Effect of local elasticity of the matrix on magnetization loops of hybrid magnetic elastomers

Abstract

To model magnetization loops of magnetorheological elastomers (MREs) with magnetically hard filler, we consider an assembly of single-domain particles possessing inversion-symmetrical shape and embedded in a soft polymer matrix. To describe the intrinsic behavior of the particle magnetic moments under an applied field, the Stoner-Wohlfarth approach is employed. Unlike the case of solid matrix, the particle in a MRE is able to rotate relative to its elastic environment, so that its equilibrium orientation results from the balance between the magnetic torque (exerted by the applied field at any magnetic moment that does not point along the field) and the elastic torque generated by the matrix. We assume that elastic resistance to the field-induced particle rotation could be presented as comprising of two contributions. The first one is valid for any not perfectly spherical particle and is independent of the particle–matrix adhesion. It reflects the fact that, when trying to rotate, the particle has to “shoulder its way” by deforming the adjoining regions of the matrix. In that case, for angular deviations up to 90° from the initial position, the resistance torque increases. However, as soon as the rotation angle grows up to the value but infinitesimally exceeding 90°, the elastic torque changes its sign and from now on forces the particle to rotate to 180°, where it attains the geometrical position that coincides with the initial one. Evidently, this process is of the barrier type: both orientations of the particle are equal in elastic energy. The second mechanism stems from the “memory” that a given particle has of its initial state and may be caused, for example, by some macromolecules grafted to its surface while curing the matrix of the MRE. This restoring force always tends to drag the particle to its 0° (initial) position. Therefore, the observed magnetization of a MRE sample comes out as a result of joint interplay of the Stoner-Wohlfarth and the two elastic mechanisms. In our simulations, we show that the proposed model in a natural way accounts for the two essential features observed (solely or together) in experiment, namely: (i) weak net coercivity of the MREs, whose filler particles as themselves are highly coercive, and (ii) asymmetric positions of the magnetization loops with respect to H=0 point at the (M-H) plane.