Self-diffusion in sheared suspensions by dynamic simulation

Abstract

The behaviour of the long-time self-diffusion tensor in concentrated colloidal dispersions is studied using dynamic simulation. The simulations are of a suspension of monodisperse Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of shear and Brownian forces, and the volume fraction, φ. Here, Pe =γa 2 /D 0 , whereγ is the shear rate, a the particle size and D 0 = kT /6πηa is the Stokes-Einstein diffusivity of an isolated particle of size a with thermal energy kT in a solvent of viscosity η. Two simulations algorithms are used: Stokesian Dynamics for inclusion of the many-body hydrodynamic interactions, and Brownian Dynamics for suspensions without hydrodynamic interactions. A new procedure for obtaining high-quality diffusion data based on averaging the results of many short simulations is presented and utilized. At low shear rates, low Pe, Brownian diffusion due to a random walk process dominates and the characteristic scale for diffusion is the Stokes-Einstein diffusivity, D 0 . At zero Pe the diffusivity is found to be a decreasing function of φ. As Pe is slowly increased, O(Pe) and O(Pe 3/2 ) corrections to the diffusivity due to the flow are clearly seen in the Brownian Dynamics system in agreement with the theoretical results of Morris & Brady (1996). At large shear rates, large Pe, both systems exhibit diffusivities that grow linearly with the shear rate by the non-Brownian mechanism of shear-induced diffusion. In contrast to the behaviour at low Pe, this shear-induced diffusion mode is an increasing function of φ. Long-time rotational self-diffusivities are of interest in the Stokesian Dynamics system and show similar behaviour to their translational analogues. An off-diagonal long-time self-diffusivity, D xy , is reported for both systems. Results for both the translational and rotational D xy show a sign change from low Pe to high Pe due to different mechanisms in the two regimes. A physical explanation for the off-diagonal diffusivities is proposed.

Shear-induced self-diffusion in non-colloidal suspensions

“…In their approach, very small strains were also used, strains that according to figure 6(b) should overestimate the resulting mean-square displacement coefficient significantly. As is apparent from figure 4, however, this is not the case and our results are in general agreement with the results reported by Foss & Brady (1999). We believe that this is the result of the very small systems used by Foss & Brady (1999); N = 27 for all of their simulations.…”

“…As is apparent from figure 4, however, this is not the case and our results are in general agreement with the results reported by Foss & Brady (1999). We believe that this is the result of the very small systems used by Foss & Brady (1999); N = 27 for all of their simulations. As was mentioned in the previous subsection, a very strong N dependence is present for such a small number of particles and the diffusivities are underestimated significantly.…”

“…Although qualitatively most sets of experimental or simulation results are in agreement, quantitatively discrepancies are present that need to be explained and examined in more detail. As was mentioned in § 3.1, the notion that the long-time self-diffusivities can be calculated for strains that are not particularly long has been used, and most of the recent experimental and simulation results presented in figures 4 and 5 are calculated over a large number of configurations, but for relatively small strains (Breedveld et al 1998(Breedveld et al , 2001Foss & Brady 1999). Unfortunately, as shown below and as was discussed by Marchioro & Acrivos (2001), the strains used for these studies are too small and therefore do not correspond to the linear regime, as a result they overestimate the diffusivity.…”

“…In a simulation where the position of each particle is known exactly and the particle velocities are already decomposed into an affine (U ∞ ) and a non-affine (U h ) part, each term of equation (4.2) is known exactly and can be manipulated separately. Foss & Brady (1999) used this decomposition to claim that the long-time self-diffusion coefficient in the x-direction can be calculated from only the non-affine part of the motion (denoted X h in (4.2)), specifically from :…”

“…The presence of an external velocity field makes the determination of diffusive displacements in the flow, or longitudinal, direction difficult, and only very limited experimental results are available for these components (Breedveld 2000). Foss & Brady (1999) calculated the longitudinal selfdiffusion coefficient for a system of Brownian particles at very high Péclet numbers by following the theoretical analysis of Morris & Brady (1996) which suggested that the affine shearing motion could be subtracted directly at each instant in time and the diffusivity could be calculated by taking into account only the non-affine displacements. Unfortunately, as we shall show, this is not correct and the affine displacement couples with the non-affine displacement to give an additional contribution to the longitudinal long-time self-diffusivity.…”