Modeling the magnetic interactions between paramagnetic beads in magnetorheological fluids

In this study, we develop and compare new and existing methods for computing the magnetic interactions between paramagnetic particles in magnetorheological (MR) fluids. The commonly employed point-dipole methods are outlined and the inter-particle magnetic forces, given by these representations, are compared with exact values. An alternative finite-dipole model, where the magnetization of a particle is represented as a distribution of current density, is described and the associated computational effort is shown to scale as O(N)O(N). As the dipole moments and forces given by this model depend on the length scale of the current distribution, a sensitivity analysis is performed to reveal a proper choice of this length scale. While the dipole models give a good estimation of the far-field interactions, as two particles come into contact, higher order multipoles are needed to properly resolve their interaction. We present the exact two-body calculation and describe a procedure to include the higher multipoles arising in a pairwise interaction into a dipole model. This inclusion procedure can be integrated with any dipole or higher-multipole calculation. Results from relevant three-body problems are compared to exact solutions to provide information as to how well the inclusion procedure performs in simulations of self-assembly and estimating the yield strength of structures.