Reversible magnetomechanical collapse: virtual touching and detachment of rigid inclusions in a soft elastic matrix

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.

Density functional approach to elastic properties of three-dimensional dipole-spring models for magnetic gels

Magnetic gels are composite materials consisting of a polymer matrix and embedded magnetic particles. Those are mechanically coupled to each other, giving rise to the magnetostrictive effects as well as to a controllable overall elasticity responsive to external magnetic fields. Due to their inherent composite and thereby multiscale nature, a theoretical framework bridging different levels of description is indispensable for understanding the magnetomechanical properties of magnetic gels. In this study, we extend a recently developed density functional approach from two spatial dimensions to more realistic three-dimensional systems. Along these lines, we connect a mesoscopic characterization resolving the discrete structure of the magnetic particles to macroscopic continuum parameters of magnetic gels. In particular, we incorporate the long-range nature of the magnetic dipole-dipole interaction and consider the approximate incompressibility of the embedding media and relative rotations with respect to an external magnetic field breaking rotational symmetry. We then probe the shape of the model system in its reference state, confirming the dependence of magnetostrictive effects on the configuration of the magnetic particles and on the shape of the considered sample. Moreover, calculating the elastic and rotational coefficients on the basis of our mesoscopic approach, we examine how the macroscopic types of behavior are related to the mesoscopic properties. Implications for real systems of random particle configurations are also discussed.

Variations in the thermal conductivity of magnetosensitive elastomers by magnetically induced internal restructuring

Magnetosensitive elastomers respond to external magnetic fields by changing their stiffness and shape. These effects result from interactions among magnetized inclusions that are embedded within an elastic matrix. Strong external magnetic fields induce internal restructuring, for example the formation of chain-like aggregates. However, such reconfigurations affect not only the overall mechanical properties of the elastomers but also the transport through such systems. We concentrate here on the transport of heat, that is thermal conductivity. For flat, thin model systems representing thin films or membranes and modeled by bead-spring discretizations, we evaluate the internal restructuring in response to magnetization of the particles. For each resulting configuration, we evaluate the associated thermal conductivity. We analyze the changes in heat transport as a function of the strength of magnetization, particle number, density of magnetizable particles (at fixed overall particle number), and aspect ratio of the system. We observe that varying any one of these parameters can induce pronounced changes in the bulk thermal conductivity. Our results motivate future experimental and theoretical studies of systems with magnetically tunable thermal but also electric conductivity-both of which have only rarely been addressed so far.

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.

The effect of polymer stiffness on magnetization reversal of magnetorheological elastomers

Ultrasoft magnetorheological elastomers (MREs) offer convenient real-time magnetic field control of mechanical properties that provides a means to mimic mechanical cues and regulators of cells in vitro. Here, we systematically investigate the effect of polymer stiffness on magnetization reversal of MREs using a combination of magnetometry measurements and computational modeling. Poly-dimethylsiloxane-based MREs with Young's moduli that range over two orders of magnitude were synthesized using commercial polymers Sylgard™ 527, Sylgard 184, and carbonyl iron powder. The magnetic hysteresis loops of the softer MREs exhibit a characteristic pinched loop shape with almost zero remanence and loop widening at intermediate fields that monotonically decreases with increasing polymer stiffness. A simple two-dipole model that incorporates magneto-mechanical coupling not only confirms that micrometer-scale particle motion along the applied magnetic field direction plays a defining role in the magnetic hysteresis of ultrasoft MREs but also reproduces the observed loop shapes and widening trends for MREs with varying polymer stiffnesses.

Magnetic field controlled behavior of magnetic gels studied using particle-based simulations

This contribution provides an overview of the study of soft magnetic materials using particle-based simulation models. We focus in particular on systems where thermal fluctuations are important. As a basis for further discussion, we first describe two-dimensional models which demonstrate two deformation mechanisms of magnetic gels in a homogeneous field. One is based on the change of magnetic interactions between magnetic particles as a response to an external field; the other is the result of magnetically blocked particles acting as cross-linkers. Based on the qualitative behavior directly observable in the two-dimensional models, we extend our description to three-dimensions. We begin with particle-cross-linked gels, as for those, our three-dimensional model also includes explicitly resolved polymer chains. Here, the polymer chains are represented by entropic springs, and the deformation of the gel is the result of the interaction between magnetic particles. We use this model to examine the influence of the magnetic spatial configuration of magnetic particles (uniaxial or isotropic) on the gel’s magnetomechanical behavior. A further part of the article will be dedicated to scale-bridging approaches such as systematic coarse-graining and models located at the boundary between particle-based and continuum modeling. We will conclude our article with a discussion of recent results for modeling time-dependent phenomena in magnetic-polymer composites. The discussion will be focused on a simulation model suitable for obtaining AC-susceptibility spectra for dilute ferrofluids including hydrodynamic interactions. This model will be the basis for studying the signature of particle–polymer coupling in magnetic hybrid materials. In the long run, we aim to compare material properties probed locally via the AC-susceptibility spectra to elastic moduli obtained for the system at a global level.

Collision and separation of nickel particles embedded in a polydimethylsiloxan matrix under a rotating magnetic field: A strong magneto active function

In order to function as soft actuators, depending on their field of use, magnetorheological elastomers (MREs) must fulfill certain criteria. To name just a few, these can include rapid response to external magnetic fields, mechanical durability, mechanical strength, and/or large deformation. Of particular interest are MREs which produce macroscopic deformation for small external magnetic field variations. This work demonstrates how this can be achieved by just a small change in magnetic field orientation. To achieve this, (super)paramagnetic nickel particles of size ≈ 160 μm were embedded in a non-magnetic polydimethylsiloxan (PDMS) (661–1301 Pa) and their displacement in a stepwise rotated magnetic field (170 mT) recorded using a video microscope. Changes in particle aggregation resulting from very small variations in magnetic field orientation led to the observation of a new strongly magneto-active effect. This configuration is characterized by an interparticle distance in relation to the angle difference between magnetic field and particle axis. This causes a strong matrix deformation which in turn demonstrates hysteresis on relaxation. It is shown that the occurrence strongly depends on the particle size, particle distance, and stiffness of the matrix. Choosing the correct parameter combination, the state can be suppressed and the particle-matrix system demonstrates no displacement or hysteresis. In addition, evidences of non-negligible higher order magnetization effects are experimentally ascertained which is qualitatively in agreement with similar, already theoretically described, particle systems. Even at larger particle geometries, the new strongly magneto-active configuration is preserved and could create macroscopic deformation changes.

Stimuli-responsive twist actuators made from soft elastic composite materials-linking mesoscopic and macroscopic descriptions

Very recently, the construction of twist actuators from magnetorheological gels and elastomers has been suggested. These materials consist of magnetizable colloidal particles embedded in a soft elastic polymeric environment. The twist actuation is enabled by a net chirality of the internal particle arrangement. Upon magnetization by a homogeneous external magnetic field, the systems feature an overall torsional deformation around the magnetization direction. Starting from a discrete minimal mesoscopic model setup, we work toward a macroscopic characterization. The two scales are linked by identifying expressions for the macroscopic system parameters as functions of the mesoscopic model parameters. In this way, the observed behavior of a macroscopic system can, in principle, be mapped to and illustratively be understood from an appropriate mesoscopic picture. Our results apply equally well to corresponding soft electrorheological gels and elastomers.

A Cascading Mean-Field Approach to the Calculation of Magnetization Fields in Magnetoactive Elastomers

We consider magnetoactive elastomer samples based on the elastic matrix and magnetizable particle inclusions. The application of an external magnetic field to such composite samples causes the magnetization of particles, which start to interact with each other. This interaction is determined by the magnetization field, generated not only by the external magnetic field but also by the magnetic fields arising in the surroundings of interacting particles. Due to the scale invariance of magnetic interactions (O(r−3) in d=3 dimensions), a comprehensive description of the local as well as of the global effects requires a knowledge about the magnetization fields within individual particles and in mesoscopic portions of the composite material. Accordingly, any precise calculation becomes technically infeasible for a specimen comprising billions of particles arranged within macroscopic sample boundaries. Here, we show a way out of this problem by presenting a greatly simplified, but accurate approximation approach for the computation of magnetization fields in the composite samples. Based on the dipole model to magnetic interactions, we introduce the cascading mean-field description of the magnetization field by separating it into three contributions on the micro-, meso-, and macroscale. It is revealed that the contributions are nested into each other, as in the Matryoshka’s toy. Such a description accompanied by an appropriate linearization scheme allows for an efficient and transparent analysis of magnetoactive elastomers under rather general conditions.

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.

Multiscale modeling and simulation of magneto-active elastomers based on experimental data

In this contribution, we present a framework for the multiscale modeling and simulation of magneto-active elastomers (MAEs). It enables us to consider these materials on the microscopic scale, where the heterogeneous microstructure consisting of magnetizable particles and elastomer matrix is explicitly resolved, as well as the macroscopic scale, where the MAE is considered to be a homogeneous magneto-active body. On both scales, a general continuum formulation of the coupled magneto-mechanical boundary value problem is applied and the finite element method is used to solve the governing equations. Starting with an experimental characterization of the individual constituents, i.e. particles and matrix, microscopic constitutive models for both are formulated and adjusted to the experimental data separately. With that, properties of MAEs resulting from the microscopic constitutive behavior can be captured within the presented modeling approach. Secondly, to discuss general macroscopic properties of magnetically soft and hard MAEs, a computational homogenization scheme is used to calculate the composites’ effective behavior for different geometrical arrangements of the particles on the microscale. Finally, the calculated effective response of a magnetically soft composite system is used to identify the parameters of a macroscopic magneto-elastic model. Using the calibrated model, the behavior of macroscopic MAEs is simulated for different sample geometries.

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 torque-driven deformation of Ni-nanorod/hydrogel nanocomposites

Nickel (Ni) nanorods were prepared by the anodized aluminum oxide (AAO) template method and dispersed in poly(acrylamide) (PAM) hydrogels. The deformation of the magnetoresponsive composites was studied with particular attention to the consequences of finite magnetic shape anisotropy as compared to rigid dipoles on the field-dependent torque. For comparison with experiments, the composite was described as an elastic continuum with a local magnetic torque density, applied by discrete particles and determined by the local orientation of their magnetic anisotropy axis with respect to the magnetic field. The mean magnetic moment of the single domain particles m and their volume density in the composite φ vol were derived from the static field-dependent optical transmission (SFOT) of linear polarized light. The mechanical coupling between the particles and their viscoelastic environment was retrieved from the rotational dynamics of the nanorods using oscillating field-dependent optical transmission (OFOT) measurements. Field- and orientation-dependent magnetization measurements were analyzed using the Stoner–Wohlfarth (SW) model and a valid parameter range was identified by introducing an effective anisotropy constant K A as a new empirical model parameter. This adapted SW-model for quantitative description of the field- and orientation dependence of the magnetic torque was validated by measuring the local rotation of nanorods in a soft elastic hydrogel. Finally, torsional and bending deformation of thin magnetically textured composite filaments were computed and compared with experiments.

Displacement field around a rigid sphere in a compressible elastic environment, corresponding higher-order Faxén relations, as well as higher-order displaceability and rotateability matrices

An efficient route to the displacement field around a rigid spherical inclusion in an infinitely extended homogeneous elastic medium is presented in a slightly alternative way when compared to some common textbook methods. Moreover, two Faxén relations of next-higher order beyond the stresslet are calculated explicitly for compressible media. They quantify higher-order moments involving the force distribution on a rigid spherical particle in a deformed elastic medium. As a consequence, additional contributions to the distortions of the deformed elastic medium are identified that are absent to lower order. Furthermore, the displaceability and rotateability matrices for an ensemble of rigid spheres are calculated up to (including) sixth order in inverse particle separation distance. These matrices describe the interactions mediated between the rigid embedded particles by the elastic environment. In this way, additional coupling effects are identified that are absent to lower order, particularly when rotations and torques are involved. All methods and results can formally be transferred to the corresponding case of incompressible hydrodynamic low-Reynolds-number Stokes flow by considering the limit of an incompressible environment. The roles of compressibility of the embedding medium and of the here additionally derived higher-order contributions are highlighted by some selected example configurations.

Memory-based mediated interactions between rigid particulate inclusions in viscoelastic environments

Many practically relevant materials combine properties of viscous fluids and elastic solids to viscoelastic behavior. Our focus is on the induced dynamic behavior of damped finite-sized particulate inclusions in such substances. We explicitly describe history-dependent interactions that emerge between the embedded particles. These interactions are mediated by the viscoelastic surroundings. They result from the flows and distortions of the viscoelastic medium when induced by the rigid inclusions. Both viscoelastic environments of terminal fluidlike flow and of completely reversible damped elastic behavior are covered. For illustration and to highlight the role of the formalism in potential applications, we briefly address the relevant examples of dragging a rigid sphere through a viscoelastic environment together with subsequent relaxation dynamics, the switching dynamics of magnetic fillers in elastic gel matrices, and the swimming behavior of active microswimmers in viscoelastic solutions. The approach provides a basis for more quantitative and extended investigations of these and related systems in the future.