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The long-time self-diffusivity in concentrated colloidal dispersions

Published online by Cambridge University Press:  26 April 2006

John F. Brady
John F. Brady*
Affiliation:
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA91125, USA
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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. Ds∞D0(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 ≫ 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).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 272 , 10 August 1994 , pp. 109 - 134
Copyright
Copyright © Cambridge University Press 1994

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References

Ackerson, B. J. 1978 Correlations for interacting Brownian particles. II. J. Chem. Phys. 69, 684.Google Scholar
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.Google Scholar
Beenakker, C. W. J. 1984 The effective viscosity of a concentrated suspension of spheres (and its relation to diffusion). Physica A 128, 48.Google Scholar
Beenakker, C. W. J. & Mazur, P. 1984 Diffusion of spheres in a concentrated dispersion II. Physica A 126, 349.Google Scholar
Berne, B. J. & Pecora, R. 1976 Dynamic Light Scattering. Wiley.Google Scholar
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.Google Scholar
Bossis, G., Brady, J. F. & Mathis, C. 1988 Shear-induced structure in colloidal suspensions. I. Numerical simulations. J. Colloid Interface Sci. 126, 1.10.1016/0021-9797(88)90093-8CrossRefGoogle Scholar
Brady, J. F. 1993a Brownian motion, hydrodynamics and the osmotic pressure. J. Chem. Phys. 98, 3335.Google Scholar
Brady, J. F. 1993b The rheological behavior of concentrated colloidal dispersions. J. Chem. Phys. 99, 567.Google Scholar
Brady, J. F. 1994 Hindered diffusion in porous media. J. Colloid Interface Sci. (to be submitted).Google Scholar
Carnahan, N. F. & Starling, K. E. 1969 Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51, 635.Google Scholar
Cichocki, B. & Felderhof, B. U. 1992 Time-dependent self-diffusion in a semidilute suspension of Brownian particles. J. Chem. Phys. 96, 4669.Google Scholar
Cichocki, B. & Hinsen, K. 1990 Self and collective diffusion coefficients of hard sphere suspensions. Ber. Bunsenges. Phys. Chem. 94, 243.Google Scholar
Cichocki, B. & Hinsen, K. 1992 Dynamic computer simulation of concentrated hard sphere suspensions. Physica A 187, 145.Google Scholar
Cohen, E. G. D. & Schepper de, I. M. 1991 Note on transport processes in dense colloidal suspensions. J. Statist. Phys. 63, 241.Google Scholar
Hess, W. & Klein, R. 1983 Generalized hydrodynamics of systems of Brownian particles. Adv. Phys. 32, 173.Google Scholar
Kops-Werkhoven, M. M. & Fijnaut, H. M. 1982 Dynamic behavior of silica dispersions studied near the optical matching point. J. Chem. Phys. 77, 2242.Google Scholar
Ladd, A. J. C. 1990 Hydrodynamic transport coefficients of random dispersions of hard spheres. J. Chem. Phys. 93, 3483.Google Scholar
Leegwater, J. A. & Szamel, G. 1992 Dynamical properties of hard-sphere suspensions. Phys. Rev. A 46, 4999.Google Scholar
Medina-Noyola, M. 1988 Long-time self-diffusion in concentrated colloidal dispersion. Phys. Rev. Lett. 60, 2705.Google Scholar
Megen van, W., Underwood, S. M. & Snook, I. 1986 Tracer diffusion in concentrated colloidal dispersions. J. Chem. Phys. 85, 4065.Google Scholar
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.Google Scholar
Ottewill, R. H. & Williams, N. St. J. 1987 Study of particle motion in concentrated dispersions by tracer diffusion. Nature 325, 232.Google Scholar
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.Google Scholar
Phung, T. N. 1993 Behavior of concentrated colloidal dispersions by Stokesian dynamics simulation. PhD thesis, California Institute of Technology.Google Scholar
Pusey, P. N. 1991 Colloidal suspensions. In Liquids, Freezing and Glass Transition (ed. Hansen, J. P., Levesque, D. & Zinn-Justin, J.). Elsevier.Google Scholar
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.Google Scholar
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.Google Scholar
Rallison, J. M. 1988 Brownian diffusion in concentrated suspensions of interacting particles. J. Fluid Mech. 186, 471.Google Scholar
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.Google Scholar
Russel, W. B. & Glendinning, A. B. 1981 The effective diffusion coefficient detected by dynamic light scattering. J. Chem. Phys. 74, 948.Google Scholar
Selim, M. S., Al-Naafa, M. A. & Jones, M. C. 1993 Brownian diffusion of hard spheres at finite concentrations. AIChE J. 39, 3.10.1002/aic.690390103CrossRefGoogle Scholar
Szamel, G. & Leegwater, J. A. 1992 Long-time self-diffusion coefficients of suspensions. Phys. Rev. A 46, 5012.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186.Google Scholar
Veluwen van, A. & Lekkerkerker, H. N. W. 1988 NonGaussian behavior of the displacement statistics of interacting colloidal particles. Phys. Rev. A 38, 3758.Google Scholar
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.Google Scholar
Woodcock, L. V. 1981 Glass transition in the hard sphere model and Kauzman's paradox. Ann. NY Acad. Sci. 37, 274.Google Scholar