The forces between magnetizable particles embedded in an elastomer

Abstract

The magnetostatic forces between two spherical particles made of an isotropic linearly polarizable magnetic substance and subjected to a uniform external field are studied. Under these conditions, the magnetization distribution within a given particle depends on the position and magnetization of the neighboring one. The solution for the magnetic energyof the pair has the form of power series with respect to the ratio of the particle radius to the particle center-to-center distance. The coefficients of this series are found numerically taking in about hundred terms. Evaluation of the interparticle forces, which requires differentiation of the energy over coordinates, is facilitated by proposing an approximate formula for, which enables one to avoid the laborious numeric procedure. In such a way, the distribution of interparticle forces is obtained and compared to that provided by the point magnetic moment model. It is shown that at close neighboring of the particles, the magnetic force differs substantially from that predicted by the dipole model. When analyzing the sign of the force, it is found that the angular interval between the field and the particle center-to-center vector, which corresponds to repulsion, is much more narrow than that for point dipoles. This means that in a system of magnetizable particles attraction is the dominating type of interaction. The problem is extended to magnetomechanics by considering the same pair of particles in an elastic matrix. There, the state of the system, when magnetized, is determined by interplay of magnetic and elastic forces. It is shown that in the field applied along the center-to-center direction, the configuration of the pair essentially depends on the field strength. In a low field the equilibrium interparticle gap changes but slightly since weak magnetic attraction induces weak restoring elastic forces. However, in stronger fields the system becomes bistable: besides the energy minimum located close to the initial interparticle gap, another minimum emerges, which corresponds to a close approach (clusterizing) of the particles. In the latter state both the magnetic and elastic forces are strong. This bistability effect has important implications in mechanics of soft magnetic elastomers.