Explicit expression for the Stokes-Einstein relation for pure Lennard-Jones liquids

An explicit expression of the Stokes-Einstein (SE) relation in molecular scale has been determined for pure Lennard-Jones (LJ) liquids on the saturated vapor line using a molecular dynamics calculation with the Green-Kubo formula, as Dη(sv)=kTξ(-1)(N/V)(1/3), where D is the self-diffusion coefficient, η(sv) the shear viscosity, k the Boltzmann constant, T the temperature, ξ the constant, and N the particle number included in the system volume V. To this end, the dependence of D and η(sv) on packing fraction, η, and T has been determined so as to complete their scaling equations. The equations for D and η(sv) in these states are m(-1/2)(N/V)(-1/3)(1-η)(4)ε(-1/2)T and m(1/2)(N/V)(2/3)(1-η)(-4)ε(1/2)T(0), respectively, where m and ε are the atomic mass and characteristic parameter of energy used in the LJ potentials, respectively. The equations can well describe the behaviors of D and η(sv) for both the LJ and the real rare-gas liquids. The obtained SE relation justifies the theoretical equation proposed by Eyring and Ree, although the value of ξ is slightly different from that given by them. The difference of the obtained expression from the original SE relation, Dη(sv)=(kT/2π)σ(-1), where σ means the particle size, is the presence of the η(1/3) term, since (N/V)(1/3)=(6/π)(1/3)σ(-1)η(1/3). Since the original SE relation is based on the fluid mechanics for continuum media, allowing the presence of voids in liquids is the origin of the η(1/3) term. Therefore, also from this viewpoint, the present expression is more justifiable in molecular scale than the original SE relation. As a result, the η(1/3) term cancels out the σ dependence from the original SE relation. The present result clearly shows that it is not necessary to attribute the deviation from the original SE relation to any temperature dependence of particle size or to introduce the fractional SE relation for pure LJ liquids. It turned out that the η dependence of both D and η(sv) is similar to that in the corresponding equations by the Enskog theory for hard sphere (HS) fluids, although the T dependence is very different, which means that the difference in the behaviors of D and η(sv) between the LJ and HS fluids are traceable simply to their temperature dependence. Although the SE relation for the HS fluids also follows Dη(sv)=kTξ(-1((N/V)(1/3), the value of ξ is significantly different from that for the LJ liquids.