Effect of microstructure on the viscosity of hard sphere dispersions and modulus of composites

Abstract

Results from theory and experiment in the literature for the viscosity of dispersions of monodisperse hard spheres are contrasted to highlight the effects of particle microstructure, such as ordered spatial distributions versus random or partially aggregated dispersions. Hard spheres, comprising a simple ideal limit with no interparticle forces other than infinite repulsion at contact, are achieved experimentally by either minimizing van der Waals attractions or negating them with short range repulsions. For dispersions, the balance between viscous forces and Brownian motion, as gauged by the Peclet number Pe, determines the microstructure and, hence, the viscosity. This results in a progression from isotropic equilibrium at Pe = 0, a small perturbation oriented in the principle direction of strain for Pe⪡ 1, two-dimensional anisotropy for Pe⪢ 1, and a return to isotropy, albeit hydrodynamicaUy dominated, at Pe = ∞. The viscosities for hard spheres vary in the order η hyd( Pe = ∞)⩾ η 0( Pe⪡1) ⩾ η ∞( Pe ⪢ 1)⩾η' ∞'( Pe = 0). The first three represent steady shear viscosities, while the last results from high frequency, small amplitude oscillations. At low Peclet numbers, both aggregation owing to short range attractions and long range repulsions increase the steady shear viscosity. With permanent aggregates the effect persists to Pe = ∞, with the data available for η hyd indicating a monotonic increase with degree of aggregation. Hence, these results for hard spheres generally represent limiting cases. A fundamental connection also exists between composites of hard particles in an incompressible, elastic continuous phase and dispersions of spheres with a corresponding microstructure. The analogy between Hookean elasticity and Stokes flow means that the static shear modulus of the former, normalized by the modulus of the continuous phase, equals the high frequency limiting relative viscosity of the latter. A combination of data and rigorous theory demonstrates that the modulus of a composite decreases in the order: simple cubic > random > body centered cubic > face centered cubic, that is, in order of increasing distance between nearest neighbors at the same volume fraction.