Molecular-dynamics investigation of tracer diffusion in a simple liquid: Test of the Stokes-Einstein law

In this work, we study the diffusion of solute particles in the limit of infinite dilution in a solvent. An estimate is made of the solute concentration below which this limit is attained. We determine the range of the size and mass values of the solute particles where the solute diffusion coefficient is well estimated from the Stokes-Einstein formula. For these aims, extensive molecular-dynamics simulations are carried out for a model tracer-solvent system made up of 5324 molecules including solvent and tracer molecules interacting through Lennard-Jones potentials. The values of the viscosity coefficient, corrected for long time tail contributions, and the diffusion coefficients are obtained with high precision. Positive deviations from the Stokes-Einstein formula are observed as the size ratio or the mass ratio of the tracer to solvent molecules is lowered. For equal solvent and tracer molecular masses, the crossover to the hydrodynamics regime is found to occur when the size ratio is similar to 4. The results show a strong coupling between the size and mass effects on the tracer diffusivity, with the latter being predominant. An analysis of the molecular-dynamics data in the hydrodynamic regime shows that the Stokes-Einstein formula holds for this system with slip boundary conditions and the hydrodynamic radius equal to the cross radius between the tracer-solvent molecules. The friction coefficient is evaluated from the computed autocorrelation function of the force exerted by the fluid on the tracer molecule, following a scheme proposed by Lagar'kov and Sergeev; it is found that the latter criterion gives the correct diffusion coefficient only in the limits of high sizes and high masses.