The long-time self-diffusivity in concentrated colloidal dispersions

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, D^s_∞, is shown to be expressible as the product of the hydrodynamically determined short-time self-diffusivity, D^s_0(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 D^s_∞(φ)~ D^s_0(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, φ_m ≈ 0.63, i.e. D^s_∞D_0(1-φ/φ_m)^2 as φ → φ_m, where D_0 = kT/6πηα is the diffusivity of a single isolated particle of radius α in a fluid of viscosity η. This scaling occurs because Ds_0(φ) vanishes linearly at random close packing and the radial-distribution function at contact diverges as (1 -φ/φ_m)^(−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: D^s_∞(φ) = D^s_∞(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(> α), 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 φ_b based on the length b. When b » α, the long-time self-diffusivity vanishes linearly at random close packing based on the 'thermodynamic' volume fraction φ_(bm). 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 D_0. 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 α^2/D^s_∞(φ).

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