Numerical calculation of interaction forces between paramagnetic colloids in two-dimensional systems

Abstract

Typically the force between paramagnetic particles in a uniform magnetic field is described using the dipolar model, which is inaccurate when particles are in close proximity to each other. Instead, the exact force between paramagnetic particles can be determined by solving a three-dimensional Laplace's equation for magnetostatics under specified boundary conditions and calculating the Maxwell stress tensor. The analytical solution to this multi-boundary-condition Laplace's equation can be obtained by using a solid harmonics expansion in conjunction with the Hobson formula. However, for a multibody system, finite truncation of the Hobson formula does not lead to convergence of the expansion at all points, which makes the approximation physically unrealistic. Here we present a numerical method for solving this Laplace's equation for magnetostatics. This method uses a smoothed representation to replace all the boundary conditions. A two-step propagation is used to dramatically accelerate the calculation without losing accuracy. Using this method, we calculate the force between two paramagnetic particles in a uniform and a rotational external field and compare our results with other models. Furthermore, the many-body effects for three-particle, ten-particle, and 24-particle systems are examined using the same method. We also calculate the interaction between particles with different magnetic susceptibilities and particle diameters. The Laplace's equation solver method described in this article that is used to determine the force between paramagnetic particles is shown to be very useful for dynamic simulations for both two-particle systems and a large cluster of particles.

Directing Assembly of DNA-Coated Colloids with Magnetic Fields To Generate Rigid, Semiflexible, and Flexible Chains

“…However, it has also been shown that the many-body effect becomes negligible once you are sufficiently far away from the edge of a cluster or chain. 42 Thus, we analyze only chains of greater than 30 beads.…”

Micro-mutual-dipolar model for rapid calculation of forces between paramagnetic colloids

“…. ,N. Nevertheless the MDM's neglect of multipolar effects leads to a significant deviation in the calculated force in the near field [17]. The multipolar effects are caused by the asymmetric magnetization inside the spheres when they are placed close 1539-3755/2014/90(3)/033310 (5) 033310-1…”

“…The solution to the Laplace equation can be analytically approximated by a solid harmonics expansion with the Hobson formula applied to unify the coordinate system [14,16]. This coordinate unification is very computationally expensive and suffers from singularity-related issues [17,18]. A numerical solution to Laplace's equation can be obtained by using a smoothed representation of susceptibility to replace the boundary conditions [17]; this method is referred to as the Laplace equation solver (LES) method.…”

“…This coordinate unification is very computationally expensive and suffers from singularity-related issues [17,18]. A numerical solution to Laplace's equation can be obtained by using a smoothed representation of susceptibility to replace the boundary conditions [17]; this method is referred to as the Laplace equation solver (LES) method. This numerical approach is stable in terms of error propagation but still computationally time consuming.…”

Dipolar magnetism and electrostatic repulsion of colloidal interacting nanoparticle system

“…Owing to such a complex dipolar magnetism involving all existing particles in a ferrofluid, the corresponding numerical analysis is often simplified into a two-particle system. [27][28][29] Therefore, we present an alternative experimental method of indirectly identifying the change in potential energy [ÁUð" rÞ] by characterizing relaxation responses of suspended particles. Numerical analysis of this parameter enables the estimation of (b) hydrodynamic size distributions indicating the secondary particle clustering at 30 wt % ferrofluid sample.…”