Molecular insights into the boundary conditions in the Stokes-Einstein relation

In order to mimic the Brownian particle in liquid, molecular dynamics calculations of dilute solutions of spherical fullerene molecules with various sizes in liquid Ar were performed. To establish the scaling equation for self-diffusion coefficient, $D$, of the fullerenes, the dependence of $D$ was examined on the mass ratio of solute to solvent and on the energy-parameter ratio used in the Lennard-Jones potentials. The dependence on the energy-parameter ratio remains up to ${\mathrm{C}}_{540}$, whereas $D$ rapidly becomes independent of the mass ratio with increasing mass of the solute. The product of the scaling equations obtained for the $D$ of the solute and for shear viscosity, ${\eta }_{\mathrm{sv}}$, for the solvent gives a relation which replaces the Stokes-Einstein relation based on the hydrodynamics. The present expression does not need both the boundary conditions and the hydrodynamic particle size, but instead the energy-parameter ratio, packing fraction of solvent, and bare size of solute. From the viewpoint of the tackiness at the boundary, the cage correlation function around the diffusing particle was examined; it was found that the decay time of the function depends mainly on the the energy-parameter ratio. Therefore, the energy-parameter ratio accounts for the main part of both the boundary conditions and the hydrodynamic particle size in the Stokes equation, which have so far been ill-defined in any molecular theories.

Figure 1

Dependence of the self-diffusion coefficient of some sizes of fullerenes and Kr in liquid Ar on (a) the mass ratio and (b) the energy-parameter ratio of solute to solvent. In (b), the plots at ${\epsilon }_1/{\epsilon }_2$ = 1 mean simply those for the extrapolated values of $D_1$.

Figure 2

Plots of $D_1{\eta }_{\mathrm{sv}}/T$ vs (a) ${\sigma }_1^{-1}$ and (b) ${({\sigma }_1/{\sigma }_2)}^{-1}{({\epsilon }_1/{\epsilon }_2)}^{-0.2}{(N/V)}^{1/3}$. The multiple plots for the same symbol mean the different temperatures shown in Table 1.

Figure 3

Plots of $D_1{\eta }_{\mathrm{sv}}/T$ using $D_1$ extrapolated at ${\epsilon }_1/{\epsilon }_2=1$ vs ${({\sigma }_1/{\sigma }_2)}^{-1}{(N/V)}^{1/3}$. The multiple plots for the same symbol mean the different temperatures shown in Table 1.

Figure 4

Hydrodynamic size of the solute molecules in liquid Ar at 120 K. The left and right axes correspond to the slip and stick boundary conditions, respectively.

Figure 5

Dependence of the decay time of half of the cage surrounding the fullerene molecules or rare gas atoms in liquid Ar on ${({\epsilon }_1/{\epsilon }_2)}^{-0.2}$. The decay time was evaluated for the same values of ${({\epsilon }_1/{\epsilon }_2)}^{-0.2}$, which is used in Fig. 1. The inset shows the decay time simply for all the kinds of solutes.