Hardening transition in a one-dimensional model for ferrogels

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.

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.

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.

Magneto-mechanical coupling of single domain particles in soft matter systems

Combining inorganic magnetic particles with complex soft matrices such as liquid crystals, biological fluids, gels, or elastomers, allows access to a plethora of magnetoactive effects that are useful for sensing and actuation perspectives, allowing inter alia to explore and manipulate material properties on the nanoscale. The article provides a comprehensive summary of recent advancement on employing magnetic nanoparticles either as tracers for dynamic processes, or as nanoscopic actuating units. By variation of the particle characteristics in terms of size, shape, surface functionality, and magnetic behavior, the interaction between the probe or actuator particles and their environment can be systematically tailored in wide ranges, giving insight into the relevant structure–property relationships.

Adsorption transition of a grafted ferromagnetic filament controlled by external magnetic fields

Extensive Langevin dynamics simulations are used to characterize the adsorption transition of a flexible magnetic filament grafted onto an attractive planar surface. Our results identify different structural transitions at different ratios of the thermal energy to the surface attraction strength: filament straightening, adsorption, and the magnetic flux closure. The adsorption temperature of a magnetic filament is found to be higher in comparison to an equivalent nonmagnetic chain. The adsorption has been also investigated under the application of a static homogeneous external magnetic field. We found that the strength and the orientation of the field can be used to control the adsorption process, providing a precise switching mechanism. Interestingly, we have observed that the characteristic field strength and tilt angle at the adsorption point are related by a simple power law.

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

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

Classical density functional theory for a two-dimensional isotropic ferrogel model with labeled particles

In this study, we formulate a density functional theory (DFT) for systems of labeled particles, considering a two-dimensional bead-spring lattice with a magnetic dipole on every bead as a model for ferrogels. On the one hand, DFT has been widely studied to investigate fluidlike states of materials, in which constituent particles are not labeled as they can exchange their positions without energy cost. On the other hand, in ferrogels consisting of magnetic particles embedded in elastic polymer matrices, the particles are labeled by their positions as their neighbors do not change over time. We resolve such an issue of particle labeling, introducing a mapping of the elastic interaction mediated by springs onto a pairwise additive interaction (pseudosprings) between unlabeled particles. We further investigate magnetostriction and changes in the elastic constants under altered magnetic interactions employing the pseudospring potential. It is revealed that there are two different response scenarios in the mechanical properties of the dipole-spring systems: While systems at low packing fractions are hardened as the magnetic moments increase in magnitude, at high packing fractions softening due to diminishing effects from the steric force, associated with increases in the volume, is observed. The validity of the theory is also verified by Monte Carlo simulations with both real springs and pseudosprings. We expect that our DFT approach may promote our understanding of materials with particle inclusions.

Surface relief of magnetoactive elastomeric films in a homogeneous magnetic field: molecular dynamics simulations

The structure of a thin magnetoactive elastomeric (MAE) film adsorbed on a solid substrate is studied by molecular dynamics simulations. Within the adopted coarse-grained approach, a MAE film consists of magnetic particles modeled as soft-core spheres, carrying point dipoles, connected by elastic springs representing a polymer matrix. MAE films containing 20, 25 and 30 vol% of randomly distributed magnetic particles are simulated. Once a magnetic field is applied, the competition between dipolar, elastic and Zeeman forces leads to the restructuring of the layer. The distribution of the magnetic particles as well as elastic strains within the MAE films are calculated for various magnetic fields applied perpendicular to the film surface. It is shown that the surface roughness increases strongly with growing magnetic field. For a given magnetic field, the roughness is larger for the softer polymeric matrix and exhibits a nonmonotonic dependence on the magnetic particle concentration. The obtained results provide a better understanding of the MAE surface structuring as well as possible guidelines for fabrication of MAE films with a tunable surface topology.

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

Soft elastic composite materials containing particulate rigid inclusions in a soft elastic matrix are candidates for developing soft actuators or tunable damping devices. The possibility to reversibly drive the rigid inclusions within such a composite together to a close-to-touching state by an external stimulus would offer important benefits. Then, a significant tuning of the mechanical properties could be achieved due to the resulting mechanical hardening. For a long time, it has been argued whether a virtual touching of the embedded magnetic particles with subsequent detachment can actually be observed in real materials, and if so, whether the process is reversible. Here, we present experimental results that demonstrate this phenomenon in reality. Our system consists of two paramagnetic nickel particles embedded at finite initial distance in a soft elastic polymeric gel matrix. Magnetization in an external magnetic field tunes the magnetic attraction between the particles and drives the process. We quantify our experimental results by different theoretical tools, i.e., explicit analytical calculations in the framework of linear elasticity theory, a projection onto simplified dipole-spring models, as well as detailed finite-element simulations. From these different approaches, we conclude that in our case the cycle of virtual touching and detachment shows hysteretic behavior due to the mutual magnetization between the paramagnetic particles. Our results are important for the design and construction of reversibly tunable mechanical damping devices. Moreover, our projection on dipole-spring models allows the formal connection of our description to various related systems, e.g., magnetosome filaments in magnetotactic bacteria.

Ground state of dipolar hard spheres confined in channels

We investigate the ground state of a classical two-dimensional system of hard-sphere dipoles confined between two hard walls. Using lattice sum minimization techniques we reveal that at fixed wall separations, a first-order transition from a vacuum to a straight one-dimensional chain of dipoles occurs upon increasing the density. Further increase in the density yields the stability of an undulated chain as well as nontrivial buckling structures. We explore the close-packed configurations of dipoles in detail, and we find that, in general, the densest packings of dipoles possess complex magnetizations along the principal axis of the slit. Our predictions serve as a guideline for experiments with granular dipolar and magnetic colloidal suspensions confined in slitlike channel geometry.

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.

Importance of matrix inelastic deformations in the initial response of magnetic elastomers

Being able to predict and understand the behaviour of soft magnetic materials paves the way to their technological applications. In this study we analyse the magnetic response of soft magnetic elastomers (SMEs) with magnetically hard particles. We present experimental evidence of a difference between the first and next magnetisation loops exhibited by these SMEs, which depends non-monotonically on the interplay between the rigidity of the polymer matrix, its mechanical coupling with the particles, and the magnetic interactions in the system. In order to explain the microstructural mechanism behind this behaviour, we used a minimal computer simulation model whose results evidence the importance of irreversible matrix deformations due to both translations and rotations of the particles. To confirm the simulation findings, computed tomography (CT) was used. We conclude that the initial exposure to the field triggers the inelastic matrix relaxation in the SMEs, as particles attempt to reorient. However, once the necessary degree of freedom is achieved, both the rotations and the magnetisation behaviour become stationary. We expect this scenario not only to be limited to the materials studied here, but also to apply to a broader class of hybrid SMEs.

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.

Polymer architecture of magnetic gels: a review

In this review article, we provide an introduction to ferrogels, i.e. polymeric gels with embedded magnetic particles. Due to the interplay between magnetic and elastic properties of these materials, they are promising candidates for engineering and biomedical applications such as actuation and controlled drug release. Particular emphasis will be put on the polymer architecture of magnetic gels since it controls the degrees of freedom of the magnetic particles in the gel, and it is important for the particle-polymer coupling determining the mechanisms available for the gel deformation in magnetic fields. We report on the different polymer architectures that have been realized so far, and provide an overview of synthesis strategies and experimental techniques for the characterization of these materials. We further focus on theoretical and simulational studies carried out on magnetic gels, and highlight their contributions towards understanding the influence of the gels' polymer architecture.

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.

Buckling of elastic filaments by discrete magnetic moments

We study the buckling of an idealized, semiflexible filament along whose contour magnetic moments are placed. We give analytic expressions for the critical stiffness of the filament below which it buckles due to the magnetic compression. For this, we consider various scenarios of the attachment of the magnetic particles to the filament. One possible application for this model are the magnetosome chains of magnetotactic bacteria. An estimate of the critical bending stiffness indicates that buckling may occur within the range of biologically relevant parameters and suggests a role for the bending stiffness of the filament to stabilize the filament against buckling, which would compromise the functional relevance of the bending stiffness of the used filament.

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.

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.

Elongated micro-structures in magneto-sensitive elastomers: a dipolar mean field model

Based on a dipole model for the mutual magnetic interactions among the magnetizable micro-particles in magneto-sensitive elastomers we develop a mean field approach to describe the arrangement of these particles into elongated micro-structures. If these micro-structures are oriented parallel to an external magnetic field the present approach provides an efficient calculation of the behavior of such samples, which is a result of the interplay between micro-structure and shape effects. Accordingly, we are able to draw comprehensive phase diagrams for the resultant deformation and predict for very oblate samples a discontinuous shape change in the presence of a homogeneous external field.

Ferrogels cross-linked by magnetic particles: Field-driven deformation and elasticity studied using computer simulations

Ferrogels, i.e., swollen polymer networks into which magnetic particles are immersed, can be considered as "smart materials" since their shape and elasticity can be controlled by an external magnetic field. Using molecular dynamics simulations on the coarse-grained level, we study a ferrogel in which the magnetic particles act as the cross-linkers of the polymer network. In a homogeneous external magnetic field, the direct coupling between the orientation of the magnetic moments and the polymers by means of covalent bonds gives rise to a deformation of the gel, independent of the interparticle dipole-dipole interaction. In this paper, we quantify this deformation, and, in particular, we investigate the gel's elastic moduli and its magnetic response for two different connectivities of the network nodes. Our results demonstrate that these properties depend significantly on the topology of the polymer network.

Buckling of paramagnetic chains in soft gels

We study the magneto-elastic coupling behavior of paramagnetic chains in soft polymer gels exposed to external magnetic fields. To this end, a laser scanning confocal microscope is used to observe the morphology of the paramagnetic chains together with the deformation field of the surrounding gel network. The paramagnetic chains in soft polymer gels show rich morphological shape changes under oblique magnetic fields, in particular a pronounced buckling deformation. The details of the resulting morphological shapes depend on the length of the chain, the strength of the external magnetic field, and the modulus of the gel. Based on the observation that the magnetic chains are strongly coupled to the surrounding polymer network, a simplified model is developed to describe their buckling behavior. A coarse-grained molecular dynamics simulation model featuring an increased matrix stiffness on the surfaces of the particles leads to morphologies in agreement with the experimentally observed buckling effects.

A simulation study of the electrostriction effects in dielectric elastomer composites containing polarizable inclusions with different spatial distributions

Controlled actuation of electroactive polymers with embedded high dielectric nanoparticles is theoretically analyzed.

Tunable dynamic response of magnetic gels: impact of structural properties and magnetic fields

Ferrogels and magnetic elastomers feature mechanical properties that can be reversibly tuned from outside through magnetic fields. Here we concentrate on the question of how their dynamic response can be adjusted. The influence of three factors on the dynamic behavior is demonstrated using appropriate minimal models: first, the orientational memory imprinted into one class of the materials during their synthesis; second, the structural arrangement of the magnetic particles in the materials; and third, the strength of an external magnetic field. To illustrate the latter point, structural data are extracted from a real experimental sample and analyzed. Understanding how internal structural properties and external influences impact the dominant dynamical properties helps to design materials that optimize the requested behavior.

Magnetic particles confined in a modulated channel: structural transitions tunable by tilting a magnetic field

The ground state of colloidal magnetic particles in a modulated channel are investigated as a function of the tilt angle of an applied magnetic field. The particles are confined by a parabolic potential in the transversal direction while in the axial direction a periodic substrate potential is present. By using Monte Carlo simulations, we construct a phase diagram for the different crystal structures as a function of the magnetic field orientation, strength of the modulated potential, and the commensurability factor of the system. Interestingly, we found first- and second-order phase transitions between different crystal structures, which can be manipulated by the orientation of the external magnetic field. A reentrant behavior is found between two- and four-chain configurations, with continuous second-order transitions. Novel configurations are found consisting of frozen solitons of defects. By changing the orientation and/or strength of the magnetic field and/or the strength and periodicity of the substrate potential, the system transits through different phases.