Effective viscosity of dense colloidal crystals

Abstract

An exact scheme is presented to determine the effective viscosity tensor for periodic arrays of hard spherical particles suspended in a Newtonian fluid. In the highly symmetric case of cubic lattices this tensor is characterized by only two parameters. These parameters are calculated numerically for the three cubic lattice types and for the whole range of volume fractions. The correctness of the present method and its numerical implementation is confirmed by a comparison with the numerical and analytical results known from the literature. Some regular terms are determined that enter singular perturbation expansions suitable for high concentrations. Previous results for these terms are shown to be highly inaccurate. The modified expansions approach the exact numerical results over a range of densities extending to relatively low concentrations. The effective viscosity is examined for simple tetragonal (st) lattices and the results for various structures of the st type can be qualitatively understood on the basis of the motion of the spheres in response to the ambient shear flow. The angular velocity of the spheres-relative to the shear flow-is shown to be nonzero for certain orientations of the st lattice with respect to the shear flow, in contrast to what has been known for cubic arrays. Finite viscosities are found in most cases where the particles are in contact as they are allowed to move in either rigid planar or linelike structures, or they can perform a smooth rolling motion. The only occurrence where the viscosity diverges for a st structure, or equally any other Bravais lattice, is for the case of close packing. Moreover, the concentration-dependent shear viscosity is determined for a variety of microstructures and the results are compared with recent data obtained from experiments on ordered hard-sphere suspensions.