Journal of Fluid Mechanics

The long-time self-diffusivity in concentrated colloidal dispersions

John F. Brady a1
a1 Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA

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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[infty infinity], 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[infty infinity](φ) [similar] 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 [approximate] 0.63, i.e. Ds[infty infinity]D0(1-φ/φm)2 as φ[rightward arrow]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[infty infinity](φ) = Ds[infty infinity](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 [dbl greater-than sign] 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[infty infinity](φ).

(Published Online April 26 2006)
(Received July 5 1993)
(Revised February 9 1994)