Skip to main content
×
Home
    • Aa
    • Aa

The long-time self-diffusivity in concentrated colloidal dispersions

  • John F. Brady (a1)
Abstract

The long-time self-diffusivity in concentrated colloidal dispersions is determined from a consideration of the temporal decay of density fluctuations. For hydrodynamically interacting Brownian particles the long-time self-diffusivity, Ds∞, is shown to be expressible as the product of the hydrodynamically determined short-time self-diffusivity, Ds(ϕ), and a contribution that depends on the distortion of the equilibrium structure caused by a diffusing particle. An argument is advanced to show that as maximum packing is approached the long-time self-diffusivity scales as Ds∞(ϕ) ∼ Ds0(ϕ)/g(2; ϕ), where g(2; ϕ) is the value of the equilibrium radial-distribution function at contact and ϕ is the volume fraction of interest. This result predicts that the longtime self-diffusivity vanishes quadratically at random close packing, ϕm ≈ 0.63, i.e. DsD0(1-ϕ/ϕm)2 as ϕ→ϕm, where D0 = kT/6πνa is the diffusivity of a single isolated particle of radius a in a fluid of viscosity ν. This scaling occurs because Ds0(ϕ) vanishes linearly at random close packing and the radial-distribution function at contact diverges as (1 -ϕ/ϕm)−1. A model is developed to determine the structural deformation for the entire range of volume fractions, and for hard spheres the longtime self-diffusivity can be represented by: Ds∞(ϕ) = Ds∞(ϕ)/[1 + 2ϕg(2;ϕ)]. This formula is in good agreement with experiment. For particles that interact through hard-spherelike repulsive interparticle forces characterized by a length b(> a), the same formula applies with the short-time self-diffusivity still determined by hydrodynamic interactions at the true or ‘hydrodynamic’ volume fraction ϕ, but the structural deformation and equilibrium radial-distribution function are now determined by the ‘thermodynamic’ volume fraction ϕb based on the length b. When b [Gt ] a, the long-time self-diffusivity vanishes linearly at random close packing based on the ‘thermodynamic’ volume fraction ϕbm. This change in behaviour occurs because the true or ‘hydrodynamic’ volume fraction is so low that the short-time self-diffusivity is given by its infinite-dilution value D0. It is also shown that the temporal transition from short- to long-time diffusive behaviour is inversely proportional to the dynamic viscosity and is a universal function for all volume fractions when time is nondimensionalized by a2/Ds∞(ϕ).

Copyright
References
Hide All
Ackerson, B. J. 1978 Correlations for interacting Brownian particles. II. J. Chem. Phys. 69, 684.
Batchelor, G. K. 1983 Diffusion in a dilute polydisperse system of interacting spheres. J. Fluid Mech. 131, 155 and Corrigendum J. Fluid Mech. 137, 1983, 467.
Beenakker, C. W. J. 1984 The effective viscosity of a concentrated suspension of spheres (and its relation to diffusion). Physica A 128, 48.
Beenakker, C. W. J. & Mazur, P. 1984 Diffusion of spheres in a concentrated dispersion II. Physica A 126, 349.
Berne,. B. J. & Pecora, R. 1976 Dynamic Light Scattering. Wiley.
Blaaderen van, A., Peetermans, J., Maret, G. & Dhont, J. K. G. 1992 Long-time self diffusion of spherical colloidal particles measured with fluorescence recovery after photobleaching. J. Chem. Phys. 96, 4591.
Bossis, G., Brady, J. F. & Mathis, C. 1988 Shear-induced structure in colloidal suspensions. I. Numerical simulations. J. Colloid Interface Sci. 126, 1.
Brady, J. F. 1993a Brownian motion, hydrodynamics and the osmotic pressure. J. Chem. Phys. 98, 3335.
Brady, J. F. 1993b The rheological behavior of concentrated colloidal dispersions. J. Chem. Phys. 99, 567.
Brady, J. F. 1994 Hindered diffusion in porous media. J. Colloid Interface Sci. (to be submitted).
Carnahan, N. F. & Starling, K. E. 1969 Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51, 635.
Cichocki, B. & Felderhof, B. U. 1992 Time-dependent self-diffusion in a semidilute suspension of Brownian particles. J. Chem. Phys. 96, 4669.
Cichocki, B. & Hinsen, K. 1990 Self and collective diffusion coefficients of hard sphere suspensions. Ber. Bunsenges. Phys. Chem. 94, 243.
Cichocki, B. & Hinsen, K. 1992 Dynamic computer simulation of concentrated hard sphere suspensions. Physica A 187, 145.
Cohen, E. G. D. & Schepper de, I. M. 1991 Note on transport processes in dense colloidal suspensions. J. Statist. Phys. 63, 241.
Hess, W. & Klein, R. 1983 Generalized hydrodynamics of systems of Brownian particles. Adv. Phys. 32, 173.
Kops-Werkhoven, M. M. & Fijnaut, H. M. 1982 Dynamic behavior of silica dispersions studied near the optical matching point. J. Chem. Phys. 77, 2242.
Ladd, A. J. C. 1990 Hydrodynamic transport coefficients of random dispersions of hard spheres. J. Chem. Phys. 93, 3483.
Leegwater, J. A. & Szamel, G. 1992 Dynamical properties of hard-sphere suspensions. Phys. Rev. A 46, 4999.
Medina-Noyola, M. 1988 Long-time self-diffusion in concentrated colloidal dispersion. Phys. Rev. Lett. 60, 2705.
Megen van, W., Underwood, S. M. & Snook, I. 1986 Tracer diffusion in concentrated colloidal dispersions. J. Chem. Phys. 85, 4065.
Megen van, W. & Underwood, S. M. 1989 Tracer diffusion in concentrated colloidal dispersions. III. Mean squared displacements and self-diffusion coefficients. J. Chem. Phys. 91, 552.
Ottewill, R. H. & Williams, N. St. J. 1987 Study of particle motion in concentrated dispersions by tracer diffusion. Nature 325, 232.
Phillips, R. J., Brady, J. F. & Bossis, G. 1988 Hydrodynamic transport properties of hard-sphere dispersions. I. Suspensions of freely mobile particles. Phys. Fluids 31, 3462.
Phung, T. N. 1993 Behavior of concentrated colloidal dispersions by Stokesian dynamics simulation. PhD thesis, California Institute of Technology.
Pusey, P. N. 1991 Colloidal suspensions. In Liquids, Freezing and Glass Transition (ed. J. P. Hansen, D. Levesque & J. Zinn-Justin). Elsevier.
Pusey, P. N. & Megen van, W. 1983 Measurement of the short-time self-mobility of particles in concentrated suspension. Evidence for many-particle hydrodynamic interactions. J. Phys. Paris 44, 285.
Qui, X., Ou-Yang, H. D., Pine, D. J. & Chaikin, P. M. 1988 Self-diffusion of interacting colloids far from equilibrium. Phys. Rev. Lett. 61, 2554.
Rallison, J. M. 1988 Brownian diffusion in concentrated suspensions of interacting particles. J. Fluid Mech. 186, 471.
Rallison, J. M. & Hinch, E. J. 1986 The effect of particle interactions on dynamic light scattering from a dilute suspension. J. Fluid Mech. 167, 131.
Russel, W. B. & Glendinning, A. B. 1981 The effective diffusion coefficient detected by dynamic light scattering. J. Chem. Phys. 74, 948.
Selim, M. S., Al-Naafa, M. A. & Jones, M. C. 1993 Brownian diffusion of hard spheres at finite concentrations. AIChE J. 39, 3.
Szamel, G. & Leegwater, J. A. 1992 Long-time self-diffusion coefficients of suspensions. Phys. Rev. A 46, 5012.
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186.
Veluwen van, A. & Lekkerkerker, H. N. W. 1988 NonGaussian behavior of the displacement statistics of interacting colloidal particles. Phys. Rev. A 38, 3758.
Werff van der, J. C., Kruif de, C. G., Blom, C. & Mellema, J. 1989 Linear viscoelastic behavior of dense hard-sphere dispersion. Phys. Rev. A 39, 418.
Woodcock, L. V. 1981 Glass transition in the hard sphere model and Kauzman's paradox. Ann. NY Acad. Sci. 37, 274.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 22 *
Loading metrics...

Abstract views

Total abstract views: 128 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st October 2017. This data will be updated every 24 hours.