Microscopic flow induced by a diffusing particle
R. Vuilleumier
Pôle de Physico-Chimie Théorique, Ecole Normale Supérieure, UMR 8640 CNRS-ENS-UPMC, 24 rue Lhomond, 75005 Paris, France.


The diffusion of a particle in a simple liquid is very often depicted by the Stokes-Einstein relation [1], which relates the diffusion coefficient to the hydrodynamic force felt by a sphere moving in a viscous fluid. While this relation is widely used, shadow zones still remain like the boundary conditions applied or the definition of the hydrodynamic radius. In many cases, the approach to address this problem numerically consisted in the calculation of the diffusion coefficient for different radii so as to extract the boundary conditions by adaptation of the Stokes-Einstein formula [2].
Here we propose a novel approach, a direct calculation of the hydrodynamic flux around the moving sphère from molecular dynamics. It then allows a direct comparison to the hydrodynamic approximation and to study the boundary conditions for this flux. Amazingly, we recovered the hydrodynamic solution at extremely short distances (in the few Ångströms range). “Slip” boundary conditions are found for this flux but at the microscopic level the condition of zero normal velocity at contact is violated. This arises from the fluctuations of the moving particle. We will present results on a Lennard-Jones fluid, liquid water, liquid carbon tetrachloride and for an experimental granular fluid [3]. Using Mori-Zwanzig projection operator techniques, we have further developed a theoretical model of the normal velocity at contact, which was successfully tested against simulations at different ratio of the solute and solvent molecular masses. As expected, the zero normal velocity at contact is recovered for large mass ratio.

REFERENCES
[1] L.D. Landau and E.M. Lifshitz, “Fluid Mechanics”, 2nd ed. (Pergamon, London, 1980), pp. 58-67.
[2] J.R. Schmidt and Skinner, ”Hydrodynamic boundary conditions, the Stokes-Einstain law, and long time tails in the Brownian limit”, Journal of Chemical Physics, 119, 8062 (2003).
[3] P.M. Reis, R.A. Ingale, M.D. Shattuck, Physical Review Letter, 96, 258001 (2006).