Ewald sum for hydrodynamic interactions with periodicity in two dimensions

TL;DR

This paper develops an efficient Ewald summation method for hydrodynamic interactions in quasi two-dimensional colloidal systems, correcting size-dependent artifacts and reducing computational costs in simulations.

Contribution

It introduces a two-dimensional Ewald summation approach for hydrodynamic tensors, addressing inaccuracies of 3D approximations in planar systems.

Findings

Corrects size dependence in hydrodynamic simulations

Reduces computational expense compared to 3D Ewald methods

Provides simple formulas for molecular and Brownian dynamics

Abstract

We carry out the Ewald summation for the Rotne-Prager-Yamakawa mobility tensor, the Oseen mobility tensor and further variations of both, relevant for the hydrodynamic interactions in colloidal suspensions, where all interacting particles are within a single plane, i.e., adsorbed at a fluid interface or other quasi two-dimensional systems. We use the Poisson summation formula for systems periodic in two dimensions and finite in the third dimension in order to obtain simple formulae for applications, such as molecular dynamics or Brownian dynamics simulations. We show, that for such systems, as soon as noise is taken into account, a commonly used approximate three-dimensional Ewald summation leads to a spurious system size dependence, which may considerably affect the interpretation of simulation results and will be cured within our approach. Additionally, the resulting formulae are found to be computationally much less expensive than the approximate three-dimensional Ewald summation.