5.5 Field-based coarse-graining
For specific systems such as nanoparticle and nanofluid transport, both molecular interactions (due to biomolecular recognition) and hydrodynamic interactions (due to fluid flow and boundary effects) are significant. The integration of disparate length and time scales does not fit traditional multiscale methods. The complexity lies in integrating fluid-flow and memory for multiphase flow in complex and arbitrary geometries, while simultaneously including thermal and stochastic effects to simulate quasi-equilibrium distributions correctly to enable receptor-ligand binding at the physiological temperature. This issue is ubiquitous in multivalent binding or adhesive interactions between nanoparticles and cells or between two cells. Bridging the multiple length scales (from meso to molecular) and the associated time scales relevant to the problem is essential to success herein. Multiple macroscopic and mesoscopic time scales governing the problem include (i) hydrodynamic time scale, (ii) viscous/Brownian relaxation time scale, and (iii) Brownian diffusion time scale.
5.5.1 Memory function approach to coarse-graining with hydrodynamic interactions: In the description of the dynamics of nanosized Brownian particles in an bounded and unbounded fluid domains the memory functions decay with algebraic correlations as enumerated by theoretical and computational studies [46, 85, 128]. The equation of stochastic motion for each component of the velocity of a nanoparticle immersed in a fluid in bounded and unbounded domains takes the form of a generalized Langevin equation (GLE) of the form of (Eq. 9); to account for hydrodynamic interaction, a composite GLE was introduced [129, 130].