Summary
Results from theory and experiment in the literature for the viscosity of dispersions of monodisperse hard spheres are contrasted to illustrate the effects of particle microstructure. Hard spheres comprise a simple ideal limit, with no inter-particle forces other than infinite repulsion at contact, and are achieved experimentally by either minimizing van der Waals attractions or negating them with short range repulsions. “Real” systems, with either additional longer range repulsions or significant short range attraction, will exhibit higher viscosities (but for different reasons); so the results discussed here 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.
For hard sphere 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 with isotropic equilibrium at Pe = 0, a small perturbation oriented with the principle direction of strain for Pe ccl, two dimensional anisotropy for Pe >>l, and a return to isotropy, albeit hydrodynamically dominated, at Pe = M. The corresponding viscosities vary as Pe = 0 2 Pe >> 1 I Pe << 1 I Pe = 00~ The high frequency limiting viscosity/static shear modulus depends on the degree of mechanical coupling between particles and, hence, increases in the order of face-centered cubic < body-centered cubic < random < simple cubic.
Our theoretical effort to develop means for predicting these and other aspects of the rheology of concentrated dispersions continues to progress. The tests of thermodynamic closures for handling three-body couplings through the pair potential have produced the first such predictions of viscosities exceeding that of the solvent by a couple of orders of magnitude, These pertain to both hard sphere and Yukawa inter-particle potentials and will soon be tested against Brownian dynamics simulations, The analogous closure for many-body hydrodynamics still eludes our grasp but the quest continues. .