Sedimentation in a dilute polydisperse system of interacting spheres. Part 2. Numerical results

Abstract

Analytical formulae or the effect of interaction between pairs of rigid spherical particles on the mean velocity of each species in a statistically homogeneous dilute polydisperse system were given in Part 1 (Batchelor 1982), and are here evaluated numerically. We have calculated the pair-distribution function and the associated value of the sedimentation coefficient for a wide variety of conditions of the two interacting species, including different values of the ratio of the radii of the spheres (λ), different values of the ratio of their (reduced) densities (γ), small and large values of the Péclet number of the interaction, and different forms of the potential of the mutual force exerted directly between the two spheres. Values of λ and γ such that some of the trajectories of one sphere centre moving under gravity alone relative to another are of finite length lie outside the scope of the calculations at large Péclet number, and the change of behaviour across the boundary of this excluded set of values leads to a complicated dependence of the sedimentation coefficient on λ and γ. At small Péclet number the behaviour is simpler, and a formula which represents the calculated values of the sedimentation coefficient over the whole range of values of λ and γ (on which the dependence is known to be linear) with fair accuracy in the absence of interparticle forces is devised. Our calculations of the effect of an interparticle force were based on the assumption of a high Coulomb barrier at a certain sphere separation which could be varied, and a van der Waals attractive force at larger separations. It appears that the direct contribution to the sedimentation coefficient made by gravity is always appreciably larger than that made either by relative Brownian diffusion of the two interacting spheres or by the interparticle force. However, all three of these (effective) forces normally have a significant influence on the pair-distribution function and thereby also affect the sedimentation coefficient indirectly. Some published observations of the mean particle velocity in monodisperse systems are interpreted in the light of the present calculations of the effect of interparticle forces.