Multifunctionality by dispersion of magnetic nanoparticles in anisotropic matrices

1 Aims and concepts

Magnetic fluids are composed of nanometre- to micrometre-sized solid particles in a liquid carrier. A combination of the magnetic properties of the dispersed particles with the fluidity of the matrix enables the development of broadly applicable magnetoresponsive materials. In these fluids, interactions with external magnetic fields occur with the individual magnetic particles (MPs). Thus, the interaction energies are not large with respect to thermal energies, in contrast to magnetic solids of the same material as the MPs. In the absence of external magnetic fields, ferrofluids lack a remanent magnetization, they behave like superparamagnets. Confining the magnetic liquid phase in a viscoelastic environment by gelation brings about mechanical and diffusive constraints resulting in materials with distinct sensitivity to mechanical stress and/or magnetic field.

Liquid-crystalline phases are characterized by spontaneous collective orientational order of the molecules in certain temperature or concentration ranges. However, the common liquid crystal (LC) mesogens are diamagnetic, and their interactions with magnetic fields in the mesophases are governed by the anisotropy of the diamagnetic susceptibility of the order of 10⁻⁶. The collective arrangement of the mesogens allows us to switch their preferential direction, the director n → , even in weak magnetic fields of a few hundreds mT. This allows to employ magnetic fields for magneto-optical or magnetomechanical switching even without MNPs.

It is a promising but also challenging concept to combine these two areas of soft-matter physics, magnetic fluids and LCs, to create new classes of multifunctional materials that combine a strong magnetic response with collective magneto-optical and magnetomechanical effects. A promising strategy is the dispersion of functionalized MNPs in anisotropic environment such as suspensions of shape-anisotropic (anisometric) microparticles, lyotropic nematic matrices or even thermotropic LCs. We describe how interactions of MNPs with complex anisotropic matrices drive self-assembly and affects the physical properties of the materials. Primary focus is the influence of the matrix structure on the magneto-optical and viscoelastic properties of the composites. An external magnetic field triggers self-organization of the MNPs, which affects the anisotropic environment (matrix). The feedback of the matrix determines the structure formation dynamics of the magnetic subphase, as well as optical and mechanical properties. We examine three different matrix systems: non-MPs (pigment crystallites) dispersed in an isotropic liquid, organic LCs, and organogels with filamentary internal structure. A special aspect is the transition from isotropic to anisotropic matrices.

The following section discusses mixtures of magnetic nanoparticles with shape-anisotropic (rod-like and plate-like) nanocrystallites in suspension. These suspensions may form lyotropic phases at larger particle concentrations, but even at low concentrations the steric interactions with codispersed MNPs enhance the macroscopic response to magnetic fields considerably. The idea is to exploit these interactions to command the larger nonmagnetic constituents via magnetic torques of external fields on MNPs and agglomerations of the latter. In Section 3, we briefly describe attempts to suspend surface-functionalized MNPs in thermotropic nematic LCs. Here, the mesogens are thermodynamically stable LC mesophases. The surface-functionalized MNPs interact with the director field to transfer magnetic alignment to the nematic host. In Section 4, the complexity is increased by physical cross-linking of the host phase. This leads to self-organized arrangements of the MNPs in the gel matrix and to the dependence of mechanical and optical sample parameters on the magnetic history of the material.

In the final section, we introduce a mechanical characterization technique that can be employed to all of these systems. It can be used to characterize magnetomechanical properties of the fluids, in particular the transfer of torques from the magnetic field to the carrier fluid and sample containers.

2 Colloidal suspensions of anisometric particles

Colloidal suspensions of anisometric nanoparticles particles are scientifically attractive because of their ability to form microstructured and nanostructured phases [1], [2], [3], [4], [5], [6]. Phases with broken rotational (nematics) and translational (smectics, columnar) symmetries have been discovered in various colloidal materials including clay [7], [8], [9], goethite [10], [11], TiO₂ rods [12], viruses [13], [14], [15], [16], or even microparticles. Due to a coupling between the orientational degrees of freedom and external fields, these dispersions show a complex rheological response and a distinctive behaviour in electric fields [4], [17].

The crucial role of entropy-driven self-assembly in the colloidal systems is particularly pronounced in binary mixtures of rod-shaped or platelet-shaped nonmagnetic nanoparticles (NPs) with MNPs [18]. These mixtures exhibit a sterically-induced orientation transfer (Onsager-Lekkerkerker effect) [18], [19]. A field-induced alignment of the MNP subsystem is transferred to the nonmagnetic components. This does not only work with shape-anisotropic MNPs. A uniform external magnetic field can cause the formation of small, stable clusters of spherical MNPs which, in turn, may command the orientation of the nonmagnetic constituents.

One example of the sterical orientation transfer occurs in dispersions of rod-shaped pigment nanocrystallites doped with MNPs. Pigment particles investigated in our study (C.I. Pigment Red) have an average length of 230 ± 70 nm and a diameter of 46 ± 20 nm [20]. The particles form a stable dispersion in dodecane with the commercially available dispersant Solsperse 11200 (Lubrizol, Brussels, Belgium). The magnetic dopant is a commercially available ferrofluid (APG 935, Ferrotec), containing magnetite NPs with an average diameter of about 10 nm suspended in hydrocarbons. In our study, we investigated dispersions with various concentrations of MNPs, as well as pigment nanorods (NRs).

2.1 Magneto-optical behaviour

The orientational order of the pigment NRs determines the birefringence of the dispersions, which was measured using the optical modulation technique [21]. Diluted dispersions of the ferrofluid exhibit only weak birefringence saturating above 200 mT. Dispersions of pigment NRs show even weaker magneto-optical response compared to the magnetic fluid. The alignment of the pigment NRs was demonstrated by measurements of the linear dichroism in strong magnetic fields [22]. The particles align perpendicular to the field direction in fields of about 5 T.

By contrast, mixtures of the pigment NRs and MNPs exhibit much higher birefringence in comparison to that of the pigment-only suspensions, where the birefringence is nearly zero, and also in comparison to that of pure MPs. Figure 1 shows the birefringence of the NR/MNP mixtures with pigment particle concentration c _r = 5  vol% and varying concentration of MNPs (c _s). The curves exhibit a saturating at about 200–300 mT, which is comparable to that of the ferrofluid. Nevertheless, the birefringence is strongly enhanced by the presence of the NRs. The maximum birefringence Δn _max attained at 640 mT is shown in Figure 2.

Figure 1:

Birefringence of mixtures with pigment particle concentration c _r = 5 vol% and varying MPs concentration (c _s). (Adopted from a study by May et al. [21]).

Figure 2:

Saturation birefringence: (a) at a constant magnetic particle concentration of c _s = 0.3  vol% and different pigment particle concentrations c _r, and (b) at constant pigment particle concentration c _r = 5  vol% and different MNP concentrations c _s. (Adopted from a study by May et al. [21]).

At a constant c _s and low NR concentrations, the birefringence increases linearly, while for high c _r, a different behaviour is observed (Figure 2). This dramatic increment of birefringence can be explained by an onset of an orientationally ordered phase of the pigment suspensions at high concentrations. At the same time, the maximum birefringence at a constant c _r shows a linear dependence on c _s (see Figure 2) in the investigated concentration range.

2.3 Molecular dynamics simulation

The molecular dynamics (MD) simulations were performed by Stavros D. Peroukidis and Sabine H. L. Klapp at the TU Berlin [21]. The model fluid consisted of a binary mixture of N _r uniaxial rods and N _s magnetic spheres. The rods were represented as prolate, nonmagnetic ellipsoids with a length l and the width σ ₀ (l/σ ₀=3) that interact via the Gay-Berne (GB) potential, using a standard parametrization [21]. The magnetic spheres are modelled via dipolar soft spheres (DSSs), with an embedded central dipole moment μ . They interact via a soft repulsive potential and a dipole-dipole interaction [25], [26]. The DSS particles have a diameter σ _s which was set to 1/4 the width of the rods, i.e. σ _s ^* = σ _s/σ ₀ = 0.25, which is consistent with the estimated relative sizes of the experimental species.

In the simulation, GB/DSS mixtures were studied in a uniform magnetic field H at various field strengths H ^* = μH/k _B T, where μ is the dipole moment of a particle. The reduced density is defined as ρ ^* = Nσ ₀ ³/V and V is the volume of the system with (N = N _s + N _r) , fraction of particles x _a = N _a/N (where a = r,s for rods and spheres, respectively). Five states were considered: [(ρ ^*,x _s) = (0.350,0.20)] , [(ρ ^*,x _s) = (0.560,0.50)] , [(ρ ^*,x _s) = (0.778,0.64)] , [(ρ ^*,x _s) = (1.00,0.72)] , and [(ρ ^*,x _s) = (1.40,0.80)] ,, that correspond to the isotropic phase of a field-free GB-DSS mixture. For these states the concentration of rods was kept the same (c _r = 44 %) whereas the concentration of DSS, c _s, was varied (it increases by increasing the concentration x _s).

Figure 3 shows the calculated order parameter S ^(r) and the magnetization ⟨M⟩ as functions of the magnetic field. Both magnetization and order parameter exhibit saturating behaviour. The saturation order parameter S _saturation has a linear dependence on the concentration of the magnetic spheres as shown in Figure 4. Since the birefringence is proportional to S ^(r), the result in Figure 4 demonstrates a qualitative agreement between the theory and the experiment. The simulations also show that the optical response of the binary mixtures is expected to decrease if the magnetic interactions between the spheres are reduced. Therefore, not only the concentrations of the MPs but also the interparticle interactions determine the response of the binary systems to an external magnetic field.

Figure 3:

(a) Nematic order parameter of the rod species S ^(r) and (b) magnetization ⟨M⟩ of the MPs as a function of the external magnetic field strength for a GB-DSS mixture with reduced magnetic dipole moment μ ^* = 2.4 of the MNPs. The results are taken for constant concentration of rods c _r = 44 % by varying the concentration of magnetic spheres c _s. (Adopted from a study by May et al. [21]).

Figure 4:

(a) Saturation value of the nematic order parameter of rods S ^(r) as a function of c _s for constant c _r = 44 %, and various values of dipolar moment μ ^*. (b) Mean size (number of particles) of all clusters ⟨n⟩ -open squares- (irrespective of their type) and mean size of chainlike clusters ⟨n _c ⟩ -open triangles- at the saturation value of S ^(r). The corresponding values of ⟨n⟩ and ⟨n _c ⟩ when the field is off are shown by solid squares and triangles, respectively. (Adopted from a study by May et al. [21]).

3 Dispersions of magnetic NPs in a nematic LC

Dispersions of MNPs in a nematic LC are called ferronematics (FN). We investigated the coupling between the nematic director and the magnetic order of CoFe₂O₄-based functionalized MNPs, which were synthesized in the group of Silke Behrens (KIT) [27]. The particle size was around 2.5 nm. In the MNP-doped system, a significant decrease of the magnetic Fréedericksz threshold was found. This suggested the existence of the coupling between the nematic director and the magnetization of small clusters of the MNPs in a particular manner. The MNP clusters represent inclusions in the director field that distort the surrounding director field. The behaviour of the LC dispersions was described using the Raikher-Burylov (RB) model for some of these systems and estimated the magnetization-director coupling strength [27].

At a combined application of electric and magnetic fields to a nematic LC in a planar sandwich cell, one can increase the electric Fréedericksz threshold field by choosing the direction of the magnetic field perpendicular to the electric field, in the cell plane. In absence of MNP doping, the diamagnetic torque on the director stabilizes the ground state. The addition of MNPs reverses this effect, the MNP aggregates align with the magnetic field and destabilize the director ground state. The details of these experiments are found in the contribution by S. Behrens et al. in this volume.

4 Mobile magnetic NPs in fibrillous gels

Ferrogels are composite materials consisting of MPs embedded in a viscoelastic matrix [28]. Their magnetoelastic properties enable applications in the fields of transducers, sensors and actuators [29]. Depending on the viscoelastic properties of the matrix, the magnetic properties of the particles and the strength of the coupling between them, their response to external stimuli like, e.g. magnetic fields or mechanical stress can be tuned in a wide range [30], [31]. Especially strong magnetomechanical response is expected when the MPs are micronsized and connected to the viscoelastic matrix. In this case the viscoelastic matrix directly experiences the torque of the Brownian rotation exhibited by the MPs when they align to the applied magnetic field [28]. On the other hand, a strong magneto-optical response can be expected from ferrogels where the MPs are not connected to the matrix and their alignment and chain formation along the magnetic field is only determined by the void accessibility in the confined volume of cavities in the mesh [32], [33], [34], [35], [36].

The ferrogels prepared in our laboratory belong to the latter category [37]. They consist of commercially available APG2135 ferrofluid diluted with n-dodecane and gelled by 12-hydroxyoctadecanoic acid (12-HOA) gelator. APG2135 consists of single-domain superparamagnetic magnetite particles surrounded by a surfactant layer to avoid aggregation and suspended in a synthetic hydrocarbon carrier. The average core diameter of the particles is 10 nm. The saturation magnetization is 17885 A m⁻¹ and the volume fraction of the particles is 3.9 vol%. The particle size distribution can be described by a lognormal distribution with an average hydrodynamic diameter of D ₀ = 14.5 nm and a standard deviation σ = 1.2 [38], [39]. Diluting APG2135 with the nonpolar solvent n-dodecane results in stable suspensions. The initial birefringence of those ferrofluids is insignificant and indicates the presence of some aggregates in the ferrofluid. The origin of birefringence in magnetic fluids can be explained in the following way. In a magnetic field, the MNPs are equivalents of magnetic dipoles aligning along the field in head-to-tail fashion. In the electric field of linearly polarized light they become oscillating dipoles. Depending on the orientation of the chains with respect to the direction of the polarization of the light, the oscillating dipole interaction between the particles is asymmetric, resulting in a strong optical anisotropy. Thus, the birefringence of the ferrofluid increases with increasing magnetic field [40], [41], [42], [43], [44], [45], [46], [47], [48]. The maximum birefringence at a given field strength is only dependent on the concentration of the MPs, as can be seen in Figure 7a.

Gelling the ferrofluids brings about several constraints into the system. Since the gelator network divides the volume into cavities with certain amounts of MNPs, there is a limited amount of available particles to form chains. Additionally, due to adsorption along the cavity walls MNPs become immobilized depleting the MNP supply. As a result there will be a smaller number of MPs available in the vicinity of a given particle to form chains than in a ferrofluid.

The mobilities of single particles will be heterogenous and spatially dependent because the alignment of chains along the magnetic field is restricted by the restrained void accessibility. The translational degree of freedom of the particles is constrained by the cavity walls leading to imperfect alignment of the dipoles in the confined space. The torque exhibited by the translations and rotations of the MNPs and their aggregates leads to a deformation of the gel network. These irreversible (inelastic) deformations happen when the ferrogel is exposed to a magnetic field for the first time [49] (Figure 5).

Figure 5:

Magnetically induced birefringence Δn(B) in an isotropic (black squares – 1st run, black triangles – 2nd run) and an anisotropic (red triangles) gel with 10 wt% MNP and 12.5 wt% gelator concentration at room temperature. Filled symbols correspond to increasing field and empty symbols to decreasing field. (Adopted from a study by Nádasi et al. [37]).

The gelator network is formed by 12-hydroxyoctadecanoic acid (12-HOA) molecules. This chiral low-molecular-weight gelator [50], [51], [52], [53] crystallises in organic solvents and due to the anisotropy of strong intermolecular interactions between the individual gelator molecules gives rise to a helical self-assembly in one dimension. The anisotropy of the interfacial free energy of the resulting strands leads to bundling and the evolution of fibrillar structures. These aggregates intertwine and develop into a three-dimensional gelator network. Since this self-assembled fibrillar network (SAFiN) is based on intermolecular interactions, i.e. H-bond, London dispersion forces, their formation is thermoreversible. Depending on the gelator concentration the gel-sol transition occurs around 65–80 °C. The SAFiN-mesh structure depends on the cooling rate during the preparation process. The slower the cooling rates the longer the annealing time of the fibre growth resulting in a small number of permanent nodes originating from crystalline mismatch branching. If the high temperature sol is quenched to room temperature, the branching grade intensifies, a SAFiN containing high numbers of nodes evolves [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65].

The gel structure was characterized by atomic force microscopy and scanning electron microscopy (SEM). A mosaic of spherulitic domains with a radial distribution of fibres can be seen in the low-magnification SEM images (Figure 6a), whose appearance is consistent with the textures observed by polarizing optical microscopy. High-magnification images (Figure 6b) reveal an interwoven structure of thin 32 ± 6 nm fibres in a gel with 10 wt% gelator. At this length-scale, the fibres appear disordered, forming a mesh with an average size of 60 ± 25 nm. This mesh size is significantly larger than the particle diameter (10 nm). Therefore, one mesh cell may accommodate short chains with 4–8 MNPs. The solvent-gelator compatibility of APG2135 with 12-HOA is comparable to that of a nonpolar solvent, since the ferrofluid can be considered as a nonpolar liquid with superparamagnetic properties. However, exactly the response of the sol in a magnetic field enables additional fine-tuning of the properties of the ferrogel.

Figure 6:

Scanning electron microscopy images of an isotropic gel with 7 wt% of the MNPs and 10 wt% of the gelator at (a) low magnification and (b) high magnification. (U = 1.5) kV. (c) Atomic force microscopy images of the gels with 7.5 wt% and 12.5 wt% of the gelator (4.5 wt% of MNPs). The inset shows a magnified image of a helical nanofilament. (Adopted from a study by Nádasi et al. [37]).

Figure 7:

(a) Birefringence Δn(B) of the ferrofluid APG2135 diluted with n-dodecane. (b) Saturation birefringence Δn _max as a function of the concentration of the MNPs c _MP in the ferrofluids. (c) Birefringence Δn(B) of a dispersion of 13.5 wt% MNPs and 10 wt% 12-HOA at T = 80 ^°C (sol). No significant hysteresis behaviour can be found for a dwell time of 30 s. (d) Magnetically-induced birefringence Δn(B) in an isotropic (black squares) and an anisotropic (red triangles) gel with 4.5 wt% MNP and 7.5 wt% gelator concentration at room temperature. Filled symbols correspond to increasing field and empty symbols to decreasing field. (Adopted from a study by Nádasi et al. [37]).

To prepare the isotropic ferrogels, the sample was quenched to room temperature (cooling rate > 50 K min⁻¹). In the case of anisotropic ferrogels, the heated sample was cooled down at a controlled cooling rate (1 K min⁻¹) in a magnetic field of 650 mT. The samples were prepared in 100-μm thick rectangular capillaries. Bars of ferrogels easily bend upon bringing them in a magnetic field gradient. This demonstrates that despite the MNPs are dispersed in the liquid subphase, there is coupling between magnetic and elastic degrees of freedom.

This coupling is confirmed by our magneto-optical measurements. In Figure 5, the magnetic field dependence of the birefringence of an isotropic gel measured by optical modulation technique is shown. The two distinctively characteristic features of those curves are the hysteresis originating from the difference between the birefringence measured in increasing and decreasing magnetic field and the lowering of the birefringence values in the second measurement.

In the first measurement, the isotropic gel is exposed to a magnetic field for the first time. As described above, the viscoelastic matrix hinders the perfect alignment of the embedded MPs along the magnetic field. The competition between the aligning magnetic dipoles and the viscoelastic deformation of the matrix leads to some irreversible deformations in favour of better alignment of the magnetic dipoles. In the meantime, this interaction facilitates the adsorption of the particles along the cavity walls, resulting in a fewer number of free MPs contributing to the chain formation in the second run. Therefore, the birefringence of the isotropic gel in the second run is lower. The hysteresis is the consequence of the viscoelastic deformations of the gel network. The relaxation of the network to the initial state is a long-term process. In fact, after a full measurement cycle (0 mT → 650 mT → 0 mT), it takes at least 10 h for the gel to relax in absence of the magnetic field to a constant birefringence close to the initial one. The hysteresis of the first run is always bigger than that of the second run corroborating the idea of the matrix inelastic deformations.

Anisotropic gels prepared by a slow cooling from the sol state exhibit a striped birefringent texture in polarizing microscopy, where the stripes are aligned along the field direction applied during the preparation.

The high birefringence of the anisotropic gels in Figure 7d can be explained by the formation of MNP chains in the initial high temperature sol state. As in the ferrofluid, the higher the applied field the bigger is the birefringence of the sol, and no hysteresis can be observed (Figure 7c). Since the magnetic field is applied throughout the whole preparation process, the chains do not fall apart and they give rise to a direction dependent evolution of the gel network. Additionally, the gradual slow cooling increases the annealing time of the gel fibre growth resulting in less branching. Altogether, the mesh network is expected to be looser, especially in the direction of the applied preparation field. If no magnetic field is present, the MNP chains fall apart, yet the direction dependent growth of the fibres produces an optical anisotropy, i.e. birefringence. That is why the initial birefringence of the anisotropic gels is always higher than of the isotropic ones (Figure 7d). Since the MNP chain formation in the gel is facilitated in the external field direction, i.e. the optical axis is aligned to the magnetic field, the following phenomena are observed:

• – the Δn(B) curves coincide in repeated measurements,

• – the birefringence is higher in the anisotropic gel with the same c _G and c _MP for any applied field,

• – the hysteresis of the anisotropic gels is always smaller than for the isotropic gels with the same c _G and c _MP.

The birefringence decreases, changing its sign if the measuring field is perpendicular to the anisotropy axis. Varying the concentration of the MNPs (c _MP) and the gelator (c _G ) enables fine-tuning of the optical properties of the organoferrogel (Figures 8 and 9a, b).

Figure 8:

Magnetic particle concentration dependence of the saturation birefringence Δn _max(c _MP) for isotropic and anisotropic gels with 10 wt% gelator. The straight lines are linear fits under the constraint Δn _max(0) = 0. (Adopted from a study by Nádasi et al. [37]).

Figure 9:

Dependence of the saturation birefringence Δn _max of the ferrogel on the concentration of the MPs c _MP in the isotropic (a) and the anisotropic (b) gels. Dependence of the saturation birefringence Δn _max of the ferrogel on the concentration of the gelator c _G in the isotropic (c) and the anisotropic (d) gels. (Adopted from a study by Nádasi et al. [37]).

An increasing gelator concentration provides negative feedback on the magneto-optical response: the saturating birefringence decreases with increasing gelator concentration (Figure 9c, d). However, the dependence Δn(c _G) remains linear in the investigated concentration range (Figure 11b, d).

There is a clear difference between the isotropic and the anisotropic gels if one considers the slopes s(c _G) = dΔn(c _MP)/dc _MP. The slope s for low concentration of the gelator is nearly independent on the concentration. This can be understood by assuming that at low gelator concentrations, the mesh is loose in the anisotropy direction. The gel mesh has less confinement effect on the growth of the magnetic aggregates. For higher gelator concentration (12.5 wt%), the gel network restricts the growth and the slope s(c _G) becomes reduced. The hysteresis h(c _MP) increases with increasing MNP concentration c _MP in both isotropic and anisotropic gels. However, the isotropic gels have a systematically lower birefringence and larger magneto-optical hysteresis (Figure 10).

Figure 10:

Dependence of the hysteresis parameter h on the concentration of the magnetic particles c _MP. (Adopted from a study by Nádasi et al. [37]).

Figure 11:

(a) and (c) show the slopes s of linear fits of Δn _max(c _MP) as functions of the gelator concentration in the isotropic and anisotropic gels, respectively. Figures (b) and (d) show the slopes s of the linear fits of Δn _max(c _G) as functions of the MNP concentration of the isotropic and anisotropic gels, respectively (Adopted from a study by Nádasi et al. [37])

Figure 7d shows the field dependence of the birefringence Δn(B) for a gel with a gelator concentration of 10 wt%. The field dependence of the birefringence Δn ₊(B) and Δn ₋(B) on increasing and decreasing magnetic field B, respectively, differ considerably. Only Δn ₋(B) can be satisfactory fitted with the Eq. (8) in gels with the gelator concentration below 10 wt%. To estimate the hysteresis strength, we introduce a dimensionless parameter h:

where B _max→∞. In our experiment, B _max was limited to 650 mT.

It is important to mention that these gels show no magnetic hysteresis. The magnetic relaxation in our system is governed by the Néel relaxation mechanism with corresponding relaxation rates in the range of 100 MHz. Thus, the magnetization rapidly adjusts to the applied magnetic field on a very short (compared to our experiment) time scale. The birefringence response Δn has two important contributions: the birefringence Δn _MP attributed to the aligned magnetic chains and the birefringence Δn _G caused by a deformation of the gel matrix. In a dilute limit, the relaxation times for Δn _MP can be estimated as the time an MNP requires to diffuse over a distance of particle radius R, τ = 6πηR ³/kT, where η is the viscosity of the matrix, T is the temperature. For dodecane, τ≈0.2 µs. In the gel state, the situation is much more complex. The viscosity increases by several orders of magnitude and the environment of the particles becomes strongly heterogeneous. Particles interact with the gel network. Already at c _G = 3 wt%, τ reaches the order of magnitude of a second. A slow increase of the birefringence in a magnetic field can also be attributed to slow deformations of the gel network and the adhesion of MNPs to the gel fibres. This also results in a residual birefringence which relaxes on the time scale of days. The dynamical properties of the mobile NPs in the gel matrix will be discussed in an upcoming paper.

The behaviour of the saturating birefringence Δn _max and the hysteresis parameter h show the effect of the local anisotropy of the gel, which is a surprising finding because the sizes of the particles are smaller than the mesh size of the gel. Larger Δn _max and smaller h suggests that MNPs have more space to rearrange and form larger aggregates along the anisotropy axis. Birefringence in MNP agglomerates results from the mutual polarizability of adjacent MNPs [42], [66]. The association of MNPs into chains was described in the early seventies by P.G. de Gennes and P.C. Jordan using hard-sphere models [40], [41], [67]. The formation of MNP dimers reduces the entropy of the system and shifts the equilibrium from single particles to dimers [67], [68].

The effective dielectric tensor ε ˆ eff of the composite fluid can be decomposed into the isotropic part ε ˆ host of the host fluid and the anisotropic part of the inclusions ε ˆ inc : ε ˆ eff = ε ˆ host + ε ˆ inc . The dielectric polarizability α of the anisometric MNP aggregates is anisotropic. In case of magnetic chains, the length of the chain and the intrinsic optical properties of MNPs determine the polarizability anisotropy Δα. In the thermal equilibrium, the alignment of the chains is subject to thermal fluctuations. The components of the dielectric permittivity of the chains are given by the average

where the summation runs over all chains of the length k with the volume fraction φ(k), respectively. The averaging is done over all possible orientations θ, ϕ.

The field-dependence of the birefringence induced by the dimerization in a magnetic fluid was studied by several authors [42], [44], [66], [69], [70], [71], [72]. Assuming a Boltzmann orientational distribution of the particle pairs, the birefringence attributed to the dimers can be found from Janssen’s dependence [66], [72]:

where x = μ _aggr B/kTb, and μ _aggr is the magnetic moment of the chain aggregate, T is the temperature and k _b is the Boltzmann constant. The factor A is determined by the polarizabilities of the particles.

Equation (8) describes well the experimental observations in pure ferrofluids and in the sol state (Figure 12). From the fit of Δn(B), we deduce the magnetic moment of the chain-aggregate μ _aggr = 4.28⋅10⁻¹⁹Am². This corresponds roughly to two MNPs. The volume fraction of magnetic dimers is determined by the magnetic field and the temperature. In the gel state, however, this description fails. Satisfactory fits can be found mostly for Δn with decreasing field B for low concentrations of MNP and the gelator. The reason for that is the nonequilibrium behaviour of Δn(B). The deformation of the gel network occurs on much slower time scales than the rearrangement of the MNPs.

Figure 12:

Magnetically induced birefringence Δn(B) as a function of the field strength B in (a) ferrofluid with c _MP = 10.4 wt% and (b) ferrogel with c _MP = 4.5 wt% of MNPs and c _G = 12.5 wt%. The solid red lines are fits using Eq. (8).

5 Magnetic suspensions in rotating fields

5.2 Torsion pendulum setup

When a magnetic fluid is exposed to a magnetic field that exerts a permanent torque on the MNPs, this torque is transferred to the carrier fluid to create a vortex flow, and further to the container by shear forces of the flowing liquid at the container walls. If the container is freely suspended, the rotation is continuously accelerated [82], the measurement of the torque from the angular velocity profile of the container is difficult. Therefore, a torsion pendulum is a reasonable alternative, it measures the distortion of the suspending wire in equilibrium of the magnetic field–induced torque with the restoring elastic torque of the wire. Figure 13 sketches the setup [83], [84].

Figure 13:

Torque balance setup for the measurement of the rotational effect in ferrofluids. The magnetic field is generated with two pairs of coaxial coils [83], [84]. The torque is measured from the twist of the suspending thread. The sample container is partly immersed in an oil bath to dampen the pendulum oscillations in order to reach the equilibrium deflection faster, and to attenuate influences of external noise. Figure adapted from a study by Storozhenko [84].

The magnetic field is generated by two sinusoidal currents, with 90 ^° phase shift, in two pairs of coaxial coils, and monitored with Hall probes. The samples are enclosed by spherical glass containers of approximately 1 mL volume. The sensitivity of the setup allows us to measure torque densities of the order of 10⁻⁴ N/m² (torques of 10⁻¹⁰ Nm on the samples), angular frequencies were typically of the order of 50 s⁻¹ to 5000 s⁻¹, in some experiments up to 30,000 s⁻¹, field strengths up to a few mT were applied. The sample was suspended by a thin glass fibre, with very low restoring torque, so that the eigenfrequency of the pendulum was below 100 mHz.

5.3 Diluted ferrofluids and viscosity effects

The transfer of the magnetic torque onto the suspending fluid occurs in two steps: in the first step, the external magnetic field interacts with the magnetization of the MNPs. In a second step, this torque is transferred to the carrier fluid by viscous forces. The torque excerted by a field B → on the magnetic moment m → of an individual NP is T → = m → × B → . This torque is proportional to sinα, with α being the angle between m → and B → . In the at moderate rotation rates ω = φ ˙ of B → = B ( cos φ , sin φ , 0 ) , the magnetization follows the field after a short transient with the same angular velocity but a constant phase lag α. Two different mechanisms compete in the reorientation of m → . The particles themselves may rotate, with m → being fixed relative to the MNP. This process is usually referred to as Brownian relaxation. It depends upon the viscosity of the carrier fluid that dampens the rotation of particles relative to the matrix. The higher this viscosity, the larger is the phase lag α and the larger is the magnetic torque at given B , | m → | , and ω. The second process is the reorientation of the magnetization m → relative to the particle orientation, referred to as Néel relaxation. When the particles are small, this is the prevailing mechanism. Here, α does not depend upon the rotation state of the MNP, and is thus independent of viscosity, even though the transfer of the magnetic torque from the MNPs to the carrier fluid still proceeds by viscous shear forces.

In order to explore the influence of viscosity of the magnetic fluid on the torque transfer, a commercial ferrofluid (APG2135, Ferrotec Corp. (Japan)) with a viscosity of 1.5 Pas was diluted with different concentrations of durasyn (viscosity 46 mPas). If the internal structure of the magnetic fluid were independent of the applied magnetic field, the torque would be proportional to the MP concentrations, and it would continuously increase with increasing rotation rate of the magnetic field. This is the consequence of an increasing phase lag between the sample magnetization and B → with increasing ω. This condition is in fact fulfilled only for well-diluted samples. Experimental results are seen in Figure 14.

Figure 14:

Torque density in samples with different concentrations of the original ferrofluid. The right graph is an expansion of the low-frequency region of the graph at the left. Figure adapted from a study by Storozhenko et al. [84].

The decrease of the torque density is smaller than expected from the decrease of MNP concentration in the range of up to 50 % dilution. On the other hand, the change of viscosity by roughly one order of magnitude apparently has only little effect on the graphs. A certain saturation of the torque density is found at comparable rotation rates. If the torque was primarily generated from Brownian relaxation, one would expect that it would substantially decrease in the diluted samples.

A characteristic difference between the high-concentration and diluted systems is a slight elevation of the torque characteristics at rotation frequencies below 10 Hz. This was attributed to the formation of small particle aggregates at higher fields, which would increase the magnetic response. At larger rotation rates, these agglomerates decompose since they cannot follow the field direction. Then, the torque drops in the range between 10 and 50 Hz for the highly concentrated suspensions.

Figure 15:

Torques on ternary mixtures of ferrofluid, dodecane and Permanent Rubine pigment particles. The ferrofluid concentration is 40%, the concentrations of pigment particles are given in the legend, the rest is dodecane. Printed with kind permission of A. Storoshenko [84].

The torque transfer is also influenced by the rheological properties of the carrier fluid. This was demonstrated by addition of nonmagnetic platelet-shaped pigment particles [86]. When the isotropic solvent (n-dodecane) is replaced by anisometric crystallites, the character of the torque characteristics changes substantially owing to the non-Newtonian character of the concentrated pigment particle suspensions Figure 15. The saturation of the torque shifts to lower frequencies. This reflects the interactions of the flow generated by the MNPs and the anisometric platelets.

5.4 Néel and Brownian relaxation in rotating fields

So far, we have disregarded the nature of the interactions between the MNP magnetizations and the external field. A closer analysis shows that in fact, the Néel relaxation time τ is dominant for most of the magnetic NPs contained in commercial ferrofluids. The quantitative analysis shows that the crossover from predominant Néel relaxation of small MNPs to Brownian relaxation of large MNPs in polydisperse samples occurs near 10 nm particle diameter. Smaller MNPs represent by far the majority of particles in typical ferrofluids, so that most of the particles are characterized by an internal relaxation of the magnetization. On the other hand, the magnetization of individual particles grows with the third power of their diameter, so that relatively few large particles still contribute noticeably to the magnetic torque. These processes were analysed recently in a combined theoretical and experimental study [85]. Magnetic torques were measured with the torque balance technique over an extended frequency range up to 35 kHz.

The original ferrofluid synthesized in the Scientific Research Laboratory of Applied Ferrohydrodynamics (ISPU, Ivanovo) by chemical condensation consists of single-domain spherical nanoparticles of magnetite Fe₃O₄ stabilized by oleic acid and dispersed in kerosene. It was stepwise diluted with dodecane (viscosity: 1.34 mPas) so that the concentration of MNPs ranged from 10.2 to 2.2%, while the viscosity dropped with increasing dilution from about 30 mPas down to 2 mPas. Figure 16 shows the experimental findings. At higher frequencies (beyond the torque maximum) all graphs can be scaled to a master curve. At low frequencies, there are some systematic deviations that most probably stem from aggregation of single MNPs in the higher concentration range.

Figure 16:

Left: Frequency dependence of the magnetic torque density in dispersions with various volume fractions φ _MNP of magnetic particles. Right: Scaled torque dependencies N v / N max ( f ˜ ) for dispersions with different volume fractions of magnetic particles. Note the extended frequency range as compared to Figure 14. Reprinted from a study by Usadel et al. [85].

Measurements of the magnetic torque of diluted dispersions of MNPs in rotating magnetic fields and the comparison to numerical simulations have demonstrated the crucial importance of the distribution of particle sizes. From a theoretical point of view, it was shown that the rigid dipole model (RDM), which fixes the magnetization in the particle, cannot explain the experimental observations. It predicts that the torque and other dynamical features depend only on the product of the viscosity and rotation rate of the external field. In order to understand the experimental results, it is necessary to take both Brownian and Néel relaxation into account [85]. The theoretical results show that in a rotating magnetic field a transition takes place from a state with the magnetic moment locked to the anisotropy axis of the MNPs to a state with precessional spin dynamics. This transition depends on the anisotropy energy. The dependence of the anisotropy energy on the magnetic volume of the MNPs causes an essential dependence on particle sizes. This must be taken into account in all experiments performed with diluted ferrofluids of a broad distribution of particle sizes.

For MNPs at room temperature with magnetic parameters typical for iron oxides (such as magnetite), the magnetic moment can be considered as being locked if the magnetic radius is larger than about 10⁻⁸ m [85]. Such particles can be described analytically within the RDM. For smaller particles, both Brownian and Néel dynamics are relevant, and a numerical analysis is required. The faster Néel relaxation decreases the relaxation time, and consequently the phase lag of the magnetization and the transferred torque considerably.

These conclusions apply to most of the commercial ferrofluids widely used in applications, which are commonly characterized by broad distributions of particle sizes. The theoretical analysis shows that much larger torques could be achieved with particles with larger anisotropy constant. Such particles would be similarly stable, e. g. with respect to aggregation, but the critical radius above which Brownian relaxation dominates would become much smaller, so that more particles contribute to a large torque. An experimental confirmation of this theoretical prediction is a desirable goal of future research.

6 Summary

Experimental and MD simulation of cosuspensions of non-magnetic anisotropic microcrystallites and magnetic nanoparticles have demonstrated an efficient orientation transfer from the magnetic subphase to the non-magnetic one, even with shape-isotropic MNPs. In relatively low magnetic fields (up to 700 mT), suspensions of the non-magnetic pigment particles alone show very low magnetically induced birefringence. The magneto-optical response is drastically enhanced by addition of a small volume fraction of MNPs. The form anisotropy of the non-MPs plays a crucial role in this effect. The gain of translational entropy resulting from the alignment of the elongated non-MPs with anisometric aggregates of the MNP dopants drives the reorientation of the nonmagnetic subphase. This alignment is particularly well pronounced in platelet-shaped nanocrystallites of Pigment Red 176, which shows an exceptionally large magneto-optical response [87].

Addition of MNPs to thermotropic liquid-crystalline phases also generates magneto-optical effects or alters the magneto-optical response of the doped LCs [27]. However, the measured effects are substantially weaker than with the magnetically doped suspensions. The stability of the samples is a substantial problem, the nematic host prefers to expel the dopants or to collect it in defect structures. The interactions of the MNPs with the LC matrix depend crucially on the MNP surface chemistry. The doped LC materials did not show any qualitatively new features, but quantitative effects are observed, and they can be explained satisfactorily.

In fibrillous gels, the motion of included MNPs is restricted by the gelator network with a mesh size comparable to the MNPs. We investigated the structure and the magneto-optical response of isotropic and anisotropic fibrillous organoferrogels with mobile MNPs. The presence of the gel network restricts the magneto-optical response of the ferrogel. Even though the ferrogel exhibits no magnetic hysteresis, an optical hysteresis has been found. This suggests that the optical response is primarily determined by the dynamics of self-assembly of the embedded MNPs into shape-anisotropic agglomerates. The optical anisotropy of the system can be fine-tuned by varying the concentrations of the gelator and the MNPs. The optical response in structurally anisotropic gels is orientation-dependent, revealing an intricate interplay between the confining mesh and the MNPs [37].

The torque measurement technique introduced in Section 5 can determine magnetic torques with nanonewton accuracy. It was demonstrated to provide useful results both for ferrofluids and for suspensions of nonmagnetic pigment crystallites and magnetic nanoparticles. Torques generated in conventional ferrofluids composed of spherical particles are quite small because Néel relaxation that dominates the dynamics of small nanoparticles is much faster than usual rotation rates of the magnetic field. The transferred torque depends upon the phase lag of the sample magnetization respective to the rotating external magnetic field, which is extremely small in typical spherical MNPs [85]. A strategy to achieve substantially larger torques would be the use of shape-anisotropic MNPs, in which the magnetization is pinned to the particle and the relaxation is governed by Brownian motion of the MNPs. The phase lag is substantially larger in that case even at slow rotation rates. Potential applications of this method are manifold, including the study of rectified thermal motions of MNPs in an oscillating field, using a ratchet effect [83], [84].