Surface Modification of Magnetoactive Elastomers by Laser Micromachining

It has been recently demonstrated that laser micromachining of magnetoactive elastomers is a very convenient method for fabricating dynamic surface microstructures with magnetically tunable properties, such as wettability and surface reflectivity. In this study, we investigate the impact of the micromachining process on the fabricated material's structural properties and its chemical composition. By employing scanning electron microscopy, we investigate changes in size distribution and spatial arrangement of carbonyl iron microparticles dispersed in the polydimethylsiloxane (PDMS) matrix as a function of laser irradiation. Based on the images obtained by a low vacuum secondary electron detector, we analyze modifications of the surface topography. The results show that most profound modifications occur during the low-exposure (8 J/cm2) treatment of the surface with the laser beam. Our findings provide important insights for developing theoretical models of functional properties of laser-sculptured microstructures from magnetoactive elastomers.

Transient Response of Macroscopic Deformation of Magnetoactive Elastomeric Cylinders in Uniform Magnetic Fields

Significant deformations of bodies made from compliant magnetoactive elastomers (MAE) in magnetic fields make these materials promising for applications in magnetically controlled actuators for soft robotics. Reported experimental research in this context was devoted to the behaviour in the quasi-static magnetic field, but the transient dynamics are of great practical importance. This paper presents an experimental study of the transient response of apparent longitudinal and transverse strains of a family of isotropic and anisotropic MAE cylinders with six different aspect ratios in time-varying uniform magnetic fields. The time dependence of the magnetic field has a trapezoidal form, where the rate of both legs is varied between 52 and 757 kA/(s·m) and the maximum magnetic field takes three values between 153 and 505 kA/m. It is proposed to introduce four characteristic times: two for the delay of the transient response during increasing and decreasing magnetic field, as well as two for rise and fall times. To facilitate the comparison between different magnetic field rates, these characteristic times are further normalized on the rise time of the magnetic field ramp. The dependence of the normalized characteristic times on the aspect ratio, the magnetic field slew rate, maximum magnetic field values, initial internal structure (isotropic versus anisotropic specimens) and weight fraction of the soft-magnetic filler are obtained and discussed in detail. The normalized magnetostrictive hysteresis loop is introduced, and used to explain why the normalized delay times vary with changing experimental parameters.

Effects of Filler Anisometry on the Mechanical Response of a Magnetoactive Elastomer Cell: A Single-Inclusion Modeling Approach

A finite-element model of the mechanical response of a magnetoactive elastomer (MAE) volume element is presented. Unit cells containing a single ferromagnetic inclusion with geometric and magnetic anisotropy are considered. The equilibrium state of the cell is calculated using the finite-element method and cell energy minimization. The response of the cell to three different excitation modes is studied: inclusion rotation, inclusion translation, and uniaxial cell stress. The influence of the magnetic properties of the filler particles on the equilibrium state of the MAE cell is considered. The dependence of the mechanical response of the cell on the filler concentration and inclusion anisometry is calculated and analyzed. Optimal filler shapes for maximizing the magnetic response of the MAE are discussed.

Programming and Reprogramming the Viscoelasticity and Magnetic Response of Magnetoactive Thermoplastic Elastomers

We present a novel type of magnetorheological material that allows one to restructure the magnetic particles inside the finished composite, tuning in situ the viscoelasticity and magnetic response of the material in a wide range using temperature and an applied magnetic field. The polymer medium is an A-g-B bottlebrush graft copolymer with side chains of two types: polydimethylsiloxane and polystyrene. At room temperature, the brush-like architecture provides the tissue mimetic softness and strain stiffening of the elastomeric matrix, which is formed through the aggregation of polystyrene side chains into aggregates that play the role of physical cross-links. The aggregates partially dissociate and the matrix softens at elevated temperatures, allowing for the effective rearrangement of magnetic particles by applying a magnetic field in the desired direction. Magnetoactive thermoplastic elastomers (MATEs) based on A-g-B bottlebrush graft copolymers with different amounts of aggregating side chains filled with different amounts of carbonyl iron microparticles were prepared. The in situ restructuring of magnetic particles in MATEs was shown to significantly alter their viscoelasticity and magnetic response. In particular, the induced anisotropy led to an order-of-magnitude enhancement of the magnetorheological properties of the composites.

Effect of microstructure evolution on the mechanical behavior of magneto-active elastomers with different matrix stiffness

Evolution of microstructure in magneto-active elastomers (MAEs) which can be caused by an applied magnetic field is a fascinating phenomenon with a significant impact on the mechanical behavior of the composite. To gain insight into the underlying mechanisms of this phenomenon, it is essential to create a model that can appropriately describe the field induced change in the particle distribution and its mechanical implications. The magneto-mechanical coupling is driven by magnetic interactions between the particles in the applied field. These magnetic interactions can result in macroscopic deformation of the sample and also in rearrangement of the microstructure, i.e. the local positions of the particles. In the case of initially isotropic MAEs made with a sufficiently soft matrix, this leads to the formation of chains of magnetized particles, creating a significant increase in the mechanical moduli along the field direction. In this paper, we implement a transversely isotropic Neo-Hookean material model to account for such anisotropic elastic behavior. A dipolar mean field approach is used to describe magnetic interactions between the particles. A penalty term is introduced to compensate for the micro-mechanical elastic energy required to move the particles inside the cross-linked elastomer. The resulting model can predict the huge magneto-rheological effects observed in experiments, and improves our understanding of how microstructure evolution affects magnetically induced deformation and stiffness of MAEs.

Tailoring the Magnetic and Electrical Properties of Epoxy Composites Containing Olive-Derived Biochar through Iron Modification

The combination of conductive carbon together with magnetic particles is a consolidated strategy to produce cutting-edge fillers for the production of polymer composites able to shield against microwave radiation. In this work, we developed and characterized an iron-tailored biochar obtained from the pyrolysis of olive pruning which was added as filler for the preparation of epoxy composites. The biochar-based composites were obtained by keeping the filler concentration at 10 and 40 wt.%. An extensive characterization was carried out in order to assess the electrical and magnetic properties of the composites containing biochar and iron-tailored biochar. The highest DC electrical conductivity of 59 mS/m was observed in the 40 wt.% iron-tailored biochar-loaded composite, while the reduction of the filler loading led to a drastic reduction in conductivity: 60 μS/m in the 10 wt.%-loaded composite. Ferromagnetic behavior of composites containing iron-tailored biochar is visible in the emerging hysteretic behavior, with a magnetic signal increasing with the filler concentration. Finally, both the complex permittivity (ε') and the AC conductivity (σ) are enhanced by increasing the BC filler amount in the matrix, regardless of the presence of iron.

Dynamically tunable lamellar surface structures from magnetoactive elastomers driven by a uniform magnetic field

Stimuli responsive materials are key ingredients for any application that requires dynamically tunable or on-demand responses. In this work we report experimental and theoretical investigation of magnetic-field driven modifications of soft-magnetic elastomers whose surface was processed by laser ablation into lamellar microstructures that can be manipulated by a uniform magnetic field. We present a minimal hybrid model that elucidates the associated deflection process of the lamellae and explains the lamellar structure frustration in terms of dipolar magnetic forces arising from the neighbouring lamellae. We experimentally determine the magnitude of the deflection as a function of magnetic flux density and explore the dynamic response of lamellae to fast changes in a magnetic field. A relationship between the deflection of lamellae and modifications of the optical reflectance of the lamellar structures is resolved.

Microstructured Magnetoactive Elastomers for Switchable Wettability

We demonstrate the control of wettability of non-structured and microstructured magnetoactive elastomers (MAEs) by magnetic field. The synthesized composite materials have a concentration of carbonyl iron particles of 75 wt.% (≈27 vol.%) and three different stiffnesses of the elastomer matrix. A new method of fabrication of MAE coatings on plastic substrates is presented, which allows one to enhance the response of the apparent contact angle to the magnetic field by exposing the particle-enriched side of MAEs to water. A magnetic field is not applied during crosslinking. The highest variation of the contact angle from (113 ± 1)° in zero field up to (156 ± 2)° at about 400 mT is achieved in the MAE sample with the softest matrix. Several lamellar and pillared MAE structures are fabricated by laser micromachining. The lateral dimension of surface structures is about 50 µm and the depth varies between 3 µm and 60 µm. A systematic investigation of the effects of parameters of laser processing (laser power and the number of passages of the laser beam) on the wetting behavior of these structures in the absence and presence of a magnetic field is performed. In particular, strong anisotropy of the wetting behavior of lamellar structures is observed. The results are qualitatively discussed in the framework of the Wenzel and Cassie–Baxter models. Finally, directions of further research on magnetically controlled wettability of microstructured MAE surfaces are outlined. The obtained results may be useful for the development of magnetically controlled smart surfaces for droplet-based microfluidics.

Mechanical Force Acting on Ferrogel in a Non-Uniform Magnetic Field: Measurements and Modeling

The development of magnetoactive microsystems for targeted drug delivery, magnetic biodetection, and replacement therapy is an important task of present day biomedical research. In this work, we experimentally studied the mechanical force acting in cylindrical ferrogel samples due to the application of a non-uniform magnetic field. A commercial microsystem is not available for this type of experimental study. Therefore, the original experimental setup for measuring the mechanical force on ferrogel in a non-uniform magnetic field was designed, calibrated, and tested. An external magnetic field was provided by an electromagnet. The maximum intensity at the surface of the electromagnet was 39.8 kA/m and it linearly decreased within 10 mm distance from the magnet. The Ferrogel samples were based on a double networking polymeric structure which included a chemical network of polyacrylamide and a physical network of natural polysaccharide guar. Magnetite particles, 0.25 micron in diameter, were embedded in the hydrogel structure, up to 24% by weight. The forces of attraction between an electromagnet and cylindrical ferrogel samples, 9 mm in height and 13 mm in diameter, increased with field intensity and the concentration of magnetic particles, and varied within 0.1–30 mN. The model provided a fair evaluation of the mechanical forces that emerged in ferrogel samples placed in a non-uniform magnetic field and proved to be useful for predicting the deformation of ferrogels in practical bioengineering applications.

Theory of static magnetization of magnetopolymer composites: The second virial approximation

This paper deals with a theoretical study of the static magnetization of a composite, consisting of nanodisperse single-domain ferromagnetic particles immobilized in a nonmagnetic medium. This situation is typical for magnetopolymer systems-smart materials, which found applications in high industrial and biomedical applications. It is supposed that the composite was polymerized without an external magnetic field; after curing the particles retain the spatial distribution and orientation of their axes of easy magnetization that they had before the host medium polymerization. Our results demonstrate that, in contrast to ferrofluids with a liquid host medium, a magnetic interparticle interaction provokes a decrease of the macroscopic magnetization of the system.

Magnetoelectric Response of Laminated Cantilevers Comprising a Magnetoactive Elastomer and a Piezoelectric Polymer, in Pulsed Uniform Magnetic Fields

The voltage response to pulsed uniform magnetic fields and the accompanying bending deformations of laminated cantilever structures are investigated experimentally in detail. The structures comprise a magnetoactive elastomer (MAE) slab and a commercially available piezoelectric polymer multilayer. The magnetic field is applied vertically and the laminated structures are customarily fixed in the horizontal plane or, alternatively, slightly tilted upwards or downwards. Six different MAE compositions incorporating three concentrations of carbonyl iron particles (70 wt%, 75 wt% and 80 wt%) and two elastomer matrices of different stiffness are used. The dependences of the generated voltage and the cantilever’s deflection on the composition of the MAE layer and its thickness are obtained. The appearance of the voltage between the electrodes of a piezoelectric material upon application of a magnetic field is considered as a manifestation of the direct magnetoelectric (ME) effect in a composite laminated structure. The ME voltage response increases with the increasing total quantity of the soft-magnetic filler in the MAE layer. The relationship between the generated voltage and the cantilever’s deflection is established. The highest observed peak voltage around 5.5 V is about 8.5-fold higher than previously reported values. The quasi-static ME voltage coefficient for this type of ME heterostructures is about 50 V/A in the magnetic field of ≈100 kA/m, obtained for the first time. The results could be useful for the development of magnetic field sensors and energy harvesting devices relying on these novel polymer composites.

The role of thermal diffusion, particle clusters, hydrodynamic and magnetic forces on the flow behaviour of magneto-polymer composites

In this paper, we study the shear-induced flow of magneto-polymer composites, consisting of dispersions of magnetic particles in solutions of polymers, as a competition between the colloidal forces amid particles and their bulk transport induced by the hydrodynamic forces. For this aim, we analyse the role of different experimental parameters. Firstly, by using only solutions of a well-known anionic polymer (sodium alginate), we provoke a moderate hindering of particle movement, but keeping the liquid-like state of the samples. On the contrary, a gel-like behaviour is conferred to the samples when a cationic polymer (chitosan) is additionally added, which further reduces the particle movement. We analyse the effect of an applied magnetic field, which is opposed to particle transport by hydrodynamic forces, by inducing magnetic attraction between the particles. We perform the analysis under both stationary and oscillatory shear. We show that by using dimensionless numbers the differences between samples and experimental conditions are emphasized. In all cases, as expected, the transport of particles driven by bulk hydrodynamic forces dominates at high values of the shear rate. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Internal structures and mechanical properties of magnetic gels and suspensions

We present results of theoretical and computer study of linear chain-like and complicated labyrinth structures in magnetic gels and suspensions as well as effect of these internal structures on macroscopic elastic properties of the composites. Our results show that at a certain threshold deformation, the structures experience a rupture which provokes a fall down of the macroscopic elastic stress, induced by the deformation. This effect is detected for both shear and tensile deformations. The results of calculations are compared with experimental data’s.

Dielectric Spectroscopy of Hybrid Magnetoactive Elastomers

Dielectric properties of two series of magnetoactive elastomers (MAEs) based on a soft silicone matrix containing 35 vol% of magnetic particles were studied experimentally in a wide temperature range. In the first series, a hybrid filler representing a mixture of magnetically hard NdFeB particles of irregular shape and an average size of 50 μm and magnetically soft carbonyl iron (CI) of 4.5 μm in diameter was used for MAE fabrication. MAEs of the second series contained only NdFeB particles. The presence of magnetically hard NdFeB filler made it possible to passively control MAE dielectric response by magnetizing the samples. It was shown that although the hopping mechanism of MAEs conductivity did not change upon magnetization, a significant component of DC conductivity appeared in the magnetized MAEs presumably due to denser clustering of interacting particles resulting in decreasing interparticle distances. The transition from a non-conducting to a conducting state was more pronounced for hybrid MAEs containing both NdFeB and Fe particles with a tenfold size mismatch. Hybrid MAEs also demonstrated a considerable increase in the real part of the complex relative permittivity upon magnetization and its asymmetric behavior in external magnetic fields of various directions. The effects of magnetic filler composition and magnetization field on the dielectric properties of MAEs are important for practical applications of MAEs as elements with a tunable dielectric response.

Magneto-Mechanical Coupling in Magneto-Active Elastomers

In the present work, the magneto-mechanical coupling in magneto-active elastomers is investigated from two different modeling perspectives: a micro-continuum and a particle–interaction approach. Since both strategies differ significantly in their basic assumptions and the resolution of the problem under investigation, they are introduced in a concise manner and their capabilities are illustrated by means of representative examples. To motivate the application of these strategies within a hybrid multiscale framework for magneto-active elastomers, their interchangeability is then examined in a systematic comparison of the model predictions with regard to the magneto-deformation of chain-like helical structures in an elastomer surrounding. The presented results show a remarkable agreement of both modeling approaches and help to provide an improved understanding of the interactions in magneto-active elastomers with chain-like microstructures.

Modeling and theoretical description of magnetic hybrid materials—bridging from meso- to macro-scales

Magnetic gels and elastomers consist of magnetic or magnetizable colloidal particles embedded in an elastic polymeric matrix. Outstanding properties of these materials comprise reversible changes in their mechanical stiffness or magnetostrictive distortions under the influence of external magnetic fields. To understand such types of overall material behavior from a theoretical point of view, it is essential to characterize the substances starting from the discrete colloidal particle level. It turns out that the macroscopic material response depends sensitively on the mesoscopic particle arrangement. We have utilized and developed several theoretical approaches to this end, allowing us both to reproduce experimental observations and to make theoretical predictions. Our hope is that both these paths help to further stimulate the interest in these fascinating materials.

Magnetic-field-induced stress in confined magnetoactive elastomers

We present a theoretical approach for calculating the state of stress induced by a uniform magnetic field in confined magnetoactive elastomers of arbitrary shape. The theory explicitly includes the magnetic field generated by magnetizable spherical inclusions in the sample interior assuming a non-linear magnetization behavior. The initial spatial distribution of particles and its change in an external magnetic field are considered. This is achieved by the introduction of an effective demagnetizing factor where both the sample shape and the material microstructure are taken into account. Theoretical predictions are fitted to the stress data measured using a specifically designed experimental setup. It is shown that the theory enables the quantification of the effect of material microstructure upon introducing a specific microstructural factor and its derivative with respect to the extensional strain in the undeformed state. The experimentally observed differences between isotropic and anisotropic samples, compliant and stiff elastomer matrices are explained.

Giant Extensional Strain of Magnetoactive Elastomeric Cylinders in Uniform Magnetic Fields

Elongations of magnetoactive elastomers (MAEs) under ascending-descending uniform magnetic fields were studied experimentally using a laboratory apparatus specifically designed to measure large extensional strains (up to 20%) in compliant MAEs. In the literature, such a phenomenon is usually denoted as giant magnetostriction. The synthesized cylindrical MAE samples were based on polydimethylsiloxane matrices filled with micrometer-sized particles of carbonyl iron. The impact of both the macroscopic shape factor of the samples and their magneto-mechanical characteristics were evaluated. For this purpose, the aspect ratio of the MAE cylindrical samples, the concentration of magnetic particles in MAEs and the effective shear modulus were systematically varied. It was shown that the magnetically induced elongation of MAE cylinders in the maximum magnetic field of about 400 kA/m, applied along the cylinder axis, grew with the increasing aspect ratio. The effect of the sample composition is discussed in terms of magnetic filler rearrangements in magnetic fields and the observed experimental tendencies are rationalized by simple theoretical estimates. The obtained results can be used for the design of new smart materials with magnetic-field-controlled deformation properties, e.g., for soft robotics.

Role of particle clusters on the rheology of magneto-polymer fluids and gels

Even in the absence of cross-linking, at large enough concentration, long polymer strands have a strong influence on the rheology of aqueous systems. In this work, we show that solutions of medium molecular weight (120 000-190 000 g mol^-1) alginate polymer retained a liquid-like behaviour even for concentrations as large as 20% w/v. On the contrary, solutions of alginate polymer of larger (and also polydisperse) molecular weight (up to 600 000 g mol^-1) presented a gel-like behaviour already at concentrations of 7% w/v. We dispersed micrometre-sized iron particles at a concentration of 5% v/v in these solutions, which resulted in either stable magnetic fluids or gels, depending on the type of alginate polymer employed (medium or large molecular weight, respectively). These magneto-polymer composites presented a shear-thinning behaviour that allowed injection through a syringe and recovery of the original properties afterwards. More interestingly, application of a magnetic field resulted in the formation of particle clusters elongated along the field direction. The presence of these clusters intensely affected the rheology of the systems, allowing a reversible control of their stiffness. We finally developed theoretical modelling for the prediction of the magnetic-sensitive rheological properties of these magneto-polymer colloids. This article is part of the theme issue 'Patterns in soft and biological matters'.

Rheology of magnetic colloids containing clusters of particle platelets and polymer nanofibres

Magnetic hydrogels (ferrogels) are soft materials with a wide range of applications, especially in biomedicine because (i) they can be provided with the required biocompatibility; (ii) their heterogeneous structure allows their use as scaffolds for tissue engineering; (iii) their mechanical properties can be modified by changing different design parameters or by the action of magnetic fields. These characteristics confer them unique properties for acting as patterns that mimic the architecture of biological systems. In addition, and (iv) given their high porosity and aqueous content, ferrogels can be loaded with drugs and guided towards specific targets for local (non-systemic) pharmaceutical treatments. The ferrogels prepared in this work contain magnetic particles obtained by precipitation of magnetite nanoparticles onto the porous surface of bentonite platelets. Then, the particles were functionalized by adsorption of alginate molecules and dispersed in an aqueous solution of sodium alginate. Finally, the gelation was promoted by cross-linking the alginate molecules with Ca²⁺ ions. The viscoelastic properties of the ferrogels were measured in the absence/presence of external magnetic fields, showing that these ferrogels exhibited a strong enough magnetorheological effect. This behaviour is explained considering the field-induced strengthening of the heterogeneous (particle-polymer) network generated inside the ferrogel. This article is part of the theme issue 'Patterns in soft and biological matters'.

Patterns in soft and biological matters

The issue is devoted to theoretical, computer and experimental studies of internal heterogeneous patterns, their morphology and evolution in various soft physical systems-organic and inorganic materials (e.g. alloys, polymers, cell cultures, biological tissues as well as metastable and composite materials). The importance of these studies is determined by the significant role of internal structures on the macroscopic properties and behaviour of natural and manufactured tissues and materials. Modern methods of computer modelling, statistical physics, heat and mass transfer, statistical hydrodynamics, nonlinear dynamics and experimental methods are presented and discussed. Non-equilibrium patterns which appear during macroscopic transport and hydrodynamic flow, chemical reactions, external physical fields (magnetic, electrical, thermal and hydrodynamic) and the impact of external noise on pattern evolution are the foci of this issue. Special attention is paid to pattern formation in biological systems (such as drug transport, hydrodynamic patterns in blood and pattern dynamics in protein and insulin crystals) and to the development of a scientific background for progressive methods of cancer and insult therapy (magnetic hyperthermia for cancer therapy; magnetically induced drug delivery in thrombosed blood vessels). The present issue includes works on pattern growth and their evolution in systems with complex internal structures, including stochastic dynamics, and the influence of internal structures on the external static, dynamic magnetic and mechanical properties of these systems. This article is part of the theme issue 'Patterns in soft and biological matters'.

Field-induced surface deformation of magnetoactive elastomers with anisometric fillers: a single-particle model

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.

The loss of mechanical stability and the critical magnetization of a ferromagnetic particle in an elastomer

The rotation of a spherical magnetically soft ferromagnetic particle with uniaxial magnetic anisotropy and located in an elastomer through a certain angle under the action of a magnetic field has been studied theoretically. It was found that if the particle loses its mechanical stability, its rotation angle becomes critically dependent on the magnetic field magnitude. The magnetically induced critical rotation of the particle in the elastomer has a magneto-elastic origin and is accompanied by a critical change in the magnetization of the particle multidomain state, as occurs in the case of second-order magnetic phase transitions. The transition of a particle in a soft-elastic elastomer from the multidomain state into a uniformly magnetized one was shown to be independent of the magnitude of the particle magnetic anisotropy field. The particle rotation was found to result in the appearance of a maximum in the field dependence of the magnetic susceptibility.

Mechanical properties of magnetic gels containing rod-like composite particles

Magnetic gels (ferrogels) are heterogeneous systems structured at the nanoscale that contains magnetic particles dispersed in three-dimensional networks of polymer chains. In the present work, the magnetic particles were synthesized with a core-shell structure, consisting of sepiolite particles covered by magnetite nanoparticles. These composite particles had a rod-like shape with a high aspect ratio. The obtained sepiolite-magnetite particles showed a high enough susceptibility and saturation magnetization. The magneto-rheological (MR) properties, and the intensity of the MR effect, of aqueous suspensions of the synthesized particles were studied. The particles, functionalized by adsorption of alginate molecules, were imbedded in alginate hydrogels to get homogeneous soft materials. The particles were linked to the polymer chains as the knots in a network and dominated in a great extent the mechanical properties of the materials. After determining the optimal compositions of the ferrogels, their viscoelastic properties were measured in the absence/presence of magnetic fields. The results pointed out that the MR effect provided by the clay-magnetite particles was considerably more intense than those achieved in ferrogels that contain spherical magnetic microparticles. Therefore, the imbedding of rod-shaped magnetic particles in hydrogels allows controlling the mechanical properties in a wider range than in conventional ferrogels. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Effect of Material Composition on Tunable Surface Roughness of Magnetoactive Elastomers

We investigated magnetic-field-induced modifications of the surface roughness of magnetoactive elastomers (MAEs) with four material compositions incorporating two concentrations of ferromagnetic microparticles (70 wt% and 80 wt%) and exhibiting two shear storage moduli of the resulting composite material (about 10 kPa and 30 kPa). The analysis was primarily based on spread optical reflection measurements. The surfaces of all four materials were found to be very smooth in the absence of magnetic field (RMS roughness below 50 nm). A maximal field-induced roughness modification (approximately 1 μm/T) was observed for the softer material with the lower filler concentration, and a minimal modification (less than 50 nm/T) was observed for the harder material with the higher filler concentration. All four materials showed a significant decrease in the total optical reflectivity with an increasing magnetic field as well. This effect is attributed to the existence of a distinct surface layer that is depleted of microparticles in the absence of a magnetic field but becomes filled with particles in the presence of the field. We analyzed the temporal response of the reflective properties to the switching on and off of the magnetic field and found switching-on response times of around 0.1 s and switching-off response times in the range of 0.3–0.6 s. These observations provide new insight into the magnetic-field-induced surface restructuring of MAEs and may be useful for the development of magnetically reconfigurable elastomeric optical surfaces.

Non-ergodic tube structures in magnetic gels and suspensions

We present results of a study of internal structures, which can appear in magnetic suspensions and gels filling a flat gap under the influence of a magnetic field applied perpendicular to the gap walls. The considered system consists of magnetizable microparticles with a mean diameter of ∼35 μm. Experimental observation demonstrates that the particles can form stable tube shaped structures elongated along the field direction. These structures have internal cavities. The theoretical analysis, performed in this study, shows that the tubes do not correspond to a thermodynamic equilibrium state of the system and rather present transitive non-ergodic structures. These structures are stacked in a state of local energetic minima because of the relatively large size of the particles and negligible Brownian effects. Our theoretical model is suggested to explain the physical reason of the appearance of tube-like structures.

Tunable dynamic moduli of magnetic elastomers: from characterization by x-ray micro-computed tomography to mesoscopic modeling

Ferrogels and magnetorheological elastomers are composite materials obtained by embedding magnetic particles of mesoscopic size in a crosslinked polymeric matrix. They combine the reversible elastic deformability of polymeric materials with the high responsivity of ferrofluids to external magnetic fields. These materials stand out, for example, for significant magnetostriction as well as a pronounced increase of the elastic moduli in the presence of external magnetic fields. By means of x-ray micro-computed tomography, the position and size of each magnetic particle can be measured with a high degree of accuracy. We here use data extracted from real magnetoelastic samples as input for coarse-grained dipole-spring modeling and calculations to investigate internal restructuring, stiffening, and changes in the normal modes spectrum. More precisely, we assign to each particle a dipole moment proportional to its volume and set a randomized network of springs between them that mimics the behavior of the polymeric elastic matrix. Extending our previously developed methods, we compute the resulting structural changes in the systems as well as the frequency-dependent elastic moduli when magnetic interactions are turned on. Particularly, with increasing magnetization, we observe the formation of chain-like aggregates. Interestingly, the static elastic moduli can first show a slight decrease with growing amplitude of the magnetic interactions, before a pronounced increase appears upon the chain formation. The change of the dynamic moduli with increasing magnetization depends on the frequency and can even feature nonmonotonic behavior. Overall, we demonstrate how theory and experiments can complement each other to learn more about the dynamic behavior of this interesting class of materials.

Renormalization of the critical exponent for the shear modulus of magnetoactive elastomers

It is shown that the critical exponent for the effective shear modulus of a composite medium where a compliant polymer matrix is filled with ferromagnetic particles may significantly depend on the external magnetic field. The physical consequence of this dependence is the critical behavior of the relative magnetorheological effect.

Dynamics in a one-dimensional ferrogel model: relaxation, pairing, shock-wave propagation

Based on theory and simulations, we elucidate the relaxation dynamics of a one-dimensional ferrogel model and provide classification scenarios.

A density functional approach to ferrogels

Ferrogels consist of magnetic colloidal particles embedded in an elastic polymer matrix. As a consequence, their structural and rheological properties are governed by a competition between magnetic particle-particle interactions and mechanical matrix elasticity. Typically, the particles are permanently fixed within the matrix, which makes them distinguishable by their positions. Over time, particle neighbors do not change due to the fixation by the matrix. Here we present a classical density functional approach for such ferrogels. We map the elastic matrix-induced interactions between neighboring colloidal particles distinguishable by their positions onto effective pairwise interactions between indistinguishable particles similar to a 'pairwise pseudopotential'. Using Monte-Carlo computer simulations, we demonstrate for one-dimensional dipole-spring models of ferrogels that this mapping is justified. We then use the pseudopotential as an input into classical density functional theory of inhomogeneous fluids and predict the bulk elastic modulus of the ferrogel under various conditions. In addition, we propose the use of an 'external pseudopotential' when one switches from the viewpoint of a one-dimensional dipole-spring object to a one-dimensional chain embedded in an infinitely extended bulk matrix. Our mapping approach paves the way to describe various inhomogeneous situations of ferrogels using classical density functional concepts of inhomogeneous fluids.

Magnetorheological response of highly filled magnetoactive elastomers from perspective of mechanical energy density: Fractal aggregates above the nanometer scale?

The dynamic shear modulus of magnetoactive elastomers containing 70 and 80 mass % of carbonyl iron microparticles is measured as a function of strain amplitude via dynamic torsion oscillations in various magnetic fields. The results are presented in terms of the mechanical energy density and considered in the framework of the conventional Kraus model. The form exponent of the Kraus model is further related to a physical model of Huber et al. [Huber et al., J. Phys.: Condens. Matter 8, 409 (1996)10.1088/0953-8984/8/29/003] that uses a realistic representation for the cluster network possessing fractal structure. Two mechanical loading regimes are identified. At small strain amplitudes the exponent β of the Kraus model changes in an externally applied magnetic field due to rearrangement of ferromagnetic-filler particles, while at large strain amplitudes, the exponent β seems to be independent of the magnetic field. The critical mechanical energy characterizing the transition between these two regimes grows with the increasing magnetic field. Similarities between agglomeration and deagglomeration of magnetic filler under simultaneously applied magnetic field and mechanical shear and the concept of jamming transition are discussed. It is proposed that the magnetic field should be considered as an additional parameter to the jamming phase diagram of rubbers filled with magnetic particles.

Effect of single-particle magnetostriction on the shear modulus of compliant magnetoactive elastomers

The influence of an external magnetic field on the static shear strain and the effective shear modulus of a magnetoactive elastomer (MAE) is studied theoretically in the framework of a recently introduced approach to the single-particle magnetostriction mechanism [V. M. Kalita et al., Phys. Rev. E 93, 062503 (2016)10.1103/PhysRevE.93.062503]. The planar problem of magnetostriction in an MAE with magnetically soft inclusions in the form of a thin disk (platelet) having the magnetic anisotropy in the plane of this disk is solved analytically. An external magnetic field acts with torques on magnetic filler particles, creates mechanical stresses in the vicinity of inclusions, induces shear strain, and increases the effective shear modulus of these composite materials. It is shown that the largest effect of the magnetic field on the effective shear modulus should be expected in MAEs with soft elastomer matrices, where the shear modulus of the matrix is less than the magnetic anisotropy constant of inclusions. It is derived that the effective shear modulus is nonlinearly dependent on the external magnetic field and approaches the saturation value in magnetic fields exceeding the field of particle anisotropy. It is shown that model calculations of the effective shear modulus correspond to a phenomenological definition of effective elastic moduli and magnetoelastic coupling constants. The obtained theoretical results compare well with known experimental data. Determination of effective elastic coefficients in MAEs and their dependence on magnetic field is discussed. The concentration dependence of the effective shear modulus at higher filler concentrations has been estimated using the method of Padé approximants, which predicts that both the absolute and relative changes of the magnetic-field-dependent effective shear modulus will significantly increase with the growing concentration of filler particles.