Skip to main content Accessibility help

Mobility of an axisymmetric particle near an elastic interface

  • Abdallah Daddi-Moussa-Ider (a1), Maciej Lisicki (a2) (a3) and Stephan Gekle (a1)


Using a fully analytical theory, we compute the leading-order corrections to the translational, rotational and translation–rotation coupling mobilities of an arbitrary axisymmetric particle immersed in a Newtonian fluid moving near an elastic cell membrane that exhibits resistance towards stretching and bending. The frequency-dependent mobility corrections are expressed as general relations involving separately the particle’s shape-dependent bulk mobility and the shape-independent parameters such as the membrane–particle distance, the particle orientation and the characteristic frequencies associated with shearing and bending of the membrane. This makes the equations applicable to an arbitrary-shaped axisymmetric particle provided that its bulk mobilities are known, either analytically or numerically. For a spheroidal particle, these general relations reduce to simple expressions in terms of the particle’s eccentricity. We find that the corrections to the translation–rotation coupling mobility are primarily determined by bending, whereas shearing manifests itself in a more pronounced way in the rotational mobility. We demonstrate the validity of the analytical approximations by a detailed comparison with boundary integral simulations of a truly extended spheroidal particle. They are found to be in a good agreement over the whole range of applied frequencies.


Corresponding author

Email address for correspondence:


Hide All

These authors contributed equally to this work.



Hide All
Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions, vol. 1. Dover.
Agudo-Canalejo, J. & Lipowsky, R. 2015 Critical particle sizes for the engulfment of nanoparticles by membranes and vesicles with bilayer asymmetry. ACS Nano 9 (4), 37043720.
Al-Obaidi, H. & Florence, A. T. 2015 Nanoparticle delivery and particle diffusion in confined and complex environments. J. Drug Deliv. Sci. Technol. 30, 266277.
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44 (03), 419440.
Bhaduri, B., Neild, A. & Ng, T. W. 2008 Directional Brownian diffusion dynamics with variable magnitudes. Appl. Phys. Lett. 92 (8), 084105.
Bickel, T. 2006 Brownian motion near a liquid-like membrane. Eur. Phys. J. E 20, 379385.
Bickel, T. 2007 Hindered mobility of a particle near a soft interface. Phys. Rev. E 75, 041403.
Bickel, T. 2014 Probing nanoscale deformations of a fluctuating interface. Europhys. Lett. 106 (1), 16004.
Blake, J. R. 1971 A note on the image system for a Stokeslet in a no-slip boundary. Math. Proc. Camb. Phil. Soc. 70 (02), 303310.
Blake, J. R. & Fulford, G. R. 1981 Force distribution on a slender body close to an interface. Bull. Austral. Math. Soc. 24 (01), 2736.
Bławzdziewicz, J., Ekiel-Jeżewska, M. L. & Wajnryb, E. 2010 Motion of a spherical particle near a planar fluid-fluid interface: the effect of surface incompressibility. J. Chem. Phys. 133 (11), 114702.
Boatwright, T., Dennin, M., Shlomovitz, R., Evans, A. A. & Levine, A. J. 2014 Probing interfacial dynamics and mechanics using submerged particle microrheology. II. Experiment. Phys. Fluids 26 (7), 071904.
Bracewell, R. 1999 The Fourier Transform and its Applications. McGraw-Hill.
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242251.
Cervantes-Martínez, A. E., Ramírez-Saito, A., Armenta-Calderón, R., Ojeda-López, M. A. & Arauz-Lara, J. L. 2011 Colloidal diffusion inside a spherical cell. Phys. Rev. E 83 (3), 030402–4.
Cheong, F. C. & Grier, D. G. 2010 Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy. Opt. Express 18 (7), 65556562.
Chwang, A. T. & Wu, T. Y.-T. 1975 Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech. 67 (04), 787815.
Cichocki, B. & Jones, R. B. 1998 Image representation of a spherical particle near a hard wall. Physica A 258 (3), 273302.
Cichocki, B., Jones, R. B., Kutteh, R. & Wajnryb, E. 2000 Friction and mobility for colloidal spheres in Stokes flow near a boundary: The multipole method and applications. J. Chem. Phys. 112 (5), 25482561.
Colin, R., Yan, M., Chevry, L., Berret, J.-F. & Abou, B. 2012 3d rotational diffusion of micrometric wires using 2d video microscopy. Europhys. Lett. 97 (3), 30008.
Conn, A. R., Gould, N. I. M. & Toint, Ph. L. 2000 Trust Region Methods, vol. 1. SIAM.
Daddi-Moussa-Ider, A. & Gekle, S. 2016 Hydrodynamic interaction between particles near elastic interfaces. J. Chem. Phys. 145 (1), 014905.
Daddi-Moussa-Ider, A., Guckenberger, A. & Gekle, S. 2016a Long-lived anomalous thermal diffusion induced by elastic cell membranes on nearby particles. Phys. Rev. E 93, 012612.
Daddi-Moussa-Ider, A., Guckenberger, A. & Gekle, S. 2016b Particle mobility between two planar elastic membranes: Brownian motion and membrane deformation. Phys. Fluids 28 (7), 071903.
De Corato, M., Greco, F., Davino, G. & Maffettone, P. L. 2015 Hydrodynamics and Brownian motions of a spheroid near a rigid wall. J. Chem. Phys. 142 (19), 194901.
De Mestre, N. J. & Russel, W. B. 1975 Low-Reynolds-number translation of a slender cylinder near a plane wall. J. Engng Maths 9 (2), 8191.
Dettmer, S. L., Pagliara, S., Misiunas, K. & Keyser, U. F. 2014 Anisotropic diffusion of spherical particles in closely confining microchannels. Phys. Rev. E 89 (6), 062305.
Doherty, G. J. & McMahon, H. T. 2009 Mechanisms of endocytosis. Annu. Rev. Biochem. 78 (1), 857902.
Dufresne, E. R., Altman, D. & Grier, D. G. 2001 Brownian dynamics of a sphere between parallel walls. Europhys. Lett. 53 (2), 264270.
Duggal, R. T & Pasquali, M. 2006 Dynamics of individual single-walled carbon nanotubes in water by real-time visualization. Phys. Rev. Lett. 96 (24), 246104.
Eral, H. B., Oh, J. M., van den Ende, D., Mugele, F. & Duits, M. H. G. 2010 Anisotropic and hindered diffusion of colloidal particles in a closed cylinder. Langmuir 26 (22), 1672216729.
Faucheux, L. P. & Libchaber, A. J. 1994 Confined Brownian motion. Phys. Rev. E 49, 51585163.
Felderhof, B. U. 2006a Effect of surface elasticity on the motion of a droplet in a viscous fluid. J. Chem. Phys. 125 (12), 124904.
Felderhof, B. U. 2006b Effect of surface tension and surface elasticity of a fluid-fluid interface on the motion of a particle immersed near the interface. J. Chem. Phys. 125 (14), 144718.
Felderhof, B. U. 2012 Hydrodynamic force on a particle oscillating in a viscous fluid near a wall with dynamic partial-slip boundary condition. Phys. Rev. E 85, 046303.
Felderhof, B. U. 2013 Velocity relaxation of an ellipsoid immersed in a viscous incompressible fluid. Phys. Fluids 25 (1), 013101.
Franosch, T. & Jeney, S. 2009 Persistent correlation of constrained colloidal motion. Phys. Rev. E 79 (3), 031402.
Freund, J. B. 2014 Numerical simulation of flowing blood cells. Annu. Rev. Fluid Mech. 46 (1), 6795.
Goldman, A. J., Cox, R. G. & Brenner, H. 1967a Slow viscous motion of a sphere parallel to a plane wall- I Motion through a quiescent fluid. Chem. Engng Sci. 22, 637651.
Goldman, A. J., Cox, R. G. & Brenner, H. 1967b Slow viscous motion of a sphere parallel to a plane wall-II Couette flow. Chem. Engng Sci. 22, 653660.
Guckenberger, A., Schraml, M. P., Chen, P. G., Leonetti, M. & Gekle, S. 2016 On the bending algorithms for soft objects in flows. Comput. Phys. Commun. 207, 123.
Han, Y., Alsayed, A., Nobili, M. & Yodh, A. G. 2009 Quasi-two-dimensional diffusion of single ellipsoids: Aspect ratio and confinement effects. Phys. Rev. E 80, 011403.
Han, Y., Alsayed, A. M., Nobili, M., Zhang, J., Lubensky, T. C. & Yodh, A. G. 2006 Brownian motion of an ellipsoid. Science 314 (5799), 626630.
Happel, J. & Brenner, H. 2012 Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, vol. 1. Springer Science & Business Media.
Helfrich, W. 1973 Elastic properties of lipid bilayers – theory and possible experiments. Z. Naturf. C. 28, 693.
Holmqvist, P., Dhont, J. K. G. & Lang, P. R. 2007 Colloidal dynamics near a wall studied by evanescent wave light scattering: experimental and theoretical improvements and methodological limitations. J. Chem. Phys. 126 (4), 044707.
Hsu, R. & Ganatos, P. 1989 The motion of a rigid body in viscous fluid bounded by a plane wall. J. Fluid Mech. 207, 2972.
Irmscher, M., de Jong, A. M., Kress, H. & Prins, M. W. J. 2012 Probing the cell membrane by magnetic particle actuation and euler angle tracking. Biophys. J. 102 (3), 698708.
Jünger, F., Kohler, F., Meinel, A., Meyer, T., Nitschke, R., Erhard, B. & Rohrbach, A. 2015 Measuring local viscosities near plasma membranes of living cells with photonic force microscopy. Biophys. J. 109 (5), 869882.
Kim, S. & Karrila, S. J. 2013 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.
Kress, H., Stelzer, E. H. K., Griffiths, G. & Rohrbach, A. 2005 Control of relative radiation pressure in optical traps: application to phagocytic membrane binding studies. Phys. Rev. E 71 (6), 061927.
Lauga, E. & Squires, T. M. 2005 Brownian motion near a partial-slip boundary: A local probe of the no-slip condition. Phys. Fluids 17 (10), 103102.
Lee, S. H., Chadwick, R. S. & Leal, L. G. 1979 Motion of a sphere in the presence of a plane interface. part 1. an approximate solution by generalization of the method of lorentz. J. Fluid Mech. 93, 705726.
Li, G. & Tang, J. X. 2004 Diffusion of actin filaments within a thin layer between two walls. Phys. Rev. E 69 (6), 061921.
Lin, B., Yu, J. & Rice, S. A. 2000 Direct measurements of constrained Brownian motion of an isolated sphere between two walls. Phys. Rev. E 62, 39093919.
Lisicki, M.2015 Evanescent wave dynamic light scattering by optically anisotropic brownian particles. PhD thesis, University of Warsaw.
Lisicki, M., Cichocki, B., Dhont, J. K. G. & Lang, P. R. 2012 One-particle correlation function in evanescent wave dynamic light scattering. J. Chem. Phys. 136 (20), 204704.
Lisicki, M., Cichocki, B., Rogers, S. A., Dhont, J. K. G. & Lang, P. R. 2014 Translational and rotational near-wall diffusion of spherical colloids studied by evanescent wave scattering. Soft Matt. 10 (24), 43124323.
Lisicki, M., Cichocki, B. & Wajnryb, E. 2016 Near-wall diffusion tensor of an axisymmetric colloidal particle. J. Chem. Phys. 145, 034904.
Liu, J., Wei, T., Zhao, J., Huang, Y., Deng, H., Kumar, A., Wang, C., Liang, Z., Ma, X. & Liang, X.-J. 2016 Multifunctional aptamer-based nanoparticles for targeted drug delivery to circumvent cancer resistance. Biomaterials 91, 4456.
Lorentz, H. A. 1907 Ein allgemeiner satz, die bewegung einer reibenden flüssigkeit betreffend, nebst einigen anwendungen desselben. Abh. Theor. Phys. 1, 23.
Meinel, A., Tränkle, B., Römer, W. & Rohrbach, A. 2014 Induced phagocytic particle uptake into a giant unilamellar vesicle. Soft Matt. 10 (20), 36673678.
Michailidou, V. N., Petekidis, G., Swan, J. W. & Brady, J. F. 2009 Dynamics of concentrated hard-sphere colloids near a wall. Phys. Rev. Lett. 102, 068302.
Michailidou, V. N., Swan, J. W., Brady, J. F. & Petekidis, G. 2013 Anisotropic diffusion of concentrated hard-sphere colloids near a hard wall studied by evanescent wave dynamic light scattering. J. Chem. Phys. 139 (16), 164905.
Mitchell, W. H. & Spagnolie, S. E. 2015 Sedimentation of spheroidal bodies near walls in viscous fluids: glancing, reversing, tumbling, and sliding. J. Fluid Mech. 772, 600.
Mukhija, D. & Solomon, M. J. 2007 Translational and rotational dynamics of colloidal rods by direct visualization with confocal microscopy. J. Colloid Interface Sci. 314 (1), 98106.
Naahidi, S., Jafari, M., Edalat, F., Raymond, K., Khademhosseini, A. & Chen, P. 2013 Biocompatibility of engineered nanoparticles for drug delivery. J. Control. Release 166 (2), 182194.
Neild, A., Padding, J. T., Yu, L., Bhaduri, B., Briels, W. J. & Ng, T. W. 2010 Translational and rotational coupling in Brownian rods near a solid surface. Phys. Rev. E 82 (4), 041126.
Padding, J. T. & Briels, W. J. 2010 Translational and rotational friction on a colloidal rod near a wall. J. Chem. Phys. 132, 054511.
Perrin, F. 1934 Mouvement brownien d’un ellipsoide – I. Dispersion diélectrique pour des molécules ellipsoidales. J. Phys. Radium 5, 497511.
Perrin, F. 1936 Mouvement brownien d’un ellipsoide – II. Rotation libre et dépolarisation des fluorescences. translation et diffusion de molécules ellipsoidales. J. Phys. Radium 7, 111.
Power, H. & Miranda, G. 1987 Second kind integral equation formulation of Stokes’ flows past a particle of arbitrary shape. SIAM J. Appl. Maths 47 (4), 689698.
Pozrikidis, C. 2001 Interfacial dynamics for Stokes flow. J. Comput. Phys. 169, 250.
Rogers, S. A., Lisicki, M., Cichocki, B., Dhont, J. K. G. & Lang, P. R. 2012 Rotational diffusion of spherical colloids close to a wall. Phys. Rev. Lett. 109 (9), 098305.
Saintyves, B., Jules, T., Salez, T. & Mahadevan, L. 2016 Self-sustained lift and low friction via soft lubrication. Proc. Natl Acad. Sci. USA 113 (21), 58475849.
Salez, T. & Mahadevan, L. 2015 Elastohydrodynamics of a sliding, spinning and sedimenting cylinder near a soft wall. J. Fluid Mech. 779, 181196.
Schäffer, E., Nørrelykke, S. F. & Howard, J. 2007 Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers. Langmuir 23 (7), 36543665.
Schiby, D. & Gallily, I. 1980 On the orderly nature of the motion of nonspherical aerosol particles. III. The effect of the particle–wall fluid-dynamic interaction. J. Colloid Interface Sci. 77 (2), 328352.
Shlomovitz, R., Evans, A., Boatwright, T., Dennin, M. & Levine, A. 2013 Measurement of monolayer viscosity using noncontact microrheology. Phys. Rev. Lett. 110 (13), 137802.
Shlomovitz, R., Evans, A. A., Boatwright, T., Dennin, M. & Levine, A. J. 2014 Probing interfacial dynamics and mechanics using submerged particle microrheology. I. Theory. Phys. Fluids 26 (7), 071903.
Skalak, R., Tozeren, A., Zarda, R. P. & Chien, S. 1973 Strain energy function of red blood cell membranes. Biophys. J. 13 (3), 245264.
Swan, J. W. & Brady, J. F. 2007 Simulation of hydrodynamically interacting particles near a no-slip boundary. Phys. Fluids 19 (11), 113306.
Tränkle, B., Ruh, D. & Rohrbach, A. 2016 Interaction dynamics of two diffusing particles: contact times and influence of nearby surfaces. Soft Matt. 12 (10), 27292736.
Waigh, T. A. 2016 Advances in the microrheology of complex fluids. Rep. Prog. Phys. 79 (7), 074601.
Wang, G. M., Prabhakar, R. & Sevick, E. M. 2009 Hydrodynamic mobility of an optically trapped colloidal particle near fluid-fluid interfaces. Phys. Rev. Lett. 103, 248303.
Wang, W. & Huang, P. 2014 Anisotropic mobility of particles near the interface of two immiscible liquids. Phys. Fluids 26 (9), 092003.
Zhao, H. & Shaqfeh, E. S. G. 2011 Shear-induced platelet margination in a microchannel. Phys. Rev. E 83, 061924.
Zhao, H., Shaqfeh, E. S. G. & Narsimhan, V. 2012 Shear-induced particle migration and margination in a cellular suspension. Phys. Fluids 24 (1), 011902.
Zheng, Z. & Han, Y. 2010 Self-diffusion in two-dimensional hard ellipsoid suspensions. J. Chem. Phys. 133 (12), 124509.
Zhu, L.2014 Simulation of individual cells in flow. PhD thesis, KTH Royal Institute of Technology in Stockholm.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Type Description Title
Supplementary materials

Daddi-Moussa-Ider supplementary material
Daddi-Moussa-Ider supplementary material 1

 Unknown (237 KB)
237 KB

Mobility of an axisymmetric particle near an elastic interface

  • Abdallah Daddi-Moussa-Ider (a1), Maciej Lisicki (a2) (a3) and Stephan Gekle (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed