Correcting the Generalized Stokes-Einstein Relation to Include Effects of Hydrodynamic Interactions from Periodic Images

In microrheology, the mean squared displacement (MSD) of micron-sized particles is measured to extract linear viscoelastic properties of entangled polymer melts using the generalized Stokes-Einstein (SE) relation. Probe rheology from molecular dynamics (MD) simulations has a significant computational cost due to the large simulation box sizes required to reduce the effect of the probe’s hydrodynamic interactions with its periodic images. Here, we use an analytical solution for Stokes flow around a body-centered cubic array of spheres to introduce a correction factor to the generalized SE relation. We show that this correction factor is a function of the ratio (R/L), where R and L are particle radius and simulation box length, and the fluid mass fraction. We then predict the dynamic moduli of an entangled melt of bead-spring polymer chains from the MSD of a probe particle at various R and L. Lattice Boltzmann simulations are also performed to calculate the transient drag force and velocity of a particle in a Newtonian fluid to account for effects of inertia at higher frequencies. This added correction factor should allow a reduction of at least one order of magnitude in MD simulation costs.