Skip to main content Accessibility help

Force-induced diffusion in suspensions of hydrodynamically interacting colloids

  • N. J. Hoh (a1) and R. N. Zia (a1)


We study the influence of hydrodynamic, thermodynamic and interparticle forces on the diffusive motion of a Brownian probe driven by a constant external force through a dilute colloidal dispersion. The influence of these microscopic forces on equilibrium self-diffusivity (passive microrheology) is well known: all three act to hinder the short- and long-time self-diffusion. Here, via pair-Smoluchowski theory, we explore their influence on self-diffusion in a flowing suspension, where particles and fluid have been set into motion by an externally forced probe (active microrheology), giving rise to non-equilibrium flow-induced diffusion. The probe’s motion entrains background particles as it travels through the bath, deforming the equilibrium suspension microstructure. The shape and extent of microstructural distortion is set by the relative strength of the external force $F^{\mathit{ext}}$ to the entropic restoring force $kT/a_{\mathit{th}}$ of the bath particles, defining a Péclet number $\mathit{Pe}\equiv F^{\mathit{ext}}/(2kT/a_{\mathit{th}})$ ; and also by the strength of hydrodynamic interactions, set by the range of interparticle repulsion ${\it\kappa}=(a_{\mathit{th}}-a)/a$ , where $kT$ is the thermal energy and $a_{\mathit{th}}$ and $a$ are the thermodynamic and hydrodynamic sizes of the particles, respectively. We find that in the presence of flow, the same forces that hinder equilibrium diffusion now enhance it, with diffusive anisotropy set by the range of interparticle repulsion ${\it\kappa}$ . A transition from hindered to enhanced diffusion occurs when diffusive and advective forces balance, $\mathit{Pe}\sim 1$ , where the exact value is a sensitive function of the strength of hydrodynamics, ${\it\kappa}$ . We find that the hindered to enhanced transition straddles two transport regimes: in hindered diffusion, stochastic forces in the presence of other bath particles produce deterministic displacements (Brownian drift) at the expense of a maximal random walk. In enhanced diffusion, driving the probe with a deterministic force through an initially random suspension leads to fluctuations in the duration of probe–bath particle entrainment, giving rise to enhanced, flow-induced diffusion. The force-induced diffusion is anisotropic for all $\mathit{Pe}$ , scaling as $D\sim \mathit{Pe}^{2}$ in all directions for weak forcing, regardless of the strength of hydrodynamic interactions. When probe forcing is strong, $D\sim \mathit{Pe}$ in all directions in the absence of hydrodynamic interactions, but the picture changes qualitatively as hydrodynamic interactions grow strong. In this nonlinear regime, microstructural asymmetry weakens while the anisotropy of the force-induced diffusion tensor increases dramatically. This behaviour owes its origins to the approach toward Stokes flow reversibility, where diffusion along the direction of probe force scales advectively while transverse diffusion must vanish.


Corresponding author

Email address for correspondence:


Hide All
Abbott, J. R., Graham, A. L., Mondy, L. A. & Brenner, H. 1998 Dispersion of a ball settling through a quiescent neutrally buoyant suspension. J. Fluid Mech. 361, 309331.
Acrivos, A., Batchelor, G. K., Hinch, E. J., Koch, D. L. & Mauri, R. 1992 Longitudinal shear-induced diffusion of spheres in a dilute suspension. J. Fluid Mech. 240, 651657.
Acrivos, A., Mauri, R. & Fan, X. 1993 Shear-induced resuspension in a Couette device. Intl J. Multiphase Flow 19, 797802.
Almog, Y. & Brenner, H. 1997 Non-continuum anomalies in the apparent viscosity experienced by a test sphere moving through an otherwise quiescent suspension. Phys. Fluids 9 (1), 1622.
Arp, P. A. & Mason, S. G. 1977 The kinetics of flowing dispersions. IX. Doublets of rigid spheres. J. Colloid Interface Sci. 61, 4461.
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52 (2), 245268.
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 74, 129.
Batchelor, G. K. 1982 Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory. J. Fluid Mech. 119, 379407.
Batchelor, G. K. 1983 Diffusion in a dilute polydisperse system of interacting spheres. J. Fluid Mech. 131, 155175.
Batchelor, G. K. & Green, J. T. 1972 The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56 (2), 375400.
Batchelor, G. K. & Wen, C.-S. 1982 Sedimentation in a dilute polydisperse system of interacting spheres. Part 2. Numerical results. J. Fluid Mech. 124, 495528.
Bergenholtz, J., Brady, J. F. & Vicic, M. 2002 The non-Newtonian rheology of dilute colloidal suspensions. J. Fluid Mech. 456, 239275.
Brady, J. F. 1994 The long-time self-diffusivity in concentrated colloidal dispersions. J. Fluid. Mech. 272, 109133.
Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid. Mech. 348, 103139.
da Cunha, F. R. & Hinch, E. J. 1996 Shear-induced dispersion in a dilute suspension of rough spheres. J. Fluid Mech. 309, 211223.
Davis, R. H. 1992 Effects of surface roughness on a sphere sedimenting through a dilute suspension of neutrally buoyant spheres. Phys. Fluids A 4, 26072619.
Davis, R. H. & Hill, N. A. 1992 Hydrodynamic diffusion of a sphere sedimenting through a dilute suspension of neutrally buoyant spheres. J. Fluid Mech. 236, 513533.
Eckstein, E. C., Bailey, D. G. & Shapiro, A. H. 1977 Self-diffusion of particles in shear flow of a suspension. J. Fluid Mech. 79, 191208.
Einstein, A. 1906 On the theory of the Brownian movement. Ann. Phys. 19 (4), 371381.
Ermak, D. L. & McCammon, J. A. 1978 Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69, 13521360.
Fåhræus, R. & Lindqvist, T. 1931 The viscosity of the blood in narrow capillary tubes. Am. J. Physiol. 96, 562568.
Fixman, M. 1978 Simulation of polymer dynamics. I. General theory. J. Chem. Phys. 69, 15271537.
Gadala-Maria, F. & Acrivos, A. 1980 Shear-induced structure in a concentrated suspension of solid spheres. J. Rheol. 24 (6), 799814.
Ham, J. M. & Homsy, G. M. 1988 Hindered settling and hydrodynamic dispersion in quiescent sedimenting suspensions. Intl J. Multiphase Flow 14, 533546.
Hasimoto, H. 1959 On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (2), 317328.
Hoh, N. J. & Zia, R. N. 2015 Hydrodynamic diffusion in active microrheology of non-colloidal suspensions: the role of interparticle forces. J. Fluid Mech. 785, 189218.
Jeffrey, D. J. & Onishi, Y. 1984 Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J. Fluid Mech. 139, 261290.
Khair, A. S. & Brady, J. F. 2006 Single particle motion in colloidal dispersions: a simple model for active and nonlinear microrheology. J. Fluid Mech. 557, 73117.
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Koch, D. L. & Shaqfeh, E. S. G. 1991 Screening in sedimenting suspensions. J. Fluid Mech. 224, 275303.
Kops-Werkhoven, M. M. & Fijnaut, H. M. 1982 Dynamic behavior of silica dispersions studied near the optical matching point. J. Chem. Phys. 77, 22422253.
Kumar, A., Henríquez Rivera, R. G. & Graham, M. D. 2014 Flow-induced segregation in confined multicomponent suspensions: effects of particle size and rigidity. J. Fluid Mech. 738, 423462.
Leighton, D. & Acrivos, A. 1986 Viscous resuspension. Chem. Engng Sci. 41, 13771384.
Leighton, D. & Acrivos, A. 1987a Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. Fluid Mech. 177, 109131.
Leighton, D. & Acrivos, A. 1987b The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415439.
Lekkerkerker, H. N. W. & Dhont, J. K. G. 1984 On the calculation of the self-diffusion coefficient of interacting Brownian particles. J. Chem. Phys. 80, 57905792.
van Megen, W. & Underwood, S. M. 1989 Tracer diffusion in concentrated colloidal dispersions. III. Mean squared displacements and self-diffusion coefficients. J. Chem. Phys. 91 (1), 552559.
van Megen, W., Underwood, S. M. & Snook, I. 1986 Tracer diffusion in concentrated colloidal dispersions. J. Chem. Phys. 85 (7), 40654072.
Morris, J. F. & Brady, J. F. 1996 Self-diffusion in sheared suspensions. J. Fluid. Mech. 312, 223252.
Nicolai, H., Herzhaft, B., Hinch, E. J., Oger, L. & Guazzelli, E. 1995 Particle velocity fluctuations and hydrodynamic self-diffusion of sedimenting non-Brownian spheres. Phys. Fluids 7, 1223.
Nicolai, H., Peysson, Y. & Guazzelli, É. 1996 Velocity fluctuations of a heavy sphere falling through a sedimenting suspension. Phys. Fluids 8, 855862.
Phillips, R. J., Armstrong, R. C., Brown, R. A., Graham, A. L. & Abbott, J. L. 1992 A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys. Fluids A 4, 3040.
Rallison, J. M. & Hinch, E. J. 1986 The effect of particle interactions on dynamic light scattering from a dilute suspension. J. Fluid Mech. 167, 131168.
Russel, W. B. 1984 The Huggins coefficient as a means for characterizing suspended particles. J. Chem. Soc. Faraday Trans. 2 80 (1), 3141.
Squires, T. M. & Brady, J. F. 2005 A simple paradigm for active and nonlinear microrheology. Phys. Fluids 17 (7), 073101.
Stillinger, F. H. & Weber, T. A. 1982 Hidden structure in liquids. Phys. Rev. A 25, 978989.
Su, Y., Chu, H. C. W. & Zia, R. N.2015a Microviscosity, normal stress and osmotic pressure of Brownian suspensions by accelerated stokesian dynamics simulation. Unpublished.
Su, Y., Gu, K. L., Hoh, N. J. & Zia, R. N.2015b Force-induced diffusion of Brownian suspensions by accelerated stokesian dynamics simulation. Unpublished.
Swan, J. W. & Zia, R. N. 2013 Active microrheology – fixed-velocity versus fixed-force. Phys. Fluids 25 (8), 083303.
Tangelder, G. J., Teirlinck, H. C., Slaaf, D. W. & Reneman, R. S. 1985 Distribution of blood platelets flowing in arterioles. Am. J. Physiol. Heart Circ. Physiol. 248 (3), H318H323.
Wang, Y., Mauri, R. & Acrivos, A. 1996 The transverse shear-induced liquid and particle tracer diffusivities of a dilute suspension of spheres undergoing a simple shear flow. J. Fluid Mech. 327, 255272.
Zhang, K. & Acrivos, A. 1994 Viscous resuspension in fully developed laminar pipe flows. Intl J. Multiphase Flow 20, 579591.
Zia, R. N. & Brady, J. F. 2010 Single-particle motion in colloids: force-induced diffusion. J. Fluid Mech. 658, 188210.
Zia, R. N. & Brady, J. F. 2012 Microviscosity, microdiffusivity, and normal stresses in colloidal dispersions. J. Rheol. 56 (5), 11751208.
Zia, R. N. & Brady, J. F. 2013 Stress development, relaxation, and memory in colloidal dispersions: transient nonlinear microrheology. J. Rheol. 57 (2), 457492.
Zia, R. N., Landrum, B. J. & Russel, W. B. 2014 A micro-mechanical study of coarsening and rheology of colloidal gels: cage building, cage hopping, and Smoluchowski’s ratchet. J. Rheol. 58 (5), 11211157.
Zwanzig, R. 1983 On the relation between self-diffusion and viscosity of liquids. J. Chem. Phys. 79, 45074508.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Force-induced diffusion in suspensions of hydrodynamically interacting colloids

  • N. J. Hoh (a1) and R. N. Zia (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.