Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T19:48:01.577Z Has data issue: false hasContentIssue false
  • English
  • Français

Single-particle motion in colloids: force-induced diffusion

Published online by Cambridge University Press:  09 June 2010

ROSEANNA N. ZIA  and
JOHN F. BRADY
ROSEANNA N. ZIA*
Affiliation:
Department of Mechanical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
JOHN F. BRADY
Affiliation:
Department of Mechanical Engineering, California Institute of Technology, Pasadena, CA 91125, USA Department of Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the fluctuating motion of a Brownian-sized probe particle as it is dragged by a constant external force through a colloidal dispersion. In this nonlinear-microrheology problem, collisions between the probe and the background bath particles, in addition to thermal fluctuations of the solvent, drive a long-time diffusive spread of the probe's trajectory. The influence of the former is determined by the spatial configuration of the bath particles and the force with which the probe perturbs it. With no external forcing the probe and bath particles form an equilibrium microstructure that fluctuates thermally with the solvent. Probe motion through the dispersion distorts the microstructure; the character of this deformation, and hence its influence on the probe's motion, depends on the strength with which the probe is forced, Fext, compared to thermal forces, kT/b, defining a Péclet number, Pe = Fext/(kT/b), where kT is the thermal energy and b the bath particle size. It is shown that the long-time mean-square fluctuational motion of the probe is diffusive and the effective diffusivity of the forced probe is determined for the full range of Péclet number. At small Pe Brownian motion dominates and the diffusive behaviour of the probe characteristic of passive microrheology is recovered, but with an incremental flow-induced ‘microdiffusivity’ that scales as Dmicro ~ DaPe2φb, where φb is the volume fraction of bath particles and Da is the self-diffusivity of an isolated probe. At the other extreme of high Péclet number the fluctuational motion is still diffusive, and the diffusivity becomes primarily force induced, scaling as (Fext/η)φb, where η is the viscosity of the solvent. The force-induced microdiffusivity is anisotropic, with diffusion longitudinal to the direction of forcing larger in both limits compared to transverse diffusion, but more strongly so in the high-Pe limit. The diffusivity is computed for all Pe for a probe of size a in a bath of colloidal particles, all of size b, for arbitrary size ratio a/b, neglecting hydrodynamic interactions. The results are compared with the force-induced diffusion measured by Brownian dynamics simulation. The theory is also compared to the analogous shear-induced diffusion of macrorheology, as well as to experimental results for macroscopic falling-ball rheometry. The results of this analysis may also be applied to the diffusive motion of self-propelled particles.

JFM classification

Complex Fluids: Colloids Multiphase and Particle-laden Flows: Particle/fluid flow Non-Newtonian Flows: Rheology Low-Reynolds-number flows: Stokesian dynamics Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 658 , 10 September 2010 , pp. 188 - 210
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abbot, J. R., Graham, A. L., Mondy, L. A. & Brenner, H. 1997 Dispersion of a ball settling through a quiescent neutrally buoyant suspension. J. Fluid Mech. 361, 309–331.CrossRefGoogle Scholar
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 74, 1–29.CrossRefGoogle Scholar
Bergenholtz, J., Brady, J. F. & Vicic, M. 2002 The non-Newtonian rheology of dilute colloidal dispersions. J. Fluid Mech. 456, 239–275.CrossRefGoogle Scholar
Brady, J. F. 1994 The long-time self-diffusivity in concentrated colloidal dispersions. J. Fluid Mech. 272, 109–133.CrossRefGoogle Scholar
Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103–139.CrossRefGoogle Scholar
Breedveld, J., van den Ende, D., Tripathi, A. & Acrivos, A. 1998 The measurement of the shear-induced particle and fluid tracer diffusivities in concentrated suspensions by a novel method. J. Fluid Mech. 375, 297–318.CrossRefGoogle Scholar
Carpen, I. C. & Brady, J. F. 2005 Microrheology of colloidal dispersions by Brownian dynamics simulations. J. Rheol. 49, 1483–1502.CrossRefGoogle Scholar
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, 513–533.CrossRefGoogle Scholar
Habdas, P., Schaar, D., Levitt, A. C. & Weeks, E. R. 2004 Forced motion of a probe particle near the colloidal glass transition. Europhys. Lett. 67, 477–483.CrossRefGoogle Scholar
Heath, J. R., Davis, M. E. & Hood, L. 2009 Nanomedicine: revolutionizing the fight against cancer. Sci. Am. 300, 44.CrossRefGoogle Scholar
Heyes, D. M. & Melrose, J. R. 1993 Brownian dynamics simulations of model hard-sphere suspensions. J. Non-Newton. Fluid Mech. 46, 1–28.CrossRefGoogle Scholar
Janke, C., Rogowski, K., Wloga, D., Regnard, C., Kajava, A. V., Strub, J. M., Temurak, N., van Dijk, J., Boucher, D., van Dorseelaer, A., Suryavanshi, S., Gaertig, J. & Edde, B. 2005 Tubulin polyglutamylase enzymes are members of the TTL domain protein family. Science 308, 1758–1762.CrossRefGoogle ScholarPubMed
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, 73–117.CrossRefGoogle Scholar
Khair, A. S. & Brady, J. F. 2008 Microrheology of colloidal dispersions: shape matters. J. Rheol. 52, 165–196.CrossRefGoogle Scholar
Leighton, D. & Acrivos, A. 1987 Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. Fluid Mech. 177, 109–131.CrossRefGoogle Scholar
MacKintosh, F. C. & Schmidt, C. F. 1999 Microrheology. Curr. Opin. Colloid Interface Sci. 4, 300–307.CrossRefGoogle Scholar
Meyer, A., Marshall, A., Bush, B. G. & Furst, E. M. 2005 Laser tweezer microrheology of a colloidal suspension. J. Rheol. 50, 77–92.CrossRefGoogle Scholar
Morris, J. F. & Brady, J. F. 1996 Self-diffusion in sheared suspensions. J. Fluid Mech. 312, 223–252.CrossRefGoogle Scholar
Shirai, Y., Osgood, A. J., Zhao, Y., Kelly, K. F. & Tour, J. M. 2005 Directional control in thermally driven single-molecule nanocars. Nano Lett. 5, 2330–2334.CrossRefGoogle ScholarPubMed
Squires, T. M. 2008 Nonlinear microrheology: bulk stresses versus direct interactions. Langmuir 24, 1147–1159.CrossRefGoogle ScholarPubMed
Squires, T. M. & Brady, J. F. 2005 A simple paradigm for active and nonlinear microrheology. Phys. Fluids 17, 073101-1–073101-2.CrossRefGoogle Scholar
Wilson, L. G., Harrison, A. W., Schofield, A. B., Arlt, J. & Poon, W. C. K. 2009 Passive and active microrheology of hard-sphere colloids. J. Phys. Chem. 113, 3806–3812.CrossRefGoogle ScholarPubMed