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A method has been proposed for calculating linear dynamic magnetization of a viscoelastic ferrocolloid in a constant magnetic field (displacement field). The magnetic phase of the colloid consists of Brownian ferromagnetic nanoparticles placed into a Jeffry’s fluid. Therefore, each particle, upon its rotation induced by an alternating (probe) field, dissipates energy via two friction “channels” operating in parallel. The usual (Newtonian) viscosity prevails at short times, while the retarded (Maxwellian) dissipative interaction plays the main role at long times. It has been shown that the retarded friction on the Jeffry’s medium gives rise to a slow magnetization relaxation mode, which must be most pronounced in ferrocolloids having substantial elasticity. As the displacement field is enhanced, this mode weakens and the friction relevant to the Newtonian viscosity becomes prevailing, because it causes small-angle orientational fluctuations of particle magnetic moments. The proposed method yields an exact solution of the model, and the results obtained using method prove that previous approximate calculations are substantially limited.

This work presents an approach to the macroscopic field-controlled mechanics of magnetoactive elastomers of mixed content, which are a special type of smart materials made of an elastic composite and a combination of two essentially different ferromagnetic fillers. High-coercive particles of NdFeB-alloy powder for the magnetically hard (MH) filler and carbonyl iron powder particles with nearly zero coercivity for the magnetically soft (MS) filler are usually used. The MH particles are tens-of-micron in size and impart to the elastomer a remanent magnetisation, whereas due to the MS particles of several microns in size, the elastomer acquires a high magnetic susceptibility. Since large MH particles once magnetised in a strong field possess their own fields to which the MS particles are susceptible, the overall elastomer magnetisation as well as its mechanical response greatly depends on the relative concentration of both fillers. This work particularly studies the bending deformation of horizontally fixed magnetoactive cantilevers with the permanent magnetisation along the length axis under the action of gravity and a vertically applied uniform magnetic field. The cantilevers of the same geometry and fixed NdFeB content but different carbonyl iron concentration are considered. The magnetomechanical model is developed based on the finite-strain theory assuming the plane-stress approximation of the two-dimensional cantilever of infinite width. The magnetic energy comprises two magnetic terms, one of which is qualitatively linear and the other one is quadratic in the applied field strength. The numerically calculated field-programmed equilibrium bending shapes of the cantilevers are compared with the experimentally observed shapes. The model provides good agreement with the experiment up to moderate concentrations of the MS filler, when the coefficients of customary interpolation formulas for the concentration dependencies of elastic modulus and magnetic susceptibility are properly adjusted.

We demonstrate that a ferronematic (a dilute suspension of ferromagnet nanoparticles in a nematogenic matrix) may display a direct analog of Neel superparamagnetism entailed by the orienting effect of the matrix on the embedded particles. The latter are assumed to be the objects with pronounced anisometricity; in magnetic aspect, they are single-domains with strong magnetic hardness (e.g. imposed by their rod-like shape) so that the magnetic moment is fixed inside the particle body. Above the isotropic–nematic transition point, the considered system is just a ferrofluid with random distribution of the particle axes. Below the transition, the particle–matrix coupling emerges that sets the axis of each particle (and, hence, its magnetic moment) under the same angle to the director. If this alignment is along the director, then each particle falls under the action of orientational potential with two equal wells (0° and 180°) separated by the energy barrier whose height is defined by the intensity of the surface particle–matrix interaction. Provided the thermal energy is of the order of the barrier height (the material estimates readily admit that), the Brownian motion makes the particle to randomly rotate between the orientational minima. This mechanism entails spontaneous inversions of the magnetic moment as well, thus ensuring relaxation of any initially established magnetization of the ferronematic; the reference time of this process depends exponentially on the hight of the energy barrier scaled with thermal energy. Treating the nematogenic matrix with the aid of mean field model and using the linear response theory to describe the magnetodynamics of the particles, we show that the “liquid-crystalline” superparamagnetism produces an easily identifiable signature in the dynamic magnetic susceptibility spectrum of a ferronematic.

Exchange bias properties of MnFe$$_2$$ 2 O$$_4$$ 4 @$$\gamma$$ γ –Fe$$_2$$ 2 O$$_3$$ 3 core–shell nanoparticles are investigated. The measured field and temperature dependencies of the magnetization point out a well-ordered ferrimagnetic core surrounded by a layer with spin glass-like arrangement. Quasi-static SQUID magnetization measurements are presented along with high-amplitude pulse ones and are cross-analyzed by comparison against ferromagnetic resonance experiments at 9 GHz. These measurements allow one to discern three types of magnetic anisotropies affecting the dynamics of the magnetic moment of the well-ordered ferrimagnetic NP’s core viz. the easy-axis (uniaxial) anisotropy, the unidirectional exchange-bias anisotropy and the rotatable anisotropy. The uniaxial anisotropy originates from the structural core–shell interface. The unidirectional exchange-bias anisotropy is associated with the spin-coupling at the ferrimagnetic/spin glass-like interface; it is observable only at low temperatures after a field-cooling process. The rotatable anisotropy is caused by partially-pinned spins at the core/shell interface; it manifests itself as an intrinsic field always parallel to the external applied magnetic field. The whole set of experimental results is interpreted in the framework of superparamagnetic theory, i.e., essentially taking into account the effect of thermal fluctuations on the magnetic moment of the particle core. In particular, it is found that the rotatable anisotropy of our system is of a uniaxial type.

Surface relief of magnetoactive elastomers (MAEs) based on soft polymer matrices filled with anisometric magnetically hard fillers is studied theoretically in magnetic fields applied perpendicular to the MAE surface. A single-particle 2D cell model describing the rotation of one individual elliptical particle in a near-surface MAE layer is developed. The equilibrium rotation angle of particles is defined by a balance between Zeeman, magnetic anisotropy and elastic (generated in the polymer matrix) energy increments. The Stoner-Wohlfarth model is used to describe magnetic properties of the filler particles while the elastic energy as a function of the particle rotation angle is evaluated numerically using FEM simulations. A representative surface MAE system is constructed via superposition of single-particle cells with field-driven magnetic particles, and surface relief characteristics are derived for various sets of geometric and statistical parameters. Limitations of the proposed approach have been discussed.

Magnetostriction effect of a spherical sample of a magnetoactive elastomer (MAE) is analyzed. In comparison with the preceding study, the consideration is done on a more realistic basis: taking into account saturation of the MAE magnetization in contrast to the former model where the magnetization was supposed to be linear whatever the field strength. This more thorough investigation has revealed that the striction-induced elongation effect, depending on the material parameters, may occur in two forms. One scenario manifests itself as tapering of the polar zones of the former sphere, where ‘beaks’ are formed, so that the shape of the object drastically deviates from a spheroidal one. The mechanism the underlies the occurrence of beaks is the surface instability of a magnetizable elastic continuum, and the beak nucleation follows the second-order transition pattern; the resulting overall elongation of the body does not display any hysteresis. Another scenario—it is related to MAEs with higher magnetic properties and softer matrices—implies that the beak formation happens simultaneously with a jump-like overall elongation of the former sphere, and this transformation resembles the first-order transition pattern. Upon assessing the chances to observe the predicted effects on the samples of now existing MAEs, one comes to a conclusion that the second scenario is hardly possible, whereas the first one, i.e., beak formation without hysteretic stretching, is much more realizable.

Chaining of magnetic nanoparticles in a microscopic ferrogel (MFG) due to interparticle interaction and external field is analyzed by the coarse-grained molecular dynamics. The embedded nanoparticles, unlike existing conventional models, are assumed to possess uniaxial magnetic anisotropy. By that, the consideration is brought closer to reality. Evolution of particle chains, both in length and straightness, is handled with the aid of “axial” radial distribution function that is sensitive to orientation of the aggregates. The effect of the particle magnetic anisotropy on the structural alterations as well as on volume changes of MFGs is demonstrated.

We propose a model of hybrid magnetic elastomers filled with a mixture of magnetically soft and magnetically hard microparticles. The magnetically hard particles are described by the Stoner–Wohlfarth model, the magnetically soft phase obeys the Frohlich–Kennelly equation. The interaction between the two types of particles is described by the mean-field approach. First-order reversal curve (FORC) diagrams were calculated for different values of the elastomer matrix elasticity. We demonstrate that the diagrams display specific new features, which identify the presence of both a deformable matrix and the two types of magnetic particles.

Small angle neutron scattering method (SANS) is used to characterize the structure properties of the polymer matrix of magnetic elastomers (MEs) of the same material content but with different magnetic textures. For that, series of silicone-rubber elastomers mixed with a ferrofluid and polymerized with/without external magnetic field were studied. In the species of pure rubber and the ME samples synthesized without field, SANS reveals a substantial number of large polymer coils (blobs) which are vertically prolate. The case of MEs polymerized under the magnetic field that is also vertically directed, is different. SANS data indicates that there the blobs are preferably elongated in the direction normal to the field.

A model has been proposed for the complete (2D) orientational Brownian motion of axially symmetric particles in a Kelvin viscoelastic medium. It has been shown that the elastic response of the carrier medium can be correctly determined by introducing an additional variable, the moment elastic forces, which, together with the particle-orientation vector, composes the complete set of phase variables of a system. A set of stochastic equations (Langevin equations) has been derived for these values. This set has been employed to construct, under the effective field approximation, a simple set of moment equations describing the magnetization dynamics of a ferrogel, i.e., a dispersion of single-domain nanoparticles in a gel matrix. The theoretical results qualitatively agree with the data of magnetic microrheology of gelatin-based ferrogels.

A consistent theory of ferromagnetic resonance in a dilute suspension of superparamagnetic particles with uniaxial anisotropy of arbitrary strength is presented. The developed approach is used for studying the high-frequency response of a magnetic fluid at different temperatures. It is shown that in a certain temperature interval the absorption line splits into two components. The width of this interval is essentially dependent on the magnitude of the particle anisotropy.

The behavior of a pair of magnetizable spherical particles embedded in a viscoelastic elastomer is studied for the case, where an AC magnetic field is applied along the center-to-center line of the particles. This system is considered as a small-scale (mesoscopic) structure element of a magnetorheological elastomer. Under a quasistatic cycle of the field, the system in question is prone to hysteretic behavior. Namely, the particles initially positioned well apart, at some finite field strength, fall onto one another (cluster) and reside in this state until the field decreases well below the value at which the cluster has been formed. Under dynamic cycling of the field, the viscous friction interferes with the particle displacement process and impedes the occurrence of the magnetodeformational hysteresis of the element. Starting from small-amplitude overdamped oscillations and enhancing the role of magnetic forces over viscous ones, we show how the anharmonicity of the system grows while it approaches the transition threshold, above which the dynamics of the element includes the cluster state. The dynamic hysteresis of magnetization that accompanies the mechanical oscillations of the particles is presented as well.

Hysteresis loops of magnetoactive elastomers (MAE) exhibit several peculiar features, including low coercivity in comparison with mechanically hard materials and a striking mismatch between first consecutive magnetization loops. The difference between loops diminishes during cycling of the applied magnetic field, thus displaying a training effect. The article presents the results of our research on a hypothesis explaining this phenomenon. Magnetization of a multigrain ferromagnetic particle is modeled at fields not exceeding the anisotropy field of the constituent grain material. It is shown that because of rotation of particles in an elastomer matrix, different branches of their hysteresis loops stem from the magnetization switching of different grain groups. This process is illustrated in detail by use of simple systems with a small number of grains. The results of simulation are supported with magnetic measurements of both MAE samples and epoxied powder. Coercivity of the modeled trained loops is in the order-of-magnitude agreement with the experiment.

The theory of orientational motion of a Brownian magnetic nanoparticle embedded in a viscoelastic medium and subjected to a time-dependent uniform magnetic field is developed. The rheology of the viscoelastic environment of the particle is modeled by the Jeffreys scheme, which under variation of a minimal number of parameters is able to resemble a wide range of soft materials: from a weakly structured (nearly Newtonian) polymer solution to a gel. It is shown that in the Jeffreys model, the diffusional orientational motion of a particle is a combination of two modes, which could be associated with a fast motion within the polymer mesh cell and a slow displacement that involves deformation of the mesh, respectively. The dependencies of the reference times of both relaxation modes on the Jeffreys viscous and elastic parameters and temperature are found. It turns out that in substantially viscoelastic media, the rate of the slow mode (it dominates in relaxation) quadratically depends on the matrix temperature. This effect does not have analogs in linearly viscous systems. For an ensemble of magnetic nanoparticles in viscoelastic and gel Jeffreys matrices: (1) the dynamic magnetic susceptibility is derived and evaluated both within an exact approach and in a simple approximation; (2) the problem of magnetic relaxometry, i.e., evolution of magnetization after step-wise turning off the field, is solved; (3) the specific power loss caused by viscous dissipation generated by the particles under an ac field is analyzed as a function of the rheological parameters. Results (1) and (2) provide simple models for magnetic nanorheology; consideration (3) advances the physics of magnetic hyperthermia in viscoelastic and gel-like media.

The microstructure of ferrogels (FG) is investigated with the aid of coarse-grained molecular dynamics. Small FGs samples are considered: about 100 magnetic nanoparticles placed in the nodes of a piece of a quasi-regular polymer mesh. The changes of internal configuration under variations of the intensity of interparticle dipolar interaction, concentration and temperature are monitored. The obtained results evidence that the radial distribution function is a sensitive indicator of self-organization of the magnetic nanoparticles in FGs, thus yielding a robust basis for interpretation of spectroscopic measurements.

A uniaxially anisotropic superparamagnetic particle suspended in a viscous fluid and subjected to an ac field is considered. Consistently taking into account both internal (Neel) and external (Brownian) magnetic relaxations, a simple expression for the dynamic susceptibility is obtained. This result, with regard to the ac field energy absorption, is compared to the common heuristic approach. This is done for a model polydisperse colloid containing maghemite nanoparticles, which are assumed to posses either bulk or surface magnetic anisotropy. It is shown that viscous losses caused by the particle motion in a fluid matrix make important contribution to the full magnetic response of a ferrocolloid and, thus, its ability to absorb the ac field energy. The obtained exact expression, which takes in both dissipation mechanisms, paves the way to correct optimization of the nanoparticle-mediated heating effect.

The semi-adiabatic theory of magnetic response of a uniaxial single-domain ferromagnetic particle is presented. The approach is developed in the context of the kinetic theory and allows for any orientation of the external field. Within this approximation, the dynamic magnetic hysteresis loops in an ac field are calculated. It is demonstrated that they very closely resemble those obtained by the full kinetic theory. The behavior of the effective coercive force is analyzed in detail, and for it a simple formula is proposed. This relation accounts not only for the temperature behavior of the coercive force, as the previous ones do, but also yields the dependence on the frequency and amplitude of the applied field.

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.

A model is presented to studying the pseudo-plasticity effect in a soft magnetic elastomer: a weakly-linked polymer matrix filled with micron-size particles of a ferromagnet with virtually zero coercivity (e.g., iron carbonyl). If the filling fraction of the composite is sufficiently high, such a material displays a remarkable ability to radically change its rheological behavior: being subjected to external magnetic field, the sample loses its elasticity and responds to external mechanical load as if being made of a ductile substance with no tendency to restore its initial shape. In our simulations we show that, considering the system at the mesoscopic scale (as an assembly of particles embedded in an elastic continuum) and taking into account just the magnetostatic and elastic interactions, one is able to qualitatively correct reproduce the field-induced pseudo-plasticity effect.

Coarse-grained molecular dynamics simulation is performed to analyse structure and deformation response of a magnetic polymersome – a submicron hollow capsule whose membrane is made of amphiphilic block-copolymer, and the intramembrane space is filled with magnetic nanoparticles. The major prospects of these objects are related to controlled drug release. The developed many-particle model is used to calculate equilibrium magnetized configurations of polymersomes with different intensity of magnetic interparticle coupling. The analysis of those data reveals that under external magnetic field, an initially spherical capsule stretches along the field direction assuming a spheroid-like shape. This behaviour is caused by the field-driven tendency of the nanoparticles to aggregate inside the polymersome membrane. At weak interparticle magnetic interaction no extended chains are observed. Accordingly, the deformation and density redistribution are relatively small. In a polymersome with strong interparticle coupling the nanoparticles self-organize in long chains aligned with the field, and this induces considerable elongation of the polymersome along with accumulation of the majority of the nanoparticles in the “equatorial” zone of the membrane of the capsule.

Orientational Brownian motion of a solid nanoparticle embedded in a viscoelastic medium modelled by the Jeffreys fluid is studied. The particle bears a magnetic moment and is subjected to a combination of a constant (bias) and a weak (probing) AC magnetic fields. First, the set of Langevin equations of the problem is presented and then on that basis a corresponding kinetic equation (KE) is derived. The solution of the KE that is linear with respect to the AC field amplitude is worked out. This solution is virtually exact since it is cast as a series that enables one to obtain the result with arbitrary accuracy. In that framework, a closed expression for the dynamic magnetic susceptibility of a statistical ensemble of magnetic particles in a viscoelastic environment is found, analyzed and compared to the previously existed approximate solution. It is shown that: (i) the dynamic elasticity of the medium has a substantial effect on the magnetic spectra; (ii) to correctly describe this effect, one has to retain a considerable number of terms in the series that presents the KE solution.

The Monte Carlo method, which is a powerful tool for finding equilibrium states (and/or their corresponding characteristics) in systems of a different nature in many cases allows one to also describe the course of kinetic processes. This work presents the Monte Carlo modeling of magnetic relaxation and dynamics of induced magnetization of an ensemble of ferromagnetic nanoparticles. It is shown by comparison with the exact solution that in both problems there is a proportionality between the number of calculation steps and the physical duration of the transition or periodic process. On this basis, relationships are proposed making it possible to estimate the time interval corresponding to a single Monte Carlo step. For the process of magnetic relaxation the obtained result is a generalization of the well-known Nowak-Chantrell-Kennedy formula for the case of particles with finite magnetic anisotropy.

A small angle X-ray scattering experiment was performed on ferrihydrite nanoparticles produced by bacteriumKlebsiella oxytoca. Structural investigations of ultrasonicated samples of different concentrations of water-dispersed particles were performed. Model calculations and fitting procedures revealed scattering objects of an elongated shape of 6.73±0.16 nm radius of gyration.

Heat generation by viscous dissipation in a dilute suspension of single-domain ferromagnetic particles in a rotating magnetic field is analyzed by assuming that the suspended particles have a high magnetic rigidity. The problem is solved by using a kinetic approach based on a rotational diffusion equation. Behavior of specific loss power (SLP) as a function of field strength H and frequency ω is examined at constant temperature. SLP increases as either of these parameters squared when the other is constant, eventually approaching a saturation value. The function SLP(H, ω) can be used to determine optimal and admissible ranges of magnetically induced heating.

The idea of establishing coherent relaxation in an assembly of magnetic moments by placing it in a passive resonator [1, 2] is applied to the case of single-domain ferromagnetic nanoparticles. The magnetodynamics of a particle is governed by the Landau-Lifshitz equation modified by adding the resonator feedback field. The particle uniaxial magnetic anisotropy and the interparticle dipole-dipole interactions are taken into account.

The method of torsion oscillations is used to measure the dynamic modulus of elasticity of magnetically controlled elastomers that comprise silicone rubber and carbonyl iron in the low-frequency (up to 100 Hz) range. The samples are synthesized in the absence of a magnetic field; therefore, they have an isotropic structure. In the measurements, a constant magnetic field (up to 24 kA/m) is superimposed along the axis of forced torsion oscillations of the sample. A simple model of the rheological behavior of magnetically controlled elastomers is proposed; the problem of torsion oscillations of a cylindrical sample is solved. From the comparison with the experiment for the materials under study, we determine the coefficients of the theoretical model and the corrections to them, which are made because of variations in the rheology of magnetically controlled elastomers under the influence of a magnetic field. The derived relations make it possible to exclude artifacts and to adequately describe dependences of the storage and loss moduli on the frequency of mechanical loading and the strength of the applied magnetic field.

A simple rheological model is introduced to describe the plasticity in soft magnetic elastomers induced by an external field. In addition to the elasticity of the polymer matrix, the effect of dry friction (plasticity), which is provided by the rearrangement of the spatial structure of a dispersed magnetic subsystem in the course of the deformation of the material by a sufficiently strong field, is taken into account in the model. Model parameters are estimated from the reported experimental data on the deformation of cylindrical samples based on a soft silicon elastomer filled with quasispherical microparticles of carbonyl iron.

The Organizers of the 3rd International School and Workshop on Complex and Magnetic Soft Matter Systems: Structure and Physico -Mechanical Properties acknowledge the financial support from the Grants of the Government Plenipotentiary Representative of Romania and scientific projects of the Cooperation Programmes JINR-Romania. Special gratitude to the JINR and FLNP administration and staff is expressed.

The results of modelling the behaviour of a pair of magnetisable spherical particles embedded in a viscoelastic elastomer with Kelvin rheology are presented for the case, where an ac magnetic field is exerted along the intercenter line of the particles. This system is considered as a small-scale (mesoscopic) structure element of a magnetorheological elastomer. The system in question is known to be able for hysteretic behaviour under the quasistatic cycle of the field. The particles initially positioned well apart, at some finite field strength, fall onto one another (cluster) and dwell in this state until the field decreases well below the value at which the cluster has been formed. Under dynamic cycling of the field, the viscous friction interferes the particle displacement process and impedes occurrence of the magnetodeformational hysteresis of the element. Starting from small-amplitude overdamped oscillations and enhancing the role of magnetic forces over viscous ones, we show how the anharmonicity of the system grows while it approaches the transition threshold, above which the dynamics of the element includes the cluster state. The dynamic hysteresis of magnetisation that accompanies the mechanical oscillations of the particles is presented as well.

Monte Carlo method with a variable number of calculation steps is used for simulations of the magnetization dynamics in a microsphere consisting of a non-magnetic core coated with a layer of magnetic nanograins. Namely, the dynamic magnetic hysteresis under an AC field in such a core-shell microsphere is considered for the cases of non-interacting particles and with allowance for the interparticle dipole-dipole interaction. The time portions corresponding to one calculation step are estimated from comparison of the Monte Carlo-Metropolis results and a numerically-exact solution of the kinetic equation.

Ferrihydrite nanoparticles (2–5 nm in size) produced by bacteria culture in the course of biomineralization of iron salt solutions from a natural medium exhibit unique magnetic properties: they are characterized by both the antiferromagnetic order inherent in a bulk ferrihydrite and the spontaneous magnetic moment due to the decompensation of spins in sublattices of a nanoparticle. The magnetic susceptibility enhanced by the superantiferromagnetism effect and the magnetic moment independent of the magnetic field provide the possibility of magnetically controlling these natural objects. This has opened up the possibilities for their use in nanomedicine and bioengineering. The results obtained from measurements of the magnetic properties of the bacterial ferrihydrite in its two main crystalline modifications are reported. This has made it possible to determine numerical values of the magnetic parameters of real biomineral nanoparticles.

The ground magnetic state of systems of finite number N of single-domain particles confined in a spherical monolayer is investigated by numeric simulations. Two model situations are considered. In the first, the particle positions are imposed and fixed, in the second, the particles are able to move freely within the layer; in the latter case the excluded volume effect is taken into account. It is found that in all the range studied ( N ⩽200) the ground state of the system retains a considerable extent of magnetic vorticity (toroid moment). Moreover, the magnetic and toroid moments are correlated: they are approximately perpendicular to one another.

Energy absorption produced by non-interacting single-domain ferromagnetic particles suspended in a liquid matrix is studied. The particles are assumed to be rigid dipoles so that all the dissipation in the system is due to the viscous flow induced in the matrix by orientational motion of the particles in response to a linearly polarized uniform AC field. The kinetic approach employed is valid for arbitrary values of the field frequency and amplitude. Comparison of the results obtained against those of the conventional phenomenological description reveals that the amplitude–frequency dependencies delivered by the latter are limited to weak field regimes.

A kinetic model is proposed to describe the low-frequency magnetodynamics of antiferromagnetic nanoparticles suspended in a fluid. Because of their small size, apart from an anisotropic magnetic susceptibility typical of antiferromagnets, these particles also have a constant magnetic moment caused by sublattice decompensation. An orientational crossover takes place in such a nanosuspension (colloid) when magnetized by a constant field: the axes of easy particle magnetization that were initially aligned along the field become oriented perpendicularly. This effect changes significantly the characteristics of the system’s magnetic response: the dynamic susceptibility spectrum and the relaxation time in a pulsed field.

The loss of the stability of a thin flat soft ferroelastic plate (membrane) fixed along its periphery in a magnetic field normal to its surface is considered using a continuum model of nonlinear magnetoelasticity. The relation between the instability threshold field and the magnetic and geometric parameters of the membrane is found, and the supercritical shape of the membrane (dome) is calculated. The calculation and experimental results agree well without fitting parameters.

The influence of a magnetic field on the viscosity of dilute dispersion of spherical magnetohard particles in viscoelastic matrix, the Maxwell liquid with a single relaxation time, is studied. Calculations are performed on the basis of the equation of rotational diffusion derived previously by the authors. It is shown that the effective viscosity of magnetic fluid decreases in the presence of a field. This unexpected result is clearly explained in physical terms.

The state of the art in the theory of dynamic response properties of single-domain particles (macrospins) is presented. If allowing for thermofluctuational effects, one admits temperature as an essential parameter. This implies substantial modification of classical results of magnetodynamic phenomenology from quasi-static hysteresis of magnetization to FMR. With a number of examples we demonstrate how this work is done on the basis of Brown's kinetic equation.

Theory of Brownian motion of a fine particle in a viscoelastic fluid continuum is developed. The rheology of the embedding medium is described in terms of classical structure (spring-and-damper) schemes. It is shown that a great variety of conceivable ramifications of such structures could be reduced to an effective Jeffreys model: a Maxwell chain shunted by a damper. This scheme comprises just three material parameters: two for viscosities (fast and slow) and one for elasticity. For the thermal motion of a particle in such a fluid, the set of Langevin equations is derived. In the cases of 1D translational and 2D orientational (rotation about fixed axis) motions, the time dependencies of mean-square displacements are found analytically. It is shown that in a Jeffreys fluid, the Brownian motion has three regimes: fast diffusion (short times), slow diffusion (long times) and a crossover of those resulting in virtual localization (dynamic confinement) of the particle. Experimental evidence for that taken from the literature is presented. The developed formalism is applied to the particles with “frozen-in” dipole moments, whose orientational motion is a combination of Brownian diffusion and regular excitation by an AC field. The dynamic susceptibilities are calculated for the ensembles of particles liable for either 2D or 3D rotations. Comparison shows that the out-of-phase part of the susceptibility χ ′′(ω ) in 3D case differs considerably from that of 2D case. Function χ (ω ) for the forced 3D rotary diffusion in a Jeffreys fluid is found for the first time. In our view, it would be useful for realistic theoretical interpretation of microrheology data and the more so for estimating the particle-mediated AC field-induced heating (magnetic hyperthermia) in viscoelastic fluid media.

The coefficient of reflection of microwave radiation from a plane layer of a magnetic fluid magne- tized in a field of 11.5 Oe is measured in a frequency range of 24 MHz in temperature intervals 293-333 K; the volume concentration of the magnetic phase (magnetite) is 15 vol %. In this range, the dynamic susceptibility of nanoparticles exhibits strong dispersion (ferromagnetic resonance). The experimental results are interpreted on the basis of the theory of high-frequency magnetization for an ensemble of noninteracting isotropic super- paramagnetic particles. It is found that this simple model provides a satisfactory description of the temperature and frequency dependences of the reflection coefficient.

A ferroparticle driven by an AC field in a Maxwell fluid is studied with regard to magnetic microrheology. Orientational kinetics is simplified by neglecting the particle rotary inertia. While in a Newtonian fluid the inertialess approximation for nanoparticles is valid practically unboundedly, in a viscoelastic matrix there exists an important restriction on the stress relaxation time. Assuming weak non-equilibrium, the magneto-orientational relaxation times and low-frequency magnetic spectra are found in the presence of a constant (bias) field. Another way of particle-mediated probing of the matrix involves the field-induced birefringence of a suspension; under zero bias field the corresponding (quadratic) susceptibility is found.

A sphere with a concentric spherical cavity is considered as a model for a magnetic vesicle: a bubble whose walls are made of a magnetiziable polymer or ferrogel. The material of the body is assumed to be homogeneous, linearly magnetizable, and linearly elastic. Using self-consistent finite element calculation we study the elongation of the sphere under the action of a uniform field and the change of the cavity volume which accompanies this deformation.

Dynamic magnetic hysteresis in a nanoparticle is treated with the aid of Brown's kinetic equation. The case in question is the one where the period of the external field falls in between the short intrawell and long interwell magnetic relaxation times. Dependencies of the hysteresis loop on the frequency and amplitude of the field and on temperature are found and analyzed.

Groups comprising few rigid magnetic dipoles (single-domain particles) organized in a regular flat N -polygon pattern are considered. The group is subjected to a constant magnetic field in the plane of the structure and responds to the field by magnetization in the same plane. In such a cluster, along with the magnetization, another order parameter does exist: the toroid moment. External field induces magnetization and suppresses the toroid moment. If the matrix of the cluster complies elastically to the interparticle forces, two types of behavior are possible. The particles may either slightly approach one another or undergo a collapse which results in their tight clotting.