Modelling the effect of particle arrangement on the magnetoelectric response of a polymer multiferroic film.

In this work the behavior of a three-component composite multiferroic (MF)-an electrically neutral polymer matrix filled with a mixture of piezoelectric and ferromagnetic micrometer-size particles-is investigated in the framework of a simple mesoscopic model. The main issue of interest is the electric polarization generated in a thin film of such an MF in response to a quasistatic magnetic field. The driving mechanism of the effect is rotation of magnetically hard particles inside the matrix which, in turn, transfers the arisen mechanical stresses to the piezoelectric grains. The model MF film is constructed as a periodic set of 2D cells each of which contains one piezoelectric and two ferromagnetic particles. The numerical simulations are performed by means of finite element method on a single cell which, however, is incorporated in an infinite film by means of periodic boundary conditions. The problem of how the spatial arrangement of the particles and the orientation of the anisotropy axis of the piezoelectric one affect the magnetoelectric response is discussed.

Magnetostrictive and Magnetoactive Effects in Piezoelectric Polymer Composites.

A mesoscopic model for a polymer-based magnetoelectric (ME) composite film is developed. The film is assumed to consist of a piezoelectric polymer matrix of the PVDF type filled with CFO-like single-domain nanoparticles. The model is treated numerically and enables one to obtain in detail the intrinsic distributions of mechanical stress, polarization and electric potential and helps to understand the influence of the main configurational parameters, viz., the poling direction and the orientational order of the particle magnetic anisotropy axes on the electric response of the film. As the model is fairly simple-it uses the RVE-like (Representative Volume Element) approach with a single-particle cell-the results obtained are rather of qualitative than quantitative nature. However, the general conclusions seem to be independent of the particularities of the model. Namely, the presented results establish that the customary ME effect in composite films always comprises at least two contributions of different origins, viz., the magnetostrictive and the magnetoactive (magnetorotational) ones. The relative proportion between those contributions is quite movable depending on the striction coefficient of the particles and the stiffness of the polymer matrix. This points out the necessity to explicitly take into account the magnetoactive contribution when modeling the ME response of composite films and when interpreting the measurements on those objects.

Effect of Piezoelectric BaTiO3 Filler on Mechanical and Magnetoelectric Properties of Zn0.25Co0.75Fe2O4/PVDF-TrFE Composites.

Polymer-based multiferroics, combining magnetic and piezoelectric properties, are studied experimentally-from synthesis to multi-parameter characterization-in view of their prospects for fabricating biocompatible scaffolds. The main advantage of these systems is facile generation of mechanical deformations and electric signals in response to external magnetic fields. Herein, we address the composites based on PVDF-TrFE polymer matrices filled with a combination of piezoelectric (BaTiO3, BTO) and/or ferrimagnetic (Zn0.25Co0.75Fe2O4, ZCFO) particles. It is shown that the presence of BTO micron-size particles favors stripe-type structuring of the ZCFO filler and enhances the magnetoelectric response of the sample up to 18.6 mV/(cm∙Oe). Besides that, the admixing of BTO particles is crucial because the mechanical properties of the composite filled with only ZCFO is much less efficient in transforming magnetic excitations into the mechanical and electric responses. Attention is focused on the local surfacial mechanical properties since those, to a great extent, determine the fate of stem cells cultivated on these surfaces. The nano-indentation tests are accomplished with the aid of scanning probe microscopy technique. With their proven suitable mechanical properties, a high level of magnetoelectric conversion and also biocompatibility, the composites of the considered type are enticing as the materials for multiferroic-based polymer scaffolds.

Magnetization of magnetoactive elastomers under the assumption of breakable adhesion at the particle/matrix interface.

In this work we study the magnetization of magnetoactive elastomers (MAE) in which the interface between the matrix and magnetic particles is unstable and allows for slipping of the particles against the wall of their elastomer cavities. The estimate of the maximal angle at which each particle can decline its axis from the initial position is made based on cyclic measurement of several consecutive hysteresis loops at different maximal magnetic fields. A model of magnetization of magnetically hard multigrain particles in an elastic environment with allowance for their possible slipping is proposed. Results of modelling is in fair agreement with the experimental data obtained on MAEs whose polymeric matrix is made of polydimethylsiloxane and the magnetic filler is NdFeB spherical particles.

Multiferroic Coupling of Ferromagnetic and Ferroelectric Particles through Elastic Polymers.

Multiferroics are materials that electrically polarize when subjected to a magnetic field and magnetize under the action of an electric field. In composites, the multiferroic effect is achieved by mixing of ferromagnetic (FM) and ferroelectric (FE) particles. The FM particles are prone to magnetostriction (field-induced deformation), whereas the FE particles display piezoelectricity (electrically polarize under mechanical stress). In solid composites, where the FM and FE grains are in tight contact, the combination of these effects directly leads to multiferroic behavior. In the present work, we considered the FM/FE composites with soft polymer bases, where the particles of alternative kinds are remote from one another. In these systems, the multiferroic coupling is different and more complicated in comparison with the solid ones as it is essentially mediated by an electromagnetically neutral matrix. When either of the fields, magnetic or electric, acts on the 'akin' particles (FM or FE) it causes their displacement and by that perturbs the particle elastic environments. The induced mechanical stresses spread over the matrix and inevitably affect the particles of an alternative kind. Therefore, magnetization causes an electric response (due to the piezoeffect in FE) whereas electric polarization might entail a magnetic response (due to the magnetostriction effect in FM). A numerical model accounting for the multiferroic behavior of a polymer composite of the above-described type is proposed and confirmed experimentally on a polymer-based dispersion of iron and lead zirconate micron-size particles.

Collective in-plane magnetization in a two-dimensional XY macrospin system within the framework of generalized Ott-Antonsen theory.

The problem of magnetic transitions between the low-temperature (macrospin ordered) phases in two-dimensional XY arrays is addressed. The system is modelled as a plane structure of identical single-domain particles arranged in a square lattice and coupled by the magnetic dipole-dipole interaction; all the particles possess a strong easy-plane magnetic anisotropy. The basic state of the system in the considered temperature range is an antiferromagnetic (AF) stripe structure, where the macrospins (particle magnetic moments) are still involved in thermofluctuational motion: the superparamagnetic blocking Tb temperature is lower than that (Taf) of the AF transition. The description is based on the stochastic equations governing the dynamics of individual magnetic moments, where the interparticle interaction is added in the mean-field approximation. With the technique of a generalized Ott-Antonsen theory, the dynamics equations for the order parameters (including the macroscopic magnetization and the AF order parameter) and the partition function of the system are rigorously obtained and analysed. We show that inside the temperature interval of existence of the AF phase, a static external field tilted to the plane of the array is able to induce first-order phase transitions from AF to ferromagnetic state; the phase diagrams displaying stable and metastable regions of the system are presented. This article is part of the theme issue 'Patterns in soft and biological matters'.

Modeling the magnetostriction effect in elastomers with magnetically soft and hard particles.

We analyze theoretically the field-induced microstructural deformations in a hybrid elastomer, that consists of a polymer matrix filled with a mixture of magnetically soft and magnetically hard spherical microparticles. These composites were introduced recently in order to obtain a material that allows the tuning of its properties by both, magnetically active and passive control. Our theoretical analysis puts forward two complementary models: a continuum magnetomechanical model and a bead-spring computer simulation model. We use both approaches to describe qualitatively the microstructural response of such elastomers to applied external fields, showing that the combination of magnetically soft and hard particles may lead to an unusual magnetostriction effect: either an elongation or a shrinking in the direction of the applied field depending on its magnitude. This behavior is observed for conditions (moderate particle densities, fields and deformations) under which the approximations of our models (linear response regime, negligible mutual magnetization between magnetically soft particles) are physically valid.

Modeling the magnetomechanical behavior of a multigrain magnetic particle in an elastic environment.

The Stoner-Wohlfarth model of a single-domain grain is applied to a complex situation: magnetization of a solid multigrain particle embedded in an elastic medium. In this situation, application of a magnetic field establishes a specific magnetomechanical process: polarization and switching of individual grains change the net energy of the particle and, as a result, make it rotate as a whole relative to the matrix. Because of that coupling, the magnetic hysteresis loop of a particle composed of highly coercive grains progressively shrinks with the increase of the matrix compliance. The effect is studied theoretically by numerical simulations on a particle comprising several hundred magnetically uniaxial grains with randomly oriented easy axes. The results of the model are discussed with regard to magnetic measurements performed on dispersions of spherical NdFeB microparticles in PDMS matrices of varied stiffness.

Structure organization and magnetic properties of microscale ferrogels: The effect of particle magnetic anisotropy.

The equilibrium structure and magnetic properties of a ferrogel object of small size (microferrogel(MFG)) are investigated by coarse-grained molecular dynamics. As a generic model of a microferrogel (MFG), a sample with a lattice-like mesh is taken. The solid phase of the MFG consists of magnetic (e.g., ferrite) nanoparticles which are mechanically linked to the mesh making some part of its nodes. Unlike previous models, the finite uniaxial magnetic anisotropy of the particles, as it is the case for real ferrogels, is taken into account. For comparison, two types of MFGs are considered: MFG-1, which dwells in virtually non-aggregated state independently of the presence of an external magnetic field, and MFG-2, which displays aggregation yet under zero field. The structure states of the samples are analyzed with the aid of angle-resolved radial distribution functions and cluster counts. The results reveal the crucial role of the matrix elasticity on the structure organization as well as on magnetization of both MFGs. The particle anisotropy, which plays insignificant role in MFG-1 (moderate interparticle magnetodipole interaction), becomes an important factor in MFG-2 (strong interaction). There, the restrictions imposed on the particle angular freedom by the elastic matrix result in notable diminution of the particle chain lengths as well as the magnetization of the sample. The approach proposed enables one to investigate a large variety of MFGs, including those of capsule type and to purposefully choose the combination of their magnetoelastic parameters.