A New Relatively Simple Approach to Multipole Interactions in Either Spherical Harmonics or Cartesians, Suitable for Implementation into Ewald Sums

We present a novel derivation of the multipole interaction (energies, forces and fields) in spherical harmonics, which results in an expression that is able to exactly reproduce the results of earlier Cartesian formulations. Our method follows the derivations of Smith (W. Smith, CCP5 Newsletter 1998, 46, 18.) and Lin (D. Lin, J. Chem. Phys. 2015, 143, 114115), who evaluate the Ewald sum for multipoles in Cartesian form, and then shows how the resulting expressions can be converted into spherical harmonics, where the conversion is performed by establishing a relation between an inner product on the space of symmetric traceless Cartesian tensors, and an inner product on the space of harmonic polynomials on the unit sphere. We also introduce a diagrammatic method for keeping track of the terms in the multipole interaction expression, such that the total electrostatic energy can be viewed as a ‘sum over diagrams’, and where the conversion to spherical harmonics is represented by ‘braiding’ subsets of Cartesian components together. For multipoles of maximum rank n, our algorithm is found to have scaling of n 3.7 vs. n 4.5 for our most optimised Cartesian implementation.

Long-range forces

A wide variety of special techniques for handling long-range forces are explained in this chapter. This includes the traditional Ewald sum, and the particle-mesh methods that use a discrete Fourier transform. A number of techniques based on spherical truncation such as the Wolf method, the isotropic periodic sum and the reaction field are also considered. Techniques for larger systems such as the fast-multipole method, the multilevel summation approach, and the direct solution of Maxwell’s equations, are explained. The advantages and disadvantages of the different approaches are reviewed, and a number of methods for tackling long-range forces in inhomogeneous systems, particularly in a slab geometry, are presented.

Stirring by periodic arrays of microswimmers

The interaction between swimming micro-organisms or artificial self-propelled colloids and passive (tracer) particles in a fluid leads to enhanced diffusion of the tracers. This enhancement has attracted strong interest, as it could lead to new strategies to tackle the difficult problem of mixing on a microfluidic scale. Most of the theoretical work on this topic has focused on hydrodynamic interactions between the tracers and swimmers in a bulk fluid. However, in simulations, periodic boundary conditions (PBCs) are often imposed on the sample and the fluid. Here, we theoretically analyse the effect of PBCs on the hydrodynamic interactions between tracer particles and microswimmers. We formulate an Ewald sum for the leading-order stresslet singularity produced by a swimmer to probe the effect of PBCs on tracer trajectories. We find that introducing periodicity into the system has a surprisingly significant effect, even for relatively small swimmer–tracer separations. We also find that the bulk limit is only reached for very large system sizes, which are challenging to simulate with most hydrodynamic solvers.