On minimal energy dipole moment distributions in regular polygonal agglomerates

Abstract

Static, regular polygonal and close-packed clusters of spherical magnetic particles and their energy-minimizing magnetic moments are investigated in a two-dimensional setting. This study focuses on a simple particle system which is solely described by the dipole-dipole interaction energy, both without and in the presence of an in-plane magnetic field. For a regular polygonal structure of n sides with n≥3, and in the absence of an external field, it is proved rigorously that the magnetic moments given by the roots of unity, i.e. tangential to the polygon, are a minimizer of the dipole–dipole interaction energy. Also, for zero external field, new multiple local minima are discovered for the regular polygonal agglomerates. The number of found local extrema is proportional to [n/2] and these critical points are characterized by the presence of a pair of magnetic moments with a large deviation from the tangential configuration and whose particles are at least three diameters apart. The changes induced by an in-plane external magnetic field on the minimal energy, tangential configurations are investigated numerically. The two critical fields, which correspond to a crossover with the linear chain minimal energy and with the break-up of the agglomerate, respectively are examined in detail. In particular, the numerical results are compared directly with the asymptotic formulas of Danilov et al. (2012) [23] and a remarkable agreement is found even for moderate to large fields. Finally, three examples of close-packed structures are investigated: a triangle, a centered hexagon, and a 19-particle close packed cluster. The numerical study reveals novel, illuminating characteristics of these compact clusters often seen in ferrofluids. The centered hexagon is energetically favorable to the regular hexagon and the minimal energy for the larger 19-particle cluster is even lower than that of the close packed hexagon. In addition, this larger close packed agglomerate has two distinctive regimes in the magnetization, which corresponds to two very different susceptibilities, in marked contrast to the behavior observed in regular polygonal structures.