Power law mass dependence of diffusion: A mode coupling theory analysis

The self-diffusion coefficient of a tagged molecule is known to exhibit a weak mass dependence, especially for solutes with size comparable to or larger than the size of the solvent molecules. Sometimes this mass dependence can be fitted to a power law, with a small exponent, less than 0.1. This weak mass dependence has often been considered as supportive of the hydrodynamic picture (that is, the Stokes-Einstein relation) of diffusion rather than the kinetic theory approach, which predicts a stronger mass dependence, for example, via the Enskog theory. Neither can explain the weak power-law mass dependence. In order to understand this, we have carried out a mode coupling theory (MCT) analysis of diffusion. It is found that a straightforward application of the existing mode coupling theory expressions lead to an inaccurate mass dependence-it predicts an increase of diffusion coefficient with an increase of the mass. We find that this is because of the inadequate description of the initial decay of the collective contributions to the friction. We have proposed a new prescription to accurately describe the short time dynamics of the density and the current term. In addition, we have modified the existing MCT by imposing the full self-consistency between the frequency-dependent friction and the mean square displacement over the whole time and frequency plane. Previously the self-consistency was performed only at the zero frequency level between the zero frequency friction and the diffusion coefficient. With these two generalizations, the mode coupling theory is found to provide a fairly accurate description of the mass dependence. In particular, the theory can correctly reproduce the power-law dependence of solvent-solute diffusion ratio on solute-solvent mass ratio, observed in computer simulations of Bearman and Jolly [Mol. Phys. 44, 665 (1981)]. Another important result is that the current mode is found to play no significant role in determining the diffusion. Thus the hydrodynamic argument of weak mass dependence has little validity for same size solute-solvent systems.