Hydrodynamic interactions in colloidal crystals - (II). Application to dense cubic and tetragonal arrays

Wavevector dependent friction factors, which together with the elastic properties determine the hydrodynamic damping of harmonic lattice waves in colloidal crystals, are calculated for cubic arrays of rigid spheres at low, intermediate and high sphere volume fractions, including simple cubic (SC), body-centred cubic (BCC) and face-centred cubic (FCC) lattices. Exact numerical data are presented for the rheological coefficient appropriate to pure vortex motion of particles, the so-called spin viscosity. These results for the spin viscosity are obtained for all three cubic lattices and for the whole range of volume fractions. An expression suitable for low volume fractions is derived that approximates the converged numerical data within an error of only 3%, for volume fractions up to 75% of the close packing concentration for all of the three cubic packing types. Spin viscosities of extremely dense cubic arrays are calculated on basis of lubrication theory which yields new expressions for BCC and FCC lattices, while reproducing a known result for the SC case. Constant contributions of many-particle hydrodynamic interactions are determined that yield a first correction to the lubrication expressions, appropriate for the three cubic lattices. Two types of body-centred tetragonal (BCT) arrays are considered. The first type (BCT1) consists of aligned rows of close-touching spheres, where variations of the volume fraction can be accomplished by changing the distance between these rows. The second type (BCT3) has a constant structure, such that the concentration can be varied by changing the size of the spheres. The two characteristic drag coefficients are calculated numerically for steadily sedimenting BCT arrays of both types and at all concentrations. Likewise two characteristic spin viscosities are computed for these structures. Analytical results suitable to dilute BCT3 arrays are presented for drag coefficients and spin viscosities. The expressions for the spin viscosities represent an approximation within 3%, for volume fractions up to 60% of the close packing concentration. (C) 1999 Elsevier Science B.V. All rights reserved.