Superelastic stress-strain behavior in ferrogels of different types of magneto-elastic coupling
SSuperelastic stress-strain behavior in ferrogels of different types of magneto-elasticcoupling
Peet Cremer, ∗ Hartmut L¨owen, and Andreas M. Menzel † Institut f¨ur Theoretische Physik II: Weiche Materie,Heinrich-Heine-Universit¨at D¨usseldorf, D-40225 D¨usseldorf, Germany (Dated: October 9, 2018)Colloidal magnetic particles embedded in an elastic polymer matrix constitute a smart materialcalled ferrogel. It responds to an applied external magnetic field by changes in elastic properties,which can be exploited for various applications like dampers, vibration absorbers, or actuators.Under appropriate conditions, the stress-strain behavior of a ferrogel can display a fascinatingfeature: superelasticity, the capability to reversibly deform by a huge amount while barely alteringthe applied load. In a previous work, using numerical simulations, we investigated this behaviorassuming that the magnetic moments carried by the embedded particles can freely reorient tominimize their magnetic interaction energy. Here, we extend the analysis to ferrogels where restoringtorques by the surrounding matrix hinder rotations towards a magnetically favored configuration.For example, the particles can be chemically cross-linked into the polymer matrix and the magneticmoments can be fixed to the particle axes. We demonstrate that these systems still feature asuperelastic regime. As before, the nonlinear stress-strain behavior can be reversibly tailored duringoperation by external magnetic fields. Yet, the different coupling of the magnetic moments causesdifferent types of response to external stimuli. For instance, an external magnetic field appliedparallel to the stretching axis hardly affects the superelastic regime but stiffens the system beyondit. Other smart materials featuring superelasticity, e.g. metallic shape-memory alloys, have alreadyfound widespread applications. Our soft polymer systems offer many additional advantages like atypically higher deformability and enhanced biocompatibility combined with high tunability.
I. INTRODUCTION
Ferrogels [1–3], also known as soft magnetic materials,magnetic gels, magnetic elastomers, or magnetorheologi-cal elastomers, are manufactured by embedding colloidalmagnetic particles into an elastic matrix that most of-ten consists of cross-linked polymer. This leads to aninterplay between magnetic and elastic interactions, al-lowing to reversibly adjust the material properties viaexternal magnetic fields [4–15]. On the other hand, dy-namically switching the elastic properties allows appli-cations as tunable dampers [16] or vibration absorbers[17]. Moreover, shape changes [5, 18–21] are interestingfor the realization of soft actuators [22–27]. Also shape-memory effects have been observed in soft magnetic ma-terials [28–30], opening the way for even more interestingapplications.Recently we have identified another fascinating fea-ture of soft magnetic materials in a simulation study[31], namely tunable superelasticity. This term was orig-inally introduced in the context of shape-memory alloys[32–34]. It addresses their special nonlinear stress-strainbehavior with a plateau-like regime, where a small addi-tional load leads to a huge additional deformation thatis, however, completely reversible. In shape-memory al-loys, the constituents are positioned on regular latticesites. The observed behavior is enabled by a stress-induced transition of the material to a more elongated ∗ [email protected] † [email protected] lattice structure that can accomodate the deformation.When the load is released, the shape-memory alloy per-forms the opposite lattice transition, which renders thewhole process reversible.In the case of anisotropic soft magnetic gels [5, 8, 35–39], the superelastic behavior is enabled by stress-inducedstructural changes. Such samples can be synthesizedby applying a strong external magnetic field during thechemical cross-linking process that forms the elastic ma-trix. Before cross-linking, when the magnetic particlesare still mobile, straight chain-like aggregates form alongthe field direction [40–43]. Cross-linking the polymerlocks the particle positions into the elastic matrix evenafter the external field is switched off. Our previous nu-merical study of stretching a magnetic gel containingchain-like aggregates along the direction of the chainsrevealed the following behavior. The strong magneticattractions within the chains first work against the elon-gation. However, once the magnetic barriers to detachchained particles are overcome, the material strongly ex-tends. A part of the stored stress working against themagnetic interactions is released, leading to additionalstrain without hardly any additional stress necessary.This behavior gives rise to to a strongly nonlinear, “su-perelastic” plateau in the stress-strain curve, similar tothe phenomenology found for shape-memory alloys. Thestrain regime that is covered by this plateau, however, issignificantly larger. Additionally, it is possible to tailorthe nonlinear stress-strain behavior by external magneticfields. Combined with the typically higher degree of bio-compatibility of soft polymeric materials [44–48], medicalapplications [49–52] might become possible. a r X i v : . [ c ond - m a t . s o f t ] A ug In this previous study [31], we restricted ourselves tothe assumption that the magnetic moments of the em-bedded particles are free to reorient. First, this is possi-ble when each magnetic moment can reorient within theparticle interior, which typically can be observed as theso-called N´eel mechanism up to particle diameters of to10–15 nm [53]. Second, the type of embedding in theelastic matrix can allow the whole particle to rotate, atleast quasi-statically, without deforming the matrix, e.g.,when in the vicinity of the particles the cross-linking ofthe polymer matrix is inhibited [54]. Finally, yolk-shellcolloidal particles feature a magnetic core that can ro-tate relatively to the nonmagnetic shell surrounding it[55, 56].Here, we mainly concentrate on the opposite scenariofor spherical, rigid magnetic particles. That is, the mag-netic moments are not free to reorient with respect tothe embedding matrix. Two ingredients are necessary forthis purpose. First, the matrix must be anchored to theparticle surfaces. In reality, this can be achieved whenchemically the particles themselves act as cross-linkers ofthe polymer matrix [57–62]. Second, the magnetic mo-ments must not rotate relatively to the particle frames.This is the case for magnetically anisotropic monodomainparticles that are large enough to block the N´eel mech-anism. Again we can observe superelastic stress-strainbehavior in such systems and again the nonlinearity canbe tuned by external magnetic fields. Yet, the response isaltered though, due to the different coupling of the mag-netic filler particles to the surrounding matrix. An ex-ternal magnetic field parallel to the chain-like aggregateslargely leaves the superelastic behavior intact. Still, asufficiently strong perpendicular field rotates the parti-cles out of the initial alignment configuration and gradu-ally removes the nonlinearity from the stress-strain curve.However, due to the covalent coupling to the elastic ma-trix counteracting particle rotations, the necessary fieldstrengths to deactivate superelasticity are much higherwhen compared to the case of freely reorientable mag-netic moments.In Sec. II we begin by introducing our numerical modeland our simulation technique for measuring the stress-strain behavior. Next, in Sec. III, we define severalferrogel systems with different coupling properties be-tween the particles and the surrounding elastic matrix.Afterwards, in Sec. IV, we analyze the resulting stress-strain behavior for these different systems and the vari-ous mechanisms and effects leading to the emerging su-perelastic features. We start with the case of vanishingexternal magnetic field and then proceed to fields par-allel and perpendicular to the chain-like aggregates. Fi-nally, in Sec. V, we conclude by reviewing our resultsand discussing possible experimental realizations as wellas prospective applications.
II. NUMERICAL MODEL AND SIMULATIONPROCEDURE
The purpose of our simulations is to determine thenonlinear stress-strain behavior of uniaxial ferrogel sys-tems containing chain-like aggregates. To achieve this,we require numerical representations of both the polymermatrix and of the embedded colloidal magnetic particles.Let us first discuss our representation of the polymermatrix. We assume that all molecular details of the cross-linked polymer can be ignored, so that we can treat thematrix as a continuous and isotropic elastic medium. Wetessellate it into a three-dimensional mesh of sufficientlysmall tetrahedra. Spherical magnetic particles are em-bedded into this mesh by approximating their surfacesas sets of planar triangles, which become faces of thetetrahedral mesh. This tessellation was enabled by themesh generation tool gmsh [63], which is based on De-launay triangulation [64]. It allows to set a characteristiclength scale parameter controlling the typical length ofthe tetrahedra edges, for which we used 0 . R , where R is the radius of the particles.Each tetrahedron of the mesh may deform affinely,which is associated with an elastic deformation energy U e given by the following nearly-incompressible Neo-Hookean hyperelastic model [65]: U e = V (cid:104)(cid:16) µ (cid:8) F t · F (cid:9) − (cid:17) − µ (det F − λ + µ F − (cid:21) . (1)Here the elastic properties of the isotropic matrix entervia the Lam´e coefficients µ and λ [66]. They can alsobe expressed in terms of the elastic modulus E and thePoisson ratio ν via µ = E ν ) and λ = Eν (1+ ν )(1 − ν ) . V denotes the volume of the tetrahedron in the undeformedstate. F is the deformation gradient tensor prescrib-ing the affine transformation that brings the tetrahedronfrom its undeformed state to the deformed state. Thedeformed state of the tetrahedron is characterized by thematrix X := ( x − x , x − x , x − x ) that contains thecurrent positions x , x , x , x of the four nodes (ver-tices), see Fig. 1 for an illustration. Similarly, the matrix ˜X := ( ˜x − ˜x , ˜x − ˜x , ˜x − ˜x ) determines the un-deformed (reference) state of the tetrahedron with nodepositions ˜x , ˜x , ˜x , ˜x . Since F is the affine transfor-mation that connects the deformed state to the referencestate, we have X = F · ˜X . Now the deformation gradienttensor F can be obtained [67] by multiplying from theright with ˜X − , yielding F ( X ) = X · ˜X − . (2)The undeformed reference state never changes, hence theinverse matrix ˜X − remains constant and has to be cal-culated only once. This allows to determine the elasticdeformation energy U e ( F ( X )) in any deformed configu-ration from the positions of the tetrahedral nodes. FIG. 1. The undeformed state ˜X of each tetrahedron is de-termined by the reference node positions ˜x , ˜x , ˜x , ˜x via ˜X = ( ˜x − ˜x , ˜x − ˜x , ˜x − ˜x ), while the deformed state X is characterized by the present node positions x , x , x , x in the form X = ( x − x , x − x , x − x ). Both states areconnected via the deformation gradient tensor F . Calculation of the force f i on each node i ( i = 0 , , , f i = −∇ x i U e ( F ) = − ∂U e ( F ) ∂ F · ∂ F ∂ x i . (3)These forces allow us to determine the displacements ofthe nodes. The characterization of the elastic matrix isthus completed. In a second step, we turn to the em-bedded rigid particles. Since they are rigid objects, wehave to treat nodes attached to particle surfaces in aspecial way. The forces on these nodes are transmittedto the corresponding particle, which leads to net forcesand torques on the particles. Rotations and translationsof the particles due to these forces and torques are cal-culated. They, in turn, determine the displacements ofthe surface nodes. We perform a parallel calculation ofall node forces in the system by slicing it into differentsections, that can be handled separately.Next we discuss our representation of the magnetic in-teractions. We assume that all N magnetic particles pos-sess permanent dipolar magnetic moments of equal mag-nitude m . This leads us to a total magnetic interactionenergy given by U m = µ π N (cid:88) i =1 i − (cid:88) j =1 m i · m j − m i · ˆr ij ) ( m j · ˆr ij ) r ij − N (cid:88) i =1 m i · B . (4)Here µ is the vacuum permeability, m i and m j are themagnetic moments of particles i and j , respectively, with | m i | = | m j | = m , r ij := r i − r j is the separation vectorbetween both particles, r ij = | r ij | is its magnitude, ˆr ij = r ij /r ij , and B is an externally applied magnetic field.The magnetic dipolar interaction can be strongly at-tractive at short distances, when the magnetic momentsof interacting particles are in a head-to-tail configura-tion. In order to prevent an unphysical interpenetrationof the particles due to such an attraction, we additionallyintroduce a steric repulsion between the particles that counteracts the attraction at short distances. The WCApotential [68] U wca = (cid:40) (cid:15) (cid:104)(cid:0) σr (cid:1) − (cid:0) σr (cid:1) + (cid:105) , if r ≤ / σ, , if r > / σ, (5)is hard and finite-ranged and commonly used to representsteric repulsions. Its strong scaling with the particle dis-tance compared to the dipolar interactions ( r − vs. r − )makes it the dominating contribution for short distances.By setting (cid:15) = µ π m R and σ = 2 R , the dipolar force be-tween two particles, with their magnetic moments alignedin the most attractive head-to-tail configuration, is ex-actly balanced by the repulsive WCA interaction whenthey are at contact.All these ingredients together express the total energyof the system. It is a function of the node positions ofthe tetrahedral mesh, the particle positions, the particleorientations, and the orientations of the magnetic mo-ments of the particles. We equilibrate our systems byperforming an energy minimization with respect to thesedegrees of freedom. As a numerical scheme, we employthe FIRE algorithm [69], using the forces and torquesresulting from Eqs. (1)–(5) to drive the system towardsits energetic minimum.
FIRE is a molecular dynamicsscheme that uses adaptive time steps and modifies the ve-locities resulting from the forces and torques to achievea quicker relaxation. There are several parameters con-trolling these modifications of velocities and time step,for which we use the values suggested in Ref. [69]. Nu-merical stability is ensured by an upper bound ∆ t max forthe time step, which we have set to ∆ t max = 0 .
01. Fromour experience, this rather simple minimization schemeis quite competitive with more sophisticated schemes likenonlinear conjugate gradient [70] that we employed in ourearlier work in Ref. [31]. In extreme situations of defor-mation, unphysical behavior may result, such as the in-version of individual tetrahedra or their penetration intothe spherical particles.The physical input parameters for our simulations arethe elastic modulus E and Poisson ratio ν of the matrix,the magnitude m of the magnetic moments and eventu-ally the external magnetic field B . We measure forces F in units of F = ER , magnetic moments in units of m = R (cid:113) πµ E , and magnetic field strength in units of B = (cid:112) µ π E . Throughout his work, we fix the materialparameters by choosing ν = 0 .
495 and m = 10 m .Besides the material properties, the behavior of a sam-ple depends on its shape [27, 71] and on the internaldistribution of particles [72–74]. Our characteristic nu-merical probes are small three-dimensional systems ofmagnetic particles embedded into an initially rectangu-lar box of elastic material. The box dimensions are22 . R × . R × . R , containing 96 identical sphericalparticles. These particles are arranged into 12 chain-likeaggregates of 8 particles each. All chains are aligned par-allel to the long edge of the box (the x -direction). Neigh-boring particles in the same chain are initially separatedby a finite gap of elastic material of thickness R/
2. Thepositions of the chains are chosen at random, with theconstraint that they shall not overlap and have a mini-mum distance of R/ . R .Since this maximum shift equals the particle diameterplus the gap thickness, there is no statistical preferenceof any particular particle-gap configuration between twochains. Our results are based on 20 different systemscreated in this manner, each with a unique particle con-figuration. About 250000 tetrahedra result in each casefrom the mesh generation.To measure the uniaxial stress-strain behavior of sucha numerical system, we quasi-statically stretch it alongthe chain direction, using the following protocol. We de-fine two numerical clamps, on the two faces where chainsstart and end. In our geometry, these faces are normalto the x -direction. All particles on the chain ends aresubject to the corresponding numerical clamp. Particleswithin the clamps may rotate. They may also translatein the y - and z -direction, however, with the constraintthat the center-of-mass displacement of all particles ina clamp is zero. This keeps the centers of the clampsfixed in the yz -plane and prevents an overall rotation ofthe long axis of the system. Finally, we prevent globalrotations of the whole system around its long axis at alltimes. For this purpose, at each timestep, we determinethe global rotational modes from which the rotation iseliminated. Overall, this definition of the clamps differsfrom our approach in Ref. [31]. There, the clamps con-sisted of the complete outer 10% of the system at bothends, that is, besides the particles also all matrix meshnodes in these volumes were included.After switching on the magnetic moments, we performan initial equilibration process. During this period, theclamps are allowed to relatively translate along the x -axis. However, the relative distance between the par-ticles in a clamp is kept constant along the x -direction.Due to this initial equilibration, we can observe an initialmatrix deformation and define the resulting state as un-stretched. This sets the equilibrium distance L betweenthe clamps as the x -separation between the innermostclamped particles. To apply a uniaxial strain, we increasethe distance between both clamps in small steps, displac-ing all clamped particles uniformly. So we can define theuniaxial strain as (cid:15) xx = ∆ L/L , where ∆ L is the momen-tary increase in the distance between both clamps. Aftereach step, we equilibrate the sample again under the con-straint of keeping the x -positions of the clamped particlesfixed. Subsequently, we can extract the force F that hasto be applied to the clamps to maintain the system in theprescribed strained state. We continue this stress-strainmeasurement up to a maximum strain of ∆ L/L = 150%and then gradually unload the system again. To checkthe reversibility of the deformation, we perform severalloading and unloading cycles. III. DEFINITION OF THE NUMERICALSYSTEMS
Within our numerical samples defined above, we dis-tinguish between two scenarios of how the magnetic mo-ments are coupled to the surrounding matrix via theircarrying particles. Systems showing the first one, whichwe term free systems, feature magnetic moments thatcan freely rotate relatively to the particle frames and sur-rounding matrix, see also Ref. [31]. In this case, a reori-entation of a magnetic moment does not directly induce adeformation of the matrix surrounding the carrying par-ticle. Computationally, we treat this system by keepingthe orientations of the magnetic moments and the ori-entations of the carrying particles as separate degrees offreedom. During the initial equilibration, within the par-ticles constituting one chain, the magnetic moments tendto align parallel to the chain axis. The magnetic momentswithin neighboring chains have the tendency to align inopposite directions to minimize the overall magnetic in-teraction energy. Figure 2a illustrates this situation byshowing a snapshot of an equilibrated free system beforestretching. A cut along the cross-sectional center planeperpendicular to the chain axes in Fig. 2a stresses thedifferent alignment of the magnetic moments in differentchains.In the opposite scenario, we assume that the magneticmoments are fixed to the axes of the carrying particles,while the particles are covalently embedded into the elas-tic matrix. A torque on a magnetic moment is thenequivalent to a torque on the carrying particle, which inturn leads to a deformation of the surrounding matrix.We mark these systems by the term cov and representthem computationally by rigidly coupling the magneticmoment orientations to the particles.Consequently, the initial orientations of the magneticmoments have a determining influence on the structureof the cov samples and, thus, on their stress-strain be-havior. We distinguish between two sub-scenarios andterm the corresponding systems cov ⇒ and cov (cid:28) . In the cov ⇒ systems, we define all magnetic moments in thesample to initially point into the same direction parallelto the chains. During the initial equilibration, the orien-tations of the magnetic moments barely change as parti-cle rotations are energetically expensive. The magneticmoments within all chains are still aligned in the samedirection, see Fig. 2b for a snapshot. This is different inthe cov (cid:28) systems: here we take the equilibrated statefrom the free systems, but then fix the magnetic momentsto the particle axes before stretching the sample. As aresult, the magnetic moments are rigidly anchored to thecarrying particles and are arranged into the chains withalternating alignment, see again Fig. 2a for a snapshot.The cov (cid:28) system constitutes an in-between case of the free and cov ⇒ systems. We can, therefore, use it to testseparately the effect of the two main modifications fromthe free to the cov ⇒ system: anchoring the magneticmoments to the particle frames ( free to cov (cid:28) ) and hav-ing all magnetic moments point into the same direction( cov (cid:28) to cov ⇒ ).If we would apply an external magnetic field beforethe magnetic moments are anchored, we would destroy (a)(b) FIG. 2. Snapshots of characteristic samples containing chain-like aggregates in the equilibrated unstretched state. The twodisplayed systems are generated from the same initial place-ment of the rigid embedded particles. Yet, the way of subse-quent anchoring of the magnetic moments, here indicated bysmall bar magnets, is different, leading to the two differentequilibrated states. The matrix was tessellated into a meshof tetrahedra, those faces of which that constitute the overallsystem boundaries are depicted explicitly. (a)
Free system,where the magnetic moments can rotate freely with respectto the carrying particles. This leads to opposite alignment ofthe magnetic moments in different chains, as indicated in thetop right for the cross-sectional center plane perpendicular tothe chain axes. (b) Snapshot for the cov ⇒ system, wherethe magnetic moments are fixed to the particle axes, likewiseincluding a cross-sectional cut. The snapshot for the cov (cid:28) system is by definition again the one shown in (a), becausein this system the magnetic moments are fixed to the particleaxes only after the initial equilibration in the free system. the alternating chain morphology that we want to study.Thus, when studying the influence of an external mag-netic field on these alternating chain systems, we applyit after the magnetic moments have been anchored. Sub-sequently, we reequilibrate the systems under these newconditions before performing the stress-strain measure-ment. IV. RESULTS AND DISCUSSION
In the following, we will present and discuss our resultsfor the three systems free , cov ⇒ , and cov (cid:28) as definedabove. We begin with vanishing external magnetic fieldand then proceed to the situation of magnetic fields ap-plied parallel and perpendicular to the stretching direc-tion. For each system and each magnetic field, we showsnapshots as well as the uniaxial stress-strain curves anddiscuss the various mechanisms that lead to our results.Important insight can be gained by statistically ana-lyzing the orientations of the magnetic moments in thesystems. We evaluate them by considering the nematicorder parameter S m , which is defined as the largest eigen-value of the nematic order parameter tensor [75] Q m = 1 N N (cid:88) i =1 (cid:18) ˆm i ⊗ ˆm i − ˆI (cid:19) . (6)Here the ˆm i are the magnetic moment orientations of the N particles in the system, ⊗ marks the dyadic product,and ˆI is the unity matrix. S m measures the degree ofalignment of the input orientations without distinguish-ing between an orientation ˆm i and its opposing orienta-tion − ˆm i . Perfect alignment leads to S m = 1, while inthe absence of global orientational order S m = 0.In addition to the magnetic order in the systems, alsothe structural order contains useful information. It canbe quantified in a very similar way by defining anothernematic order parameter S r for the orientations ˆr i ofthe separation vectors from each particle i to its nearest-neighbor.As will be revealed later in more detail, in the free system a “flipping mechanism” [31] plays an importantrole. “Flips” refer to events during which some magneticmoments suddenly change their direction with respect tothe stretching axes from parallel towards perpendicular.They are induced by the stress-induced structural changeof the system. To appropriately characterize this flippingmechanism, we define special modified nematic order pa-rameters ˜ S m and ˜ S r as described here for ˜ S m . First, toget rid of the distinction between different perpendiculardirections, we determine the projections ˆ m (cid:107) i of the mag-netic moment orientations ˆm i onto the stretching axis aswell as the projections ˆ m ⊥ i into the plane perpendicularto the stretching axis. Then we define a two-dimensionalnematic order parameter tensor as ˜Q m = 1 N N (cid:88) i =1 (cid:0) ˆ m (cid:107) i (cid:1) − m (cid:107) i ˆ m ⊥ i m (cid:107) i ˆ m ⊥ i (cid:0) ˆ m ⊥ i (cid:1) − (7)and obtain ˜ S m as the largest eigenvalue of this tensor.The calculation of ˜ S r is analogous. A. Vanishing external magnetic field (B = 0 ) We now start by quasistatically stretching the threesystems along the chain axes in the absence of an externalmagnetic field. The elongation is stepwise increased to amaximum and then, in the inverse way, reduced back tozero. The necessary forces on the clamps are recorded.Figure 3 shows the strongly nonlinear stress-strain be-haviors resulting for the three systems. In the beginning,all systems show an almost identically steep increase ofthe stress with the imposed strain. Then, from a strain ofabout ∆
L/L ≈
10% up to ∆
L/L ≈ cov ⇒ and cov (cid:28) systems the plateau is almost completely flat.However, in our strain-controlled measurements we finda regime of negative slope [76] for the free system. More-over and in contrast to the other systems, we here observeconsiderable hysteresis for the free system in the straininterval containing the superelastic plateau. In all cases,subsequent to the plateau, the slope partially recovers,becomes relatively constant, and does not differ muchamong the different systems.The main mechanism responsible for the nonlinearitiesin all systems is a stress-induced detachment mechanism[31]. We briefly illustrate how it can lead to the changefrom the steep slope at the origin of the stress-straincurve to the subsequent superelastic plateau. Consideragain the unstretched states depicted in Fig. 2. In thesestates, the chains are contracted because of the mutualattraction between the magnetic moments of neighbor-ing particles. Thus, the elastic material in the gaps be-tween particles is pre-compressed and the particles areclose to each other. In this situation, the dipolar attrac-tion is strong, since its interaction energy, see Eq. (4),scales with the inverse cube of the distance. To stretchthe system, work has to be performed against this strongattraction between the particles, which accounts for thesteep initial increase in the stress-strain curve. However,when a section of a chain is detached a little from theremainder, the attraction between both parts weakensconsiderably. Therefore, once overcoming the magneticbarrier, the displaced chain section can be detached fromthe remainder of the chain. Such a detachment event re-leases the energy stored in the gap between the detachedparticles and allows a sudden elongation of the system. FIG. 3. Uniaxial stress-strain curves for the free , cov ⇒ , and cov (cid:28) systems as well as for a corresponding system contain-ing unmagnetized particles ( unmag ) when stretching alongthe axes of the chain-like aggregates. The magnetized sam-ples show a superelastic plateau between ∆ L/L ≈
10% and∆
L/L ≈ free system, our curves show pronounced hysteresis. Figure 4a shows a snapshot of a free sample stretchedby 35%, illustrating this process. In the depicted situ-ation, some particles are detached from the chains withincreased particle separation, while smaller segments arestill intact. Each individual detachment event corre-sponds to a small localized drop in the stress-strain curve.In a very small and ordered system, this would lead to aspiky appearance of the stress-strain relation as we havedemonstrated for a single chain in Ref. [31]. However, av-eraging over the many detachment events that occur in alarger, inhomogeneous system with many parallel chainsyields a smooth superelastic plateau as in Fig. 3. Uponunloading the system, the individual particles can sim-ply reattach, reform the chains, and restore the energy intheir separating gaps, so that the detachment mechanismis reversible.The second mechanism contributing to the observedsuperelasticity is the flipping mechanism. It only playsa significant role in the free system. In the unstretchedsample, the magnetic moments align along the chain axesin a head-to-tail configuration to minimize their magneticinteraction energy, see Fig. 2. This situation changeswhen the sample is sufficiently stretched in the directionparallel to the chains. The distances between particlesin the same chain eventually increase, see Fig. 4a andRef. [31]. Meanwhile, volume preservation in our nearlyincompressible systems enforces a contraction in the per-pendicular direction, driving different chains closer toeach other. Eventually, the interparticle distances inthe parallel and perpendicular directions become approx-imately equal for subsets of particles. For the involved (a) (b)(c) (d)
FIG. 4. (a) Snapshot of a free sample stretched by 35%. The freely rotating magnetic moments in this system can minimizetheir magnetic interaction by aligning along the direction of shortest interparticle distance. When the sample is stretched, theperpendicular direction becomes more and more favored, because the interparticle distance within the chains is increased, whilenear-incompressibility of the sample forces neighboring chains to approach each other. In the depicted situation, about half ofthe particles are detached from the chains, their magnetic moments having performed a flip from a direction parallel to thestretching axis towards perpendicular. In the (b) cov ⇒ and (c) cov (cid:28) systems, rotations of the magnetic moments necessitaterotations of the carrying particles, causing restoring torques by the surrounding matrix. Still, significant particle rotations canbe observed in these samples stretched by 100% with respect to the unstretched states in Fig. 2, caused, however, primarilyby inhomogeneous deformations of the surrounding matrix due to the particle embedding. (d) Snapshot of an unmagnetized( unmag ) system starting from the same configuration. The bars indicate the initially horizontal particle axes to illustrate theparticle rotations. They show a similar pattern as the systems in (b),(c) although magnetic interactions are absent. magnetic moments this means a sudden change in theirpreferred orientation from parallel to the stretching axistowards perpendicular. In the free system, the momentscan easily seize this opportunity to minimize their mag-netic interaction energy by sudden reorientation. Thisconstitutes a flip event.Flips are associated with drops in the stress-straincurve for the following reason. As long as the magneticmoments participating in a flip event are still aligned par-allel to the stretching direction, their magnetic interac-tion energy increases with the strain. However, once theflip has occurred and they have aligned towards perpen-dicular, their magnetic interaction energy decreases withthe stretching. Therefore, during a flip event, the slopeof the magnetic interaction energy suddenly changes forthe participating magnetic moments. Since the stress isthe derivative of the energy with respect to the strain,this causes a drop in the stress-strain curve. Or, dis-cussing the situation directly in the force picture: as longas the magnetic moments align along the stretching axis,they counteract the elongation, which requires a higherstretching force; once they flip, they repel each otheralong the stretching axis, which supports the elongation.In an inhomogeneous sample, flips are local events andcan occur over a wide range of global strain magnitudes. As a result, the individual drops are smoothened out inthe stress-strain curves resulting from our characteristicsystems.Consider again the snapshot in Fig. 4a. Comparedto the particles in the still intact chain parts, the de-tached particles have a larger interparticle distance inthe stretching direction and their magnetic moments in-deed prefer an orientation towards perpendicular to thatdirection. When a detachment event occurs, the cor-responding sample section elongates, which can in turntrigger flip events. Conversely, a reorientation of mag-netic moments towards a perpendicular direction can in-duce detachment. So in our characteristic free systems,the detachment and flipping mechanisms are intertwined.Yet, considering suitable idealized model situations, bothmechanisms can be studied in isolation, see Ref. [31]. Theinterplay between both mechanisms supports the hystere-sis observed in our stress-strain curves for the free system,see Fig. 3. The magnetic attractions pull the particlestogether along the orientation of the magnetic moments,which in turn self-strengthens the magnetic interaction.In this way, an energetic barrier is created that needs tobe overcome every time the magnetic moments are pulledapart and flip, either during initial stretching, or in theflipped state during unloading. FIG. 5. Degrees of magnetic order ˜ S m and structural order˜ S r for the free system, following the definition in Eq. (7).For vanishing strain, alignment along the initial anisotropyaxis is preferred both magnetically and structurally. Whenthe strain is increased, detachment and flip events occur andthe system enters a mixed state where the parallel directionbecomes less dominant in favor of directions perpendicularto the stretching axis. The minimum is reached at a strainof ∆ L/L ≈ We can further quantify the flipping mechanism by sta-tistically analyzing the orientations of the magnetic mo-ments. Let us evaluate the nematic order parameters ˜ S m and ˜ S r defined in Eq. (7) as a function of the imposedstrains. ˜ S m measures the degree of alignment of the mag-netic moments and ˜ S r does the same for the separationvectors between nearest-neighboring particles. The resultis plotted in Fig. 5. For low strains, magnetic momentsare aligned parallel to the stretching axis, because thisis the direction of smallest interparticle distance. Con-sequently the system is in a state of high magnetic andstructural order, reflected by the high levels of ˜ S m and ˜ S r .Upon increasing the strain, the interparticle distances inthe stretching direction increase, particles are detachedand magnetic moments flip, taking the system into amixed state. ˜ S m and ˜ S r simultaneously decrease andreach a minimum at ∆ L/L ≈ S m and ˜ S r increase again until finally all particles are detached andall magnetic moments have flipped. The strain regimewhere the order parameters change significantly coincideswith the position of the superelastic plateau in the stress-strain curve in Fig. 3. Finally, at the highest strains, bothorder parameters again decrease slightly when the lateralcontraction of the system squeezes the particles together.This causes them to evade each other when they cometoo close and makes them shift relatively to each other along the stretching axis, which disturbs the perpendic-ular alignment. Also for the order parameters, we hereobserve again the hysteresis discussed already before inthe context of the stress-strain curve.Let us now come back to the cov ⇒ and cov (cid:28) systemswhere the magnetic moments cannot rotate relatively tothe particle frames. Then magnetic reorientations costa significant amount of elastic energy, as this requires acorotation of the elastic matrix directly anchored to theparticle surfaces. Figures 4b,c show snapshots of cor-responding samples at a strain of 100%. There we cannonetheless observe particle rotations. These particle ro-tations, however, do not apparently lead to a configu-ration that minimizes the magnetic interaction energy.In fact, the primary reason for these rotations is notthe magnetic interaction between particles, but inhomo-geneities in the stiffness across the system. We recall thatthe particles in our systems are rigid inclusions of finiteextension. Consequently, the particles are local sourcesof elevated rigidity within the soft elastic matrix. Al-ready in an unmagnetized system, such rigid inclusionslead to an overall stiffer elastic behavior of the whole sys-tem [77–79]. In our case, an increase of a factor of ∼ m = 0) just aswell. In Fig. 4d we show a snapshot of an unmagnetizedsystem stretched by 100% for demonstration. There weindicate the initially horizontal particle axes by bars tovisualize the particle rotations. The resulting patterns ofparticle rotation are qualitatively similar to the ones inthe cov ⇒ and cov (cid:28) systems.Again we use statistical analysis to further quantifythe particle rotations. Due to the different mechanismwhen compared to the flipping process, we are here onlyinterested in the degree of alignment along the initialanisotropy axis. Therefore, we use the nematic orderparameter S m defined in Eq. (6) for quantification. Theresults are plotted in Fig. 6a as a function of the imposedstrain for the free , cov ⇒ , and cov (cid:28) systems, as well asfor the unmagnetized ( unmag ) case. Let us first considerthe unmag system. Up to a strain of ∆ L/L ≈ S m stays closeto 1. Then, there is a crossover to a regime of approx-imately linear decay of S m . The particles rotate moreand more away from the initial axes of alignment as aconsequence of the inhomogeneous stiffness. The behav-ior in the cov ⇒ and cov (cid:28) systems is very similar, thecrossover to the regime of declining order merely occursat a higher strain of ∆ L/L ≈ (a)(b) FIG. 6. (a) Nematic order parameter S m according toEq. (6) for the magnetic moment orientations of the free , cov ⇒ , cov (cid:28) systems, as well as for an unmagnetized ( un-mag ) system as function of the imposed strain ∆ L/L . Inthe latter three systems, there is a regime of high magneticorder at low strains. At a strain of ∆ L/L ≈
35% in the unmag system and ∆
L/L ≈
50% in the cov ⇒ and cov (cid:28) systems, there is a crossover to a regime of declining order,as inhomogeneous stresses begin to rotate the particles. Inthe free system, again a minimum indicates the occurrence offlip events. The recovery of S m beyond the minimum shows,that there is one globally preferred perpendicular directionemerging subsequent to flipping. (b) Nematic order parame-ter S r for the nearest-neighbor separation vectors in the samesystems. All curves have a minimum at the point where thepreferred direction switches from parallel to the stretchingaxis towards perpendicular. In the magnetized systems, thisminimum is postponed to higher strains. In these systems,the detachment barrier and magnetic interactions along thechains stabilize the chain structure, which is then preservedup to higher strains. and stabilize the alignment up to higher strains. Whenthe detachment of the particles from the chains has beencompleted at the end of the superelastic plateau, this sta-bilizing magnetic interaction disappears, rendering theparticles susceptible to shear stresses originating from thesystem inhomogeneity. The curve for the cov (cid:28) systemis always below the one for cov ⇒ , because already theinitial unstretched state is less ordered, see again Fig. 2. The behavior of S m for the free system is obviouslycompletely different and should rather be compared with˜ S m in Fig. 5. S m shows a rapid initial decay up to a mini-mum and afterwards recovers to reach a relatively low butconstant level. This is despite the fact that S m , unlike˜ S m , distinguishes between different directions perpendic-ular to the stretching direction. Therefore, beyond thesuperelastic plateau, one particular axis perpendicularto the stretching axis must emerge along which the mag-netic moments preferably align. Such a direction formsas nearby flipped magnetic moments tend to align bymagnetic dipolar interaction. In turn, this favors furthercontraction along such an emerging axis of alignment,providing a self-supporting mechanism. Inherent struc-tural inhomogeneities will affect this mechanism.The same analysis as for S m can be conducted for thenematic order parameter S r of the separation vectors be-tween nearest-neighbors. It is plotted for all systemsin Fig. 6b. S r starts at a high value for all systems,because in the unstretched state the nearest-neighborof each particle is always along the chain. The morethe sample is stretched, the more the distances alongthe chains increase, while the distances between sepa-rate chains decrease due to volume preservation. Thus,it becomes increasingly likely that the nearest-neighborfor a particle is a member of a different chain. In the unmag system, there is no stabilizing attractive inter-action keeping the chains together. So the minimum,where nearest-neighbor directions predominantly switch,is reached relatively soon. In the other systems, how-ever, the magnetic attraction makes the chains subject tothe detachment mechanism. Segments detach from thechains, while the remainder of the chains remains intact.As a result, partial structural order is preserved up tomuch higher strains. Again, the strain regime where S r changes a lot due to the changes in structural order coin-cides with the strain interval of the superelastic plateauin the stress-strain curves in Fig. 3. B. External magnetic field along the stretchingaxis (B = B x ˆx) Applying an external magnetic field parallel to thechain and stretching axis (the x -direction) when record-ing the stress-strain behavior changes the situation fun-damentally in all three systems free , cov ⇒ , and cov (cid:28) .In the free system, turning on the field after the initialequilibration causes all magnetic moments to point intothe same direction along the field as opposed to the situa-tion in Fig. 2a. There, the magnetic moments carried byparticles in different chains can show opposite magneticalignment. In the free system as well as in the cov ⇒ system, the field also introduces an additional energeticpenalty for the rotation of magnetic moments away fromthe chain axes. The detachment mechanism is not im-peded by this, as it relies on the strong magnetic attrac-tion between neighboring particles within the same chain0 B x (a) (d)(b)(c) FIG. 7. Results for the free system under the influence of an external magnetic field of varying strength, applied parallelto the stretching axis. (a) Uniaxial stress-strain behavior . The external field gradually deactivates the flipping mechanism.As a result the superelastic plateau is flattened, the dip at ∆ L/L ≈
50% and the hysteresis are removed until the behaviorresembles the one for the cov ⇒ system in Fig. 3 for vanishing external magnetic field. (b) Snapshot showing a free systemsubject to an external field of B x = 1 B at a strain of ∆ L/L = 100%. Even in this highly strained state, the magnetic momentsassume oblique angles instead of performing full flips towards a perpendicular direction. (c) Degree of magnetic order ˜ S m and(d) degree of structural positional order ˜ S r as defined by Eq. (7), indicating the deactivation of the flipping mechanism withincreasing B x . The minimum in ˜ S m is gradually removed by the parallel external magnetic field. Meanwhile, the minimum in˜ S r is shifted slightly. and the storage of elastic energy within the compressedgap material. The magnetic moments are not rotatedaway from the alignment along the chain axes duringthis process. In contrast to that, the flipping mecha-nism is based on reorientations away from the directionof the applied magnetic field and is, therefore, affected bythe aligning magnetic field. In the cov (cid:28) system featur-ing anchored magnetic moments of opposite alignment,the external magnetic field has a particularly interest-ing effect. Roughly half of the magnetic moments arealigned with the field. The remaining moments are mis-aligned and the corresponding particles would need torotate by about 180 degrees to minimize the interactionenergy with the external magnetic field. In Ref. [31], the magnetic field strengths in the figures containingstress-strain curves were not scaled correctly. Instead of 10 B ,20 B , 30 B it should read 1 B , 2 B , 3 B , respectively. Figure 7 revisits our results for the free systems for var-ious applied magnetic field strengths. The stress-straincurves in Fig. 7a illustrate the tunability of the material .Already a small external magnetic field of B x = 1 B re-moves the dip at ∆ L/L ≈ cov ⇒ system in thecase of vanishing external magnetic field, see Fig. 3.The snapshot in Fig. 7b shows a free system for B = 1 B at a strain of ∆ L/L = 100%. It reveals,that the magnetic moments do not perform completeflips anymore and instead show oblique orientation an-gles. In summary, the flipping transition and the con-nected bumps in the superelastic plateau together withthe hysteresis can be deactivated by the field.1The plot in Fig. 7c of the nematic order parameter˜ S m quantifying the magnetic order in the system pro-vides further evidence that the field impedes the flippingmechanism. An external magnetic field of B x = 1 B issufficiently strong to smoothen the sharp local minimumin ˜ S m corresponding to the transition from a state ofparallel towards perpendicular magnetic alignment withrespect to the stretching axis. Stronger fields enforcea parallel alignment, remove the local minimum in ˜ S m and thus deactivate the flipping mechanism. Only thedetachment mechanism remains active. Meanwhile, thestructural positional order in the sample does not seemto be influenced significantly by the external magneticfield, as the plots of the nematic order parameter ˜ S r for the separation vectors between nearest-neighbors inFig. 7d suggest. The minimum where the most likelynearest-neighbor direction switches from parallel towardsperpendicular is shifted slightly. Beyond the minimum,˜ S r decreases with increasing B x . This results from anarising competition between two effects. On the onehand, due to overall volume preservation, the particlesare driven together along the direction perpendicular tothe stretching axis as before. On the other hand, flipsare hindered by the external magnetic field, or even sup-pressed completely. Therefore, the magnetic momentscannot support the perpendicular approach anymore asefficiently, or even counteract it due to the magnetic re-pulsion when the magnetic moments are forced into thedirection of the external magnetic field. This also largelyremoves the hysteresis from our curves.Let us discuss the cov ⇒ system next. The results aresummarized in Fig. 8. Figure 8a shows the correspondingstress-strain behavior. Up to the end of the superelasticplateau, the curves for different external magnetic fieldstrengths hardly differ. This is not surprising, since wehave established before that the flipping mechanism playsno role for these systems and that the detachment mech-anism is not impeded by an external magnetic field par-allel to the chains. However, beyond the plateau, wherewe have a regime of relatively constant increase of thestress with the imposed strain, we can observe a stiff-ening of the system when a higher field strength is ap-plied. Only at very high strain, the slopes for all differentfield strengths become similar again. The explanation forthis stiffening influence of the external magnetic field isthe suppression of magnetic moment reorientations and,thus, in this cov ⇒ system, of particle rotations. Wehave seen, however, in Fig. 4b that such particle rota-tions would arise in the absence of a magnetic field tominimize the elastic energy. Suppressing them increasesthe necessary mechanical energy input into the system.The snapshot in Fig. 8b shows a sample with an appliedfield of B x = 10 B at a strain of ∆ L/L = 100% forcomparison with the analogous situation in Fig. 4b for B x = 0.For a more quantitative analysis of the rotation effects,we evaluate the nematic order parameter S m of the ori-entations of the magnetic moments as a function of the imposed strain, see Fig. 8c. We can distinguish betweentwo major regimes. In the first one, the overall strainis still too low to induce significant local shear defor-mations due to the inhomogeneities, thus, the particlesrotate only slightly and S m remains on a high and rela-tively constant level. However, in the second regime, wecan observe an approximately linear decay in S m as theparticles begin to significantly rotate. In the absence ofan external magnetic field, the crossover between bothregimes occurs at the end of the superelastic plateau.There, the particles are detached from the chains. Thisreduces the aligning magnetic interactions and the parti-cles become susceptible to rotations due to the elastic in-homogeneities in the system. Interestingly, increasing thestrength of the external magnetic field can postpone thecrossover far beyond this point by supporting the mag-netic moment orientations along the field direction. Thisstiffens the system in two ways. First, the inhomogeneityshear stresses are prevented from relaxing via the favoredchannel: the rotation of particles. Second, the magneticmoments in the system keep repelling each other perpen-dicular to the stretching axis, which works against theirperpendicular approach. The stronger the external mag-netic field strength, the longer the embedded particlescan resist a rotation, maintaining the stiffening effect.For all considered magnetic field strengths, the particleseventually begin to rotate, as indicated by the crossoverin S m . Therefore, the slopes of the stress-strain curvesbecome similar again at the maximum strain.Finally, we show for completeness in Fig. 8d the ne-matic order parameter S r for the nearest-neighbor sep-aration vectors as a function of the imposed strain.Here, the curves for different magnetic field strengths arelargely similar.Now we come to the cov (cid:28) system and present the re-sults in Fig. 9. Before the external magnetic field is ap-plied, these systems are in a state like the one depictedin Fig. 2a. Roughly half of the magnetic moments arealigned along to the magnetic field direction, while theother half is oppositely aligned and tends to reorient tominimize the magnetic interaction energy with the exter-nal field. This has implications on the stress-strain be-havior, as illustrated in Fig. 9a. For small field strengths( B x = 2 B ), the behavior barely changes compared tothe case of vanishing external magnetic field. Then forintermediate fields of B x = 4 B , the steep increase atlow strains as well as the superelastic plateau becomeless pronounced. Starting from a field of B x = 6 B , thesuperelastic features vanish altogether. An explanationis given in the following. As long as the external fieldstrength is low enough ( B x = 2 B ), the energy cost ofmisalignment is not particularly large for the magneticmoments in the metastable configuration antiparallel tothe field. However, when increasing the external field,due to imperfections in the initial antiparallel alignment,at some point the magnetic particles can be rotated bya significant amount. Then, the torques due to the ex-ternal field get amplified, causing the particles to rotate2 B x (a) (b)(c) (d) FIG. 8. Same as Fig. 7, but for the cov ⇒ system. (a) The stress-strain curves for different external magnetic field strengths arealmost identical up to the end of the superelastic plateau. Beyond this point, higher field strengths increase the stiffness untilat very high strains the slopes become similar again. (b) Snapshot of a system at a strain of ∆ L/L = 100% illustrating thata field of B x = 10 B can effectively prevent the particle rotations favored by local shears due to the elastic inhomogeneities.Here, the internal shear stresses of the system cannot relax via particle rotations and the parallel magnetic moments repeleach other in the direction perpendicular to the stretching axis, both effects stiffen the system against further elongation. (c)Nematic order parameter S m for the magnetic moment orientations. The external magnetic field can postpone the crossoverto the regime of decreasing orientational order, allowing for particle rotations and magnetic moment reorientations only atvery high strains. (d) Here, the nematic order parameter S r for the nearest-neighbor separation vectors is barely sensitive to achange in the external magnetic field strength. even further. At this stage, the reorientations of themisaligned moments together with their carrying parti-cles begin to distort the sample substantially. Obviously,for the corresponding chains, the detachment mechanismwill seize to function at this point, but also the chainscontaining aligned magnetic moments in the neighbor-hood will be disturbed. This chaotic situation is depictedin the snapshot in Fig. 9b for an external magnetic fieldof B x = 6 B and a strain of 30%. One can still identifythe particles that have been aligned along the field di-rection, but the corresponding chains are distorted. Asa result, the detachment mechanism is disabled and thesuperelastic plateau vanishes.The plots of the nematic order parameters S m and S r in Figs. 9c,d support this picture. For small magneticfield strength of B x = 2 B , S m is still very similar tothe case of vanishing magnetic field. Further increasingthe field strength up to B x = 6 B promotes magnetic disorder in the system, leading to an overall low level of S m . From there on, the level of S m slightly increaseswith the magnetic field strength as the orientations ofthe aligned magnetic moments are stabilized by the field.The structural order measured by S r does not changetoo much as long as B x (cid:46) B . Starting from B x (cid:38) B ,however, the misaligned magnetic moments are rotatedsignificantly and distort the system. The increased mag-netic order indicated by a higher level of S m apparentlycannot prevent the structure from becoming more dis-turbed, so that S r is still lowered further.In conclusion, the effect of an external magnetic fieldapplied parallel to the stretching axis varies substantiallyamong the different systems. In the free system, the maineffect is the deactivation of the flipping mechanism, whichmakes the stress-strain behavior almost identical to theone of the cov ⇒ system in the absence of an externalmagnetic field. Within the cov ⇒ system the superelas-3 B x (a) (b)(c) (d) FIG. 9. Same as Fig. 7 but for the cov (cid:28) system. (a) Uniaxial stress-strain behavior. Applying an external magnetic fieldparallel to the stretching axis gradually removes the pronounced nonlinearity. (b) Snapshot of a cov (cid:28) system under theinfluence of an external magnetic field of B x = 6 B at a strain of ∆ L/L = 30%. The particles carrying the misalignedmagnetic moments are strongly rotated towards the external magnetic field and distort their environment in the process, whichalso affects the chains containing the particles of aligned magnetic moments. As a result, the detachment mechanism is mostlydeactivated. (c) Nematic order parameter S m for the magnetic moment orientations. Increasing the strength of the externalmagnetic field first lowers the overall S m due to the rotations of particles carrying misaligned magnetic moments and due tothe resulting distortions of the rest of the system. At high field strengths, S m increases slightly with B x , as the orientations ofthe aligned magnetic moments are stabilized. (d) The structural order in the system measured by S r is not influenced stronglyas long as B x (cid:46) B . Beyond that field strength, however, it significantly decreases because of the induced rotations of theparticles carrying misaligned magnetic moments. ticity is barely affected. However, the external magneticfield stabilizes the particle orientations at strains beyondthe superelastic plateau and thereby stiffens the stress-strain behavior. Finally, in the cov (cid:28) system the fieldpromotes a strongly disturbed structure by rotating par-ticles carrying magnetic moments misaligned with thefield. As a consequence, the detachment mechanism isdisabled and the superelastic plateau vanishes from thestress-strain curves. C. External magnetic field perpendicular to thestretching axis (B = B y ˆy) An external magnetic field applied perpendicular tothe stretching axis (here the y -axis) attempts to rotate the magnetic moments away from their attractive head-to-tail configuration within the chains. This influenceis strongest in the free system, where the magnetic mo-ments are free to reorient to minimize their magnetic en-ergy. In the cov ⇒ and cov (cid:28) systems, however, rotationsof the magnetic moments are counteracted by restoringtorques on the embedded particles due to the induceddeformation of the surrounding matrix.Let us again discuss the free system first. We presentthe results in the same fashion as before for the parallelfield. Figure 10a shows the resulting stress-strain behav-ior . The perpendicular field has two effects. First, itinfluences the superelasticity, causing the plateau to beconfined to a smaller strain interval. Second, it lowersthe initial slope of the stress-strain curve. At a highenough magnetic field strength, the superelastic nonlin-4 (a) (b)(c) (d) FIG. 10. Results for the free system under the influence of an external magnetic field of varying strength perpendicular tothe stretching axis. (a) The superelastic stress-strain behavior can be readily tuned . Increasing the field gradually removesthe superelasticity and lowers the slope of the initial steep increase. A field of B y = 3 B is already strong enough to removeall superelastic nonlinearities. (b) Snapshot of an unstretched sample with an applied external magnetic field of B y = 2 B .A significant portion of the particles is already detached, their carried magnetic moments already flipped. As a consequence,the detachment and flipping mechanism have less impact on the stress-strain behavior, and superelastic as well as hystereticfeatures are reduced. (c) Degree of magnetic order ˜ S m and (d) degree of structural order ˜ S r using the definition in Eq. (7).Both order parameters are again strongly correlated. Increasing the magnetic field strength shifts the local minimum markingthe regime of mixed orientations to lower strains. That is, the threshold strains for detachment and flip events are lowered,with many events having occurred already in the unstretched state. This limits the amount of events that can still take placewhen the sample is stretched. At B y = 3 B , the pronounced minima of ˜ S m and ˜ S r have vanished as all magnetic momentsare already reoriented in the unstretched state. Therefore, there are no remaining flip or detachment events already in theunstretched state and, as a consequence, superelasticity is switched off. earities are switched off completely together with the hys-teresis, and the stress-strain curve becomes ordinary.To understand this behavior, it is first noted that theperpendicular magnetic field shifts the flipping mecha-nism to smaller strains. This is intuitive, as the exter-nal magnetic field energetically supports flips to a direc-tion perpendicular to the stretching axis. Analysis ofthe nematic order parameters ˜ S m and ˜ S r in Figs. 10c,d,respectively, confirms this expectation. The regime ofmixed orientations centered around the minimum in ˜ S m is shifted to lower strains by the field. In this regime,some of the magnetic moments are still aligned along thechains, while others have already flipped. Meanwhile,˜ S r remains strongly correlated with ˜ S m . This indicates that the external magnetic field does not only influencethe flipping mechanism, but also the detachment mech-anism. As noted before, flip events trigger detachmentevents and vice versa. Reoriented magnetic moments donot feel a strong attraction along the stretching axis thatcould keep the carrying particles attached to the chains.So the threshold strains for both mechanisms are loweredat the same time.This shift of threshold strains can cause the system toenter a mixed state already without any external strainimposed. The snapshot in Fig. 10b shows a situationof B y = 2 B . Although the system is unstretched inthe depicted case, a significant amount of particles hasalready detached from the chains. Their magnetic mo-5ments are aligned along the field direction, perpendicularto the chain axis. So the fraction of particles that canstill perform detachment or flip events is lowered. Asa result, the features corresponding to both mechanismsare less pronounced in the stress-strain curves. Also theinitial slope is lower, because the overall magnetic attrac-tion along the stretching direction cannot counteract theelongation as strongly. Consequently, the superelasticplateau spans a smaller strain interval.We now proceed to the results for the cov ⇒ systemshown in Fig. 11. In the case of vanishing external mag-netic field, this system features global magnetic order inthe x -direction, see again Fig. 2b. Applying an exter-nal magnetic field perpendicular to the stretching axisleads to a new state of rotated global polar magneticorder. Figure 11b shows a snapshot of an unstretchedsystem subject to a strong external magnetic field of B y = 10 B . The magnetic moments, together with thecarrying particles, are rotated towards a configuration ofcollective polar alignment oblique to the external mag-netic field. This occurs against the strong magnetic at-tractions within each chain and the necessary elastic de-formation of the matrix between the particles. The rota-tions of individual particles are energetically expensive.In fact, the system partially avoids these expensive ro-tations by allowing chain segments to rotate as a wholetowards the field. Undulations and buckling of the chains[80] then lead to a compromise between the minimizationof the elastic and magnetic parts of the total energy.Either way, the magnetic dipolar attraction betweenneighboring particles along the stretching direction isweakened, which impedes the detachment mechanism.So the influence of the perpendicular external magneticfield on the stress-strain behavior is again a gradualremoval of the superelastic plateau, as illustrated inFig. 11a. A stiffening of the stress-strain behavior be-yond the superelastic plateau, as in the case of a paral-lel external magnetic field, however, cannot be observed.Contrary to the parallel magnetic field, the perpendicu-lar magnetic field breaks the uniaxial symmetry of thesystem and offers a distinctive direction for the particlesto rotate towards. As can be deduced from the nematicorder parameter S m of the magnetic moments plotted inFig. 11c, the perpendicular external field aligns the par-ticles very effectively even up to the highest consideredstrains. Differences in the rotations of the particles dueto elastic inhomogeneities can, thus, be prevented. Afield of B y = 2 B , is already quite successful in this re-spect, using stronger fields does not significantly increasethe effect much further. The mutual repulsion betweenthe parallel magnetic moments does not counteract anelongation of the system any more. Thus, there is nosignificant stiffening of the stress-strain behavior whenchanging the external magnetic field strength.We also plot the nematic order parameter S r of thenearest-neighbor separation vectors in Fig. 11d. For B y = 0, S r is at a high level for low strains, where itis most likely that the nearest-neighbor of a particle is located along the stretching axis within the same chain.Then S r quickly drops as the chains are stretched outand subsequently remains at a low level. When a per-pendicular magnetic field is applied, such a drop of S r never occurs. It remains likely that the nearest-neighborof a particle is within the same chain for the whole con-sidered range of strains. This reflects again the tendencyof whole chain segments to rotate as one unit towards thefield, staying structurally intact and creating the partialstructural order reflected by S r .Let us finally discuss the cov (cid:28) system under the influ-ence of a perpendicular external magnetic field. Contraryto the case of a parallel external magnetic field, there areno particles that are aligned oppositely to the externalfield. All particles can in principle rotate equally eas-ily into the external magnetic field direction. However,the initial orientation of the magnetic moment of a par-ticle determines the sense of rotation towards the field.Neighboring chains with opposing alignment of the mag-netic moments show opposing sense of rotation. As aconsequence, in contrast to the cov ⇒ system, the rota-tions of complete chain segments towards the magneticfield are largely blocked. Instead, the particles withinthe chains individually rotate towards the external field,as depicted in the snapshot of an unstretched sample inFig. 12b. Here, the external magnetic field of B y = 10 B has rotated the particles by a significant amount, butthe chains are still relatively ordered and aligned alongthe stretching axis. Depending on their initial alignment,the magnetic moments together with their carrying parti-cles rotate either clockwise or counterclockwise towardsthe field. In this way, there are two competing mag-netic polarities in the system, with roughly the same y -component but oppositely signed x -components. The re-sulting stress-strain behavior is plotted in Fig. 12a andreveals an influence of the external magnetic field verysimilar to the cov ⇒ system. Increasing the magneticfield strength rotates the particles further and weakenstheir attraction along the stretching axis. This gradu-ally disables the detachment mechanism and, therefore,removes the superelastic plateau from the stress-straincurve. Again, we cannot observe significant stiffening ofthe system at high strains when increasing the externalmagnetic field strength, for the same reasons as in the cov ⇒ system.The two competing magnetic polarities are reflectedby the nematic order parameter S m plotted in Fig. 12c.In the unstretched state, when neighboring particles ina chain are close to each other, their magnetic interac-tion intensifies an alignment of the magnetic momentsparallel to the stretching axis. The magnetic field, how-ever, urges the differently orientated magnetic momentsand their carrying particles to rotate out of their com-mon initial axis of alignment. More precisely, for mag-netic moments of opposite initial orientation, this leadsto opposite senses of rotation, which destroys the overallnematic alignment. At low field strengths the particlesrotate only slightly in the unstretched state, so that S m is6 (a) (b)(c) (d) FIG. 11. Same as Fig. 10, but for the cov ⇒ system. (a) The superelasticity in the stress-strain behavior can again bedeactivated by a perpendicular external magnetic field, but only at significantly higher field strengths. (b) Snapshot showingthe unstretched state of a system under the influence of a field of B y = 10 B . The system enters a new state of global polarmagnetic order, with magnetic moments aligned oblique to the external magnetic field. Energetically expensive rotations ofindividual particles are avoided, instead whole chain segments rotate as one unit. (c) Plot of the nematic order parameter S m for the magnetic moment orientations demonstrating that already a moderate magnetic field strength can maintain a state ofglobal polar magnetic order up to the maximum elongation. (d) Nematic order parameter S r for the nearest-neighbor separationvectors. When the external magnetic field is weak, S r is high at low strains and then drops to a low and relatively constantlevel. A strong field removes this large drop so that a relatively constant intermediate level of structural order remains at allstrains. This indicates again the tendency of whole chain segments to rotate as one unit, creating a principal axis of structuralorder oblique to the external magnetic field direction and the initial chain axes. initially high. Stronger fields are able to rotate the parti-cles further, see again Fig. 12b, leading to a lower value of S m at zero strain. With increasing strain, the magneticinteractions between neighboring particles in a chain areweakened due to their increased separation. The particlesbecome more susceptible to rotations by the magneticfield. Thus, a decline in S m can be observed. S m in-creases again when the y -direction becomes predominantfor all magnetic moments so that they again align alonga common axis. At even stronger fields of B y = 8 B and B y = 10 B , the y -direction is prevalent at all strains, sothat S m is monotoneously increasing. This is in agree-ment with the observation, that for these magnetic fieldstrengths superelastic features in the stress-strain curveare absent.Finally, we show in Fig. 12d the nematic order pa- rameter S r for the nearest-neighbor separation vectors.The minimum in each curve indicates the point where itbecomes more likely for particles to find their nearest-neighbors in a direction perpendicular to the stretchingaxis than parallel. For low field strengths, this struc-tural bias along the perpendicular axis is not very dis-tinctive. Increasing the field strength, however, shiftsthe minimum to lower strains and increases the valueof S r at higher strains. This is intuitive, because forstronger magnetic fields there is simply less attractionwithin individual chains along the stretching axis andmore attraction perpendicular to the stretching axis be-tween reoriented particles belonging to different chains.In summary, the main effect of the perpendicular ex-ternal magnetic field in all systems is the gradual removalof the superelastic plateau from the stress-strain curves.7 (a) (b)(c) (d) FIG. 12. Same as Fig. 10, but for the cov (cid:28) system. (a) The stress-strain behavior responds to the external magnetic fieldin a very similar way as for the cov (cid:28) system. Increasing the field strength gradually removes the superelastic nonlinearity.(b) Snapshot of an unstretched system with an applied external magnetic field of B y = 10 B . There are two competingpolarities for the magnetic moments, sharing a common y -component but with opposite x -components. (c) Quantificationof the magnetic order in the system via the nematic order parameter S m for the magnetic moment orientations. When themagnetic field strength and the strain are low, the two opposing polarities that are not aligned along a common axis compete,and S m is a decreasing function of the strain. The higher the magnetic field strength and the higher the strain, the more themagnetic moments are rotated. Eventually, the magnetic field direction is preferred over the stretching axis by both polaritiesand S m becomes an increasing function of the strain. For B y (cid:38) B this is already the case in the unstretched state, which isconsistent with the observation that the corresponding stress-strain curves do not show superelasticity anymore. (d) Nematicorder parameter S r for the nearest-neighbor separation vectors, quantifying the structural order. The minimum in S r shiftsto lower strains when increasing the field strength and the overall value beyond the minimum is increased. This is simply aconsequence of the particle rotations that lead to less magnetic attraction between particles along the stretching axis and tomore attraction along the magnetic field direction. This is mainly caused by the rotation of the magnetic mo-ments into the direction of the magnetic field. When themagnetic attraction between neighboring particles alongthe stretching axis disappears, the detachment mecha-nism seizes to function. In the free system, magneticmoment reorientations can be achieved exceptionally eas-ily (see the different scales for B y in Figs. 10–12), mak-ing this system highly susceptible to the perpendicularexternal magnetic field. Together with the detachmentmechanism, also the flipping mechanism is gradually de-activated. In the cov ⇒ system rotations of the magneticmoments are harder to achieve and require significantlystronger magnetic fields. We can observe collective ro- tations of the particles such that global polar magneticordering is preserved with all magnetic moments alignedoblique to the external field. Furthermore, these systemsavoid the energetically expensive rotations of individualparticles by allowing whole segments of the chains to ro-tate towards the external magnetic field as one unit. Asa result, the chains buckle and undulate as a compro-mise between minimizing the magnetic and elastic en-ergetic contributions. Finally, the cov (cid:28) system behavesquite similar concerning the influence of the external fieldon the stress-strain behavior. However, here the parti-cles do rotate individually towards the field, facilitatedby the initially opposite magnetic alignment in differ-8ent chains. During the rotation process, the opposingmagnetic alignments lead to two separate polarizationdirections of the magnetic moments. Altogether, in both cov systems, particle rotations induced by elastic inho-mogeneities of the system are effectively superseded byparticle rotations due to the external magnetic field. V. CONCLUSIONS
We have numerically investigated the stress-strain be-havior of uniaxial ferrogel systems. Our anisotropic nu-merical systems consist of chain-like aggregates of spher-ical colloidal magnetic particles that are embedded in anelastic matrix of a cross-linked polymer. The particlesare rigid and of finite size, while the matrix is treated bycontinuum elasticity theory. In experimental situations,the chain-like aggregates can be generated by applying astrong homogeneous external magnetic field during syn-thesis. We have considered three different realizations ofsuch uniaxial ferrogel systems. The free system featuresmagnetic moments that can freely reorient with respectto the frames of the carrying particles frames and the sur-rounding matrix. In contrast to that, in the cov ⇒ sys-tem the magnetic moments are fixed with respect to theaxes of the carrying particles. Additionally, the particlesare covalently embedded into the matrix: particle rota-tions require corotations of the directly surrounding elas-tic material, leading to matrix deformations and restor-ing torques. Initially, all magnetic moments point intothe same direction along the chain axes. The third sys-tem is the cov (cid:28) system, where the magnetic momentsare likewise firmly anchored. However, initially the mag-netic moments point into opposite directions along thechain axes.When we stretch these systems along the chain axes,a pronounced nonlinearity in the stress-strain behaviorappears. It has the form of a superelastic plateau, alongwhich the samples can be strongly deformed while barelyincreasing the load. The deformation is reversible and theshape and intensity of the superelastic plateau can be re-versibly tailored by external magnetic fields. There aretwo stretching-induced mechanisms that enable supere-lasticty. The main mechanism is a detachment mecha-nism and active in all systems. It relies on the strongmagnetic dipolar attraction between neighboring parti-cles within one chain as long as the magnetic momentsalign along the chain axis. At certain threshold strains,parts of the chain can detach, leading to a local elon-gation of the system. This leaves the remainder of thechain intact until the next detachment event is triggered.Besides, a flipping mechanism corresponding to reorien-tation events of magnetic moments is only active in the free system, where the magnetic moments can easily ro-tate. A flip event occurs when elongation of the systemcauses positional rearrangements such that for a subsetof magnetic moments a new orientation is suddenly ren-dered energetically more favorable. The inhomogeneous distribution of the rigid inclusionsin our samples results in regions of elevated stiffness. Athigh strains, this leads to local shears that rotate the em-bedded particles. This is especially apparent in the cov ⇒ and cov (cid:28) systems and influences their stress-strain be-havior.Our systems can be reversibly tuned by an externalmagnetic field as follows. If the field is applied parallel tothe chain axes, the detachment mechanism is not affectedin the free and cov ⇒ systems, so that the superelasticplateau remains intact. However, in the cov (cid:28) systemthe particles carrying misaligned magnetic moments areforced to rotate. The corresponding chains are stronglydistorted, which perturbs the neighboring chains carryingaligned magnetic moments as well. This weakens the re-quired magnetic attractions along the stretching axis thatare vital for a pronounced detachment mechanism andremoves the superelasticity from the stress-strain curveof the cov (cid:28) system. Moreover, in the free system theflipping mechanism can be deactivated as well, as thealigning external magnetic field hinders reorientations ofmagnetic moments. Consequently, the related featuresare removed from the stress-strain behavior, leaving onlya flat plateau caused by the detachment mechanism. Fi-nally, in the cov ⇒ system, the external field parallel tothe chains has another interesting effect. We can ob-serve a stiffening of the system when increasing the fieldstrength at high strains beyond the superelastic plateau.In this situation, all particles have been detached fromtheir chains, leaving them particularly susceptible to ro-tations due to shears caused by the elastic inhomogene-ity of the system. Since the external magnetic field in-troduces an energetic penalty for particle rotations, theintrinsic inhomogeneity-caused shear stresses cannot re-lax via particle rotations and the magnetic moments re-main parallel to each other. The parallel magnetic mo-ments repel each other in the direction perpendicular tothe stretching axis and, thus, work against a volume-conserving stretching deformation. In combination botheffects increase the stiffness of the system.When instead the magnetic field is applied perpendic-ular to the stretching axis, the detachment mechanismis weakened in all three systems due to an induced ro-tation of the magnetic moments towards a configurationwhich is repulsive along the stretching axis. In this way,the superelastic plateau can be gradually removed fromthe stress-strain curve by increasing the field strength.This works exceptionally well in the free system, wherethe magnetic moments are not anchored to the particleframes and the flipping mechanism is likewise weakened.In contrast to that, in the cov ⇒ and cov (cid:28) systems, evena strong external magnetic field cannot rotate the mag-netic moments completely. While in the cov ⇒ system,the magnetic moments feature a global magnetic align-ment oblique to the external magnetic field, the two op-posite initial magnetic alignment directions in the cov (cid:28) system lead to two separate polar alignment directions,each of them oblique to the external magnetic field.9Our effects rely on the sufficiently strong magnetic in-teractions in our systems when compared to the elasticinteractions. To achieve this experimentally, the rem-nant magnetization of the particle material should be ashigh as possible. For example, NdFeB, can easily ex-ceed 2 × A / m [81]. At the same time, the elasticmatrix into which the particles are embedded should besoft. Fabricating matrices with E (cid:46) Pa is possibleusing silicone [10, 82, 83] or polydimethylsiloxane [80].With these materials, our assumed value of m = 10 m can be achieved and is, therefore, experimentally realis-tic. Also the highest considered magnetic field strengthof B = 10 B corresponding to 100 mT is readily acces-sible. We stress that the behavior of our systems doesnot depend on the length scale of the problem. In anexperiment, this freedom can for instance be exploitedto adjust the particle size to the effect under investiga-tion. For example, the free system could be realized byrelatively small particles where the N´eel mechanism [53]is active and the magnetic moments can rotate relativelyto the particle frame. Increased particle size would benecessary to generate the cov ⇒ and cov (cid:28) systems.The free and cov ⇒ systems can be generated by ap-plying an external magnetic field during synthesis toform the embedded chains [40–43] from N´eel-type par-ticles [53] and from monodomain particles of larger size,respectively, possibly by covalently anchoring appropri-ately sized particles into the matrix [58–61]. For the smallN´eel-type particles, typically of sizes up to 10–15 nm,thermal fluctuations become important. These can sup-press the hysteretic behavior as well as the negative slopeassociated with the dip in our stress-strain curves. Over-all, these fluctuations will smoothen the bumps along theplateau, leading to a flatter appearance. Free systemsof larger particle size could be realized e.g. using so-called yolk-shell colloidal particles [55, 56] that consistof a magnetic core rotatable within a shell. To realizethe cov (cid:28) system, electro-magnetorheological fluids [84–86] could be used as a precursor of the anisotropic fer-rogel. In such a system, an external electrical field canbe applied to induce the chain formation of the electri-cally polarizable magnetic particles, while still allowingfor opposite alignments of the magnetic moments in sepa-rate chains. Subsequent cross-linking of the surroundingpolymer with covalent embedding of the particles shouldlock the chain structures together with their oppositelydirected magnetic alignments into the emerging matrix.The result would be an anisotropic ferrogel with the de-sired cov (cid:28) morphology.We have assumed permanent magnetic dipoles carriedby spherical particles in this work. The particles are arranged in characteristic chain-like structures. Possi-ble quantitative refinements comprise extensions beyondthe permanent point-dipole picture [87–89] or to elon-gated, non-spherical particles [60, 61, 90, 91]. However,the main mechanism leading to superelastic behavior inour systems is the detachment mechanism for which onlystrong attraction at short distances between the neigh-boring particles along the stretching axis is necessary.This kind of attraction can likewise be realized for softmagnetic particles magnetized by an external field. Thesame mechanism could also be realized for nonmagneticattractive interaction forces, e.g., for particles sufficientlypolarizable by an external electrical field. Moreover, alsothe flipping mechanism could be initiated for soft mag-netic particles, when the direction of a magnetizing ex-ternal magnetic field is switched at the correspondingimposed strain. Furthermore, to observe the basic phe-nomenology, the chain-like aggregates do not necessar-ily need to span the whole system. In the most basicopposite situation, embedded pair aggregates would besufficient [87]. Also the chains do not need to be as per-fectly straight as considered here but could for examplebe weakly wiggled [8]. On the theoretical side, a connec-tion to continuum descriptions on the macroscopic scaleshall be established in the future [4, 36, 92].Exploiting the described reversibly tunable nonlinearstress-strain behavior of our systems should enables amanifold of applications. When a pre-stress is appliedto the material, such that it is pre-strained to the su-perelastic regime, it becomes extremely deformable[93].This is an interesting property for easily applicable gas-kets, packagings, or valves [24]. Moreover, in such astate, the ferrogel can be operated as a soft actuator[22, 23, 25–27], as external magnetic fields can triggersignificant deformations. Passive dampers based on su-perelastic shape-memory alloys are already established[94, 95] and utilize hysteretic losses under recoverablecyclic loading to dissipate the energy. Our results for the free system might stimulate the construction of analogoussoft passive dampers with the additional benefit of beingreversibly tunable from outside. Finally, the typically el-evated biocompatibility of polymeric materials allows formedical applications exploiting the above features, e.g.,in the form of quickly fittable wound dressings, artificialmuscles [96, 97], or tunable implants [50, 51].
ACKNOWLEDGEMENTS