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

Wavevector dependent friction factors, which together with the elastic prop erties determine the hydrodynamic damping of harmonic lattice waves in coll oidal crystals, are calculated for cubic arrays of rigid spheres at low, in termediate and high sphere volume fractions, including simple cubic (SC), b ody-centred cubic (BCC) and face-centred cubic (FCC) lattices. Exact numeri cal 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 w hole range of volume fractions. An expression suitable for low volume fract ions is derived that approximates the converged numerical data within an er ror of only 3%, for volume fractions up to 75% of the close packing concent ration for all of the three cubic packing types. Spin viscosities of extrem ely 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 hydrodynami c interactions are determined that yield a first correction to the lubricat ion expressions, appropriate for the three cubic lattices. Two types of bod y-centred tetragonal (BCT) arrays are considered. The first type (BCT1) con sists of aligned rows of close-touching spheres, where variations of the vo lume fraction can be accomplished by changing the distance between these ro ws. The second type (BCT3) has a constant structure, such that the concentr ation can be varied by changing the size of the spheres. The two characteri stic drag coefficients are calculated numerically for steadily sedimenting BCT arrays of both types and at all concentrations. Likewise two characteri stic spin viscosities are computed for these structures. Analytical results suitable to dilute BCT3 arrays are presented for drag coefficients and spi n viscosities. The expressions for the spin viscosities represent an approx imation within 3%, for volume fractions up to 60% of the close packing conc entration. (C) 1999 Elsevier Science B.V. All rights reserved.