Hydrodynamic transport coefficients of random dispersions of hard spheres

Abstract

Accurate values for the hydrodynamic transport properties of random dispersions of hard spheres have been determined by numerical simulation. The many-body hydrodynamic interactions are calculated from a multipole-moment expansion of the force density on the surface of the solid particles; the singular lubrication forces are included exactly for pairs of particles near contact. It has been possible to calculate the transport properties of small periodic systems, at all packing fractions, with uncertainties of less than 1%; but for larger systems we are limited computationally to lower order, and therefore less accurate, moment approximations to the induced force density. Nevertheless, since the higher-order moment contributions are short range they are essentially independent of system size and we can use small system data to correct our results for larger systems. Numerical calculations show that this is a reliable and accurate procedure. The ensemble-averaged mobility tensors are strongly dependent on system size, with deviations from the thermodynamic limit varying as N−1/3. We show that these deviations can be accounted for by a straightforward calculation based on the length of the unit cell and the suspension viscosity and mobility. The remaining number dependencies are small. Problems involved in implementing this method for larger numbers of particles are considered, and alternative methods are discussed.

Inclusion of lubrication forces in dynamic simulations

“…This, however, is computationally expensive owing to the integration required by (3). Since f~Ub will be chosen to be nonzero only in a small region near the gap between the particles, we can approximate (3) by the first two terms in the Taylor series expansion…”

“…where and The lubrication force density rub in (3) must be determined as a function of the relative motion of the two particles through appropriate asymptotic analysis as will be done later in this article. In determining the strength of multipoles used in defining the differential operator viii,…”

“…I). Their method was also employed by Ladd,3 who showed that this approximation can be systematically improved by increasing N s • The computationally intensive step in this method for large N, the number of interacting particles, is the inversion of the mobility matrix of size N t = (3N;-1)N, which requires roughly O(Nt) computations.…”

“…It can be proved that this applies even to particles a and y. The proof rests on the observation that, while the integrand in (3) is singular at x=y, u~ub itself is continuous at the surface of particle a. Thus, for the purpose of satisfying the boundary conditions on that particle, we first move the singularities at sa radially outward from the surface by a small distance O.…”

Sedimentation of Strongly and Weakly Charged Colloidal Particles: Prediction of Fractional Density Dependence

“…This distortion leads to an additional change in the sedimentation velocity. Nevertheless, simulations of hard sphere suspensions show that the differences between the short-time and the long-time sedimentation velocities due to memory effects are rather small [35,36].…”