(474j) Shear Induced Gradient Diffusivity of Dilute Suspensions

It is well known in both colloidal and non-colloidal suspensions that the gradient diffusivity differs from the self-diffusivity. The cause of this is simple: the particle flux arising from a concentration gradient may occur with the uniform expansion of a suspension, while the random walk of a tracer leading to the self-diffusivity requires particles to exchange positions. Since such motion is hindered by adjacent particles, the gradient diffusivity due to shear induced interactions (or Brownian motion in colloidal systems) exceeds the self-diffusivity by more than an order of magnitude at high concentrations. For Brownian systems, the two diffusivities are identical in the dilute limit. In this paper we explore the relative magnitude of the shear-induced diffusivities for non-colloidal suspensions in the dilute limit.

In the past, shear induced gradient diffusivities for unbounded suspensions due to multiparticle interactions have been calculated by Wang et al., 1998 and due to surface roughness by da Cunha & Hinch, 1996. When multiparticle interactions are the dominant source of dispersion, then both self-diffusivity and gradient diffusivity scale as Φ² with the concentration Φ and the gradient diffusivity is larger than the self-diffusivity. If there is sufficient surface roughness to permit particle contact in the dilute limit, the diffusivities scale linearly with concentration and the gradient diffusivity is larger than the self-diffusivity by a factor about 6. The dispersion mechanism in bounded dilute suspensions is complicated further by the wall stresslet interactions. Zurita-Gotor et al., 2007 showed that the presence of bounding walls can result in a class of open trajectories in which particles exchange positions which leads to a random walk. This mechanism is the dominant source of self-diffusion and explains the unexpectedly large self-diffusivities measured by Zarraga & Leighton, 2002 and Beimfohr et al., 1993 which scaled linearly with the concentration. Intriguingly, because the particles exchange positions via this wall reflection mechanism, it does not lead to any net flux of particles. Thus, while it dominates the shear-induced self-diffusivity in dilute suspensions, it cannot lead to any gradient diffusivity. Theory suggests, therefore, that the self-diffusivity is much greater than the gradient diffusivity in a dilute bounded suspension.

To explore this effect, we conducted experiments in a rotating parallel plate geometry in which the motion of several marked tracers in a suspension is optically tracked. By careful substitution of ZnCl₂ with ZnBr₂ in a ternary ZnCl₂, H2O, Triton X-100 system we created two miscible fluids with identical viscosities and slightly different densities bracketing the density of the PMMA particles. Such a layered system resulted in an initial condition where the particles were confined to a narrow layer far from either wall (H/2a = 20). The concentration profile of the expansion of this layer under steady shear is shown to have a self-similar solution with position and time which is used to characterize the gradient diffusivity. The experiments were carried out at average particle concentrations of Φ = 0.010, 0.020 and 0.035, where the effective concentration over which the expansion was observed was roughly twice the average value. The gradient diffusivity measured for these experiments scaled linearly with concentration and was significantly higher than the self-diffusivity. While this is consistent with the relative magnitude expected if diffusivities result from roughness, the magnitude was far larger than would be expected for these particles. We were unable to resolve this discrepancy.

References

Beimfohr, S., Looby, T., Leighton, D. T., Measurement of the shear-induced coefficient of self-diffusion in dilute suspensions. In Proceedings of the DOE/NSF Workshop on Flow of Particles and Fluids, Ithaca, NY, 1993.

da Cunha, F. R., Hinch, E. J., Shear-induced dispersion in a dilute suspension of rough spheres. J. Fluid Mech. 309, 211-223, 1996.

Wang, Y., Mauri, R., Acrivos, A. Transverse shear-induced gradient diffusion in a dilute suspension of spheres. J. Fluid Mech. 357, 279-287, 1998.

Zarraga, I.E., Leighton, D.T., Measurement of an unexpectedly large shear-induced self-diffusivity in a dilute suspension of spheres. Phys. Fluids, 14(7), 2194-2201, 2002.

Zurita-Gotor, M., Blawzdziewicz, J., Wajnryb, E. Swapping Trajectories: a new wall-induced cross-streamline particle migration mechanism in a dilute suspension of spheres. J. Fluid. Mech. 592, 447-469, 2007.