Microrheology analysis in molecular dynamics simulations: Finite box size correction

In single-particle microrheology, the viscoelastic properties of a complex fluid can be extracted using the generalized Stokes–Einstein (GSE) relation by embedding a micrometer-sized particle and tracking its motion through the fluid. Applying the same analysis to molecular dynamics simulations can result in overestimated G∗ values because of the hydrodynamic interaction between the probe bead and its periodic images. We derive a simple correction to the GSE equation by implementing an analytical solution for Stokes drag on a periodic array of spheres, which allows smaller box sizes to be simulated while still retaining accuracy. Fluid and particle inertia are neglected, although the approach used here might be generalized to include them. The correction is applied to molecular dynamics simulations of a coarse-grained polymer melt. For several bead-to-box-size ratios R/L and two probe sizes, we measure the mean squared displacement and calculate G∗ using both the original GSE and the corrected hydrodynamic Stokes–Einstein (HSE) equations. These results are compared to small amplitude oscillatory non-equilibrium molecular dynamics (NEMD) simulations, which show that the HSE analysis correctly predicts the G∗ values at low frequencies but breaks down when either fluid inertia is important or the bead size is too small to "see" a continuum. For small R/L, the GSE and HSE results are similar, although a small correction is still required for the finite box size and computational cost is significantly larger. The HSE equation allows the use of smaller box sizes, reducing computational costs by more than an order of magnitude.