017/jfm.2015.242Google Scholar
Schwalbe, J. T., Vlahovska, P. M. & Miksis, M. J. 2011 Vesicle electrohydrodynamics. Phys. Rev. E 83 (4), 046309.Google Scholar
Segers, T., de Rond, L., de Jong, N., Borden, M. & Versluis, M. 2016 Stability of monodisperse phospholipid-coated microbubbles formed by flow-focusing at high production rates. Langmuir 32 (16), 3937–3944.10.1021/acs.langmuir.6b00616Google Scholar
Shaw, S. J., Spelt, P. D. M. & Matar, O. K. 2009 Electrically induced bubble deformation, translation and collapse. J. Engng Maths 65 (4), 291–310.10.1007/s10665-009-9314-yGoogle Scholar
Spelt, P. D. M. & Matar, O. K. 2006 Collapse of a bubble in an electric field. Phys. Rev. E 74 (4), 046309.Google Scholar
Stone, H. A. 1990 A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2 (1), 111–112.10.1063/1.857686Google Scholar
Stride, E. & Saffari, N. 2003 On the destruction of microbubble ultrasound contrast agents. Ultrasound Med. Biol. 29 (4), 563–573.10.1016/S0301-5629(02)00787-1Google Scholar
Stride, E. P. & Coussios, C. C. 2010 Cavitation and contrast: the use of bubbles in ultrasound imaging and therapy. Proc. Inst. Mech. Engng H 224 (2), 171–191.10.1243/09544119JEIM622Google Scholar
Taylor, G. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383–397.Google Scholar
Trinh, E. H., Holt, R. G. & Thiessen, D. B. 1996 The dynamics of ultrasonically levitated drops in an electric field. Phys. Fluids 8 (1), 43–61.10.1063/1.868813Google Scholar
Tsiglifis, K. & Pelekasis, N. A. 2008 Nonlinear radial oscillations of encapsulated microbubbles subject to ultrasound: the effect of membrane constitutive law. J. Acoust. Soc. Am. 123 (6), 4059–4070.10.1121/1.2909553Google Scholar
Tsukada, T., Katayama, T., Ito, Y. & Hozawa, M. 1993 Theoretical and experimental studies of circulations inside and outside a deformed drop under a uniform electric field. J. Chem. Engng Jpn. 26 (6), 698–703.10.1252/jcej.26.698Google Scholar
Tu, J., Guan, J., Qiu, Y. & Matula, T. J. 2009 Estimating the shell parameters of SonoVue® microbubbles using light scattering. J. Acoust. Soc. Am. 126 (6), 2954–2962.10.1121/1.3242346Google Scholar
Unger, E. C., Hersh, E., Vannan, M., Matsunaga, T. O. & McCreery, T. 2001 Local drug and gene delivery through microbubbles. Prog. Cardiovasc. Dis. 44 (1), 45–54.10.1053/pcad.2001.26443Google Scholar
Unnikrishnan, S. & Klibanov, A. L. 2012 Microbubbles as ultrasound contrast agents for molecular imaging: preparation and application. Am. J. Roentgenol. 199 (2), 292–299.10.2214/AJR.12.8826Google Scholar
Van Vliet, K. J., Bao, G. & Suresh, S. 2003 The biomechanics toolbox: experimental approaches for living cells and biomolecules. Acta Mater. 51 (19), 5881–5905.10.1016/j.actamat.2003.09.001Google Scholar
Vizika, O. & Saville, D. A. 1992 The electrohydrodynamic deformation of drops suspended in liquids in steady and oscillatory electric fields. J. Fluid Mech. 239, 1–21.10.1017/S0022112092004294Google Scholar
Vlahovska, P. M., Gracia, R. S., Aranda-Espinoza, S. & Dimova, R. 2009 Electrohydrodynamic model of vesicle deformation in alternating electric fields. Biophys. J. 96 (12), 4789–4803.10.1016/j.bpj.2009.03.054Google Scholar
Yamamoto, T., Aranda-Espinoza, S., Dimova, R. & Lipowsky, R. 2010 Stability of spherical vesicles in electric fields. Langmuir 26 (14), 12390–12407.10.1021/la1011132Google Scholar