Breakdown of the Stokes-Einstein relation in pure Lennard-Jones fluids: From gas to liquid via supercritical states

We have examined the conditions under which the breakdown of the Stokes-Einstein (SE) relation occurs in pure Lennard-Jones (LJ) fluids over a wide range of temperatures and packing fractions beyond the critical point. To this end, the temperature and packing-fraction dependence of the self-diffusion coefficient, $D$, and the shear viscosity, ${\eta }_{\mathrm{sv}}$, were evaluated for Xe using molecular dynamics calculations with the Green-Kubo formula. The results showed good agreement with the experimental values. The breakdown was determined in light of the SE equation which we have recently derived for pure LJ liquids: $D{\eta }_{\mathrm{sv}}=(k_{B}T/2\pi ){(N/V)}^{1/3}$, where $k_B$ is the Boltzmann constant, $T$ is the temperature, and $N$ is the particle number included in the system volume $V$. We have found that the breakdown occurs in the lower range of the packing fraction, $\eta <0.2$, and derived the SE relation in its broken form as $D{\eta }_{\mathrm{sv}}=0.007{(1-\eta )}^{-5}{\eta }^{-4/3}{(k_{B}T/\epsilon )}^n{k}_{B}T{(N/V)}^{1/3}$, where $n$ increases from 0 up to 1 with the decreasing $\eta$. The equation clearly shows that the breakdown mainly occurs because the packing-fraction dependence does not cancel out between $D$ and ${\eta }_{\mathrm{sv}}$ in this region, which is attributed to the gaseous behavior in the packing-fraction dependence of the shear viscosity under a constant number density. In addition, the gaseous behavior in the temperature dependence of the shear viscosity also partially causes the breakdown.

Figure 1

Thermodynamic states evaluated in this study, shown by the red open circles, on the phase diagram of Xe [14]. The states where the experimental results are available are indicated by the green closed circles and pink closed triangle for the self-diffusion coefficient [9] and shear viscosity [10], respectively.

Figure 2

Variation in the order parameters, $\alpha ,\beta$, and $\gamma$, for the temperature dependence of the self-diffusion coefficient, shear viscosity, and their product, respectively, for the whole range of packing fractions in fluid Xe.

Figure 3

(a) Reduced self-diffusion coefficient and (b) reduced shear viscosity of Xe as a function of the packing fraction from the super- and subcritical states to normal liquid states shown in Fig. 1. The solid lines are fitted to the quantities, respectively, in the lower packing-fraction range, i.e., $\eta <0.2$. The experimental results for Xe are taken from Refs. [9, 10] and the results for Ar and Kr are from Refs. [4, 16]. Most error bars for the present calculations are smaller than the symbols.

Figure 4

(a) Packing fraction dependence of the SE relation in the whole range from the super- and subcritical fluid to normal liquid states. (b) Plot of the SE relation vs the function of the packing fraction optimized for $\eta <0.2$. The experimental results for Xe are taken from Refs. [9, 10] and the results for Ar and Kr are from Ref. [4].

Figure 5

Variation in the Fourier transformed time-correlation function of components of the stress tensor vs packing fraction at 348 K (a) under the states shown in Table 1 and (b) under a constant number density of 0.32 ${\mathrm{nm}}^{-3}$.