## A molecular theory of the Stokes-Einstein relationship

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## Abstract

There is now much experimental evidence for a simple relationship between the translational and rotational diffusion constants of a molecule in a liquid and the liquid's shear viscosity. Such a relationship was derived by Einstein using the equations of hydrodynamics, treating the liquid as a continuous incompressible medium, the diffusing molecule as a macroscopic body and applying boundary conditions to specify their interaction. Whilst the form of this relationship is preserved in the experimental findings, the detailed dependence on the shape and size of the molecule is often at variance with the hydrodynamic prediction. To understand these effects and also to explain why the hydrodynamic theory should be as successful as it is, a microscopic theory is required. This would replace the boundary conditions by quantities depending on the liquid structure and on the intermolecular forces.

By using the Mori generalised Langevin equation, molecular expressions for the velocity auto-correlation function (VACF) and the angular velocity auto-correlation function (AVACF) for a spherical particle in a liquid were obtained. In the limit of a large and massive particle, by taking an appropriate choice of the potential of interaction between the particle and the fluid particles, we recover exactly the hydrodynamic results for the VACF and the AVACF for both slip and stick boundary conditions in compressible and incompressible fluids. Furthermore, in the case of continuous potentials, we obtain the correct short time behaviour in the VACF as opposed to the cusp predicted by hydrodynamics.

Next we consider the limit of the diffusing particle being much less massive than a fluid molecule - the Lorentz Gas limit. We show that our theory is fairly successful at predicting the diffusion constant as a function of fluid density and predicts a negative t^{-5/2} tail in the VACF, in agreement with Kinetic Theory.

In order to apply the full equations to a diffusing particle of arbitrary mass and size, numerical problems arise in solving the coupled integral equations, but by making an extra approximation commonly made in mode-coupling calculations a qualitative idea of the dependence of the Diffusion constant upon the particle's mass and radius was obtained.