Spontaneous ordering of magnetic particles in liquid crystals: From chains to biaxial lamellae

Liquid-crystal composites of carbon nanotubes in a magnetic field: Bridging continuum theory and a molecular-statistical approach

We propose an approach combining the continuum theory and molecular-statistical approach for a suspension of carbon nanotubes based on a negative diamagnetic anisotropy liquid crystal. Using the continuum theory, we show that in the case of an infinite sample in suspension it is possible to observe peculiar magnetic Fréedericksz-like transitions between three nematic phases: planar, angular, and homeotropic with different mutual orientations of liquid-crystal and nanotube directors. The transition fields between these phases are found analytically as functions of material parameters of the continuum theory. To account for the effects associated with temperature changes, we propose a molecular-statistical approach that allows obtaining the equations of orientational state for the orientation angles of the main axes of the nematic order, i.e., the liquid-crystal and carbon-nanotube directors in a similar form as was obtained within the continuum theory. Thus, it is possible to relate the parameters of the continuum theory, such as the surface-energy density of a coupling between molecules and nanotubes, to the parameters of the molecular-statistical model and the order parameters of the liquid crystal and carbon nanotubes. This approach allows determining the temperature dependencies of the threshold fields of transitions between different nematic phases, which is impossible in the framework of the continuum theory. In the framework of the molecular-statistical approach we predict the existence of an additional direct transition between the planar and homeotropic nematic phases of the suspension, which cannot be described based on the continuum theory. As the main results, the magneto-orientational response of the liquid-crystal composite is studied and a possible biaxial orientational ordering of the nanotubes in the magnetic field is shown.

Multifunctionality by dispersion of magnetic nanoparticles in anisotropic matrices

Interactions between magnetic nanoparticles and an anisotropic environment give rise to a variety of new magneto-optical, rheological and mechanical phenomena. This opens new avenues for developing novel multifunctional materials. In the course of this project, we investigated three types of anisotropic systems: dispersions of shape-anisotropic nanocrystals, magnetically doped molecular and colloidal liquid crystals, and organoferrogels. They were investigated by means of magneto-optical observations and by a magneto-mechanical torsion pendulum method.

Superparamagnetic nanoparticles with LC polymer brush shell as efficient dopants for ferronematic phases

Liquid crystal (LC) based magnetic materials consisting of LC hosts doped with functional magnetic nanoparticles enable optical switching of the mesogens at moderate magnetic field strengths and thereby open the pathway for the design of novel smart devices. A promising route for the fabrication of stable ferronematic phases is the attachment of a covalently bound LC polymer shell onto the surface of nanoparticles. With this approach, ferronematic phases based on magnetically blocked particles and the commercial LC 4-cyano-4'-pentylbiphenyl (5CB) liquid crystal were shown to have a sufficient magnetic sensitivity, but the mechanism of the magneto-nematic coupling is unidentified. To get deeper insight into the coupling modes present in these systems, we prepared ferronematic materials based on superparamagnetic particles, which respond to external fields with internal magnetic realignment instead of mechanical rotation. This aims at clarifying whether the hard coupling of the magnetization to the particle's orientation (magnetic blocking) is a necessary component of the magnetization-nematic director coupling mechanism. We herein report the fabrication of a ferronematic phase consisting of surface-functionalized superparamagnetic Fe₃O₄ particles and 5CB. We characterize the phase behavior and investigate the magneto-optical properties of the new ferronematic phase and compare it to the ferronematic system containing magnetically blocked CoFe₂O₄ particles to get information about the origin of the magneto-nematic coupling.

Defect Structures of Magnetic Nanoparticles in Smectic A Liquid Crystals

Topological defects in anisotropic fluids like liquid crystals serve as a playground for the research of various effects. In this study, we concentrated on a hybrid system of chiral rod-like molecules doped by magnetic nanoparticles. In textures of the smectic A phase, we observed linear defects and found that clusters of nanoparticles promote nucleation of smectic layer defects just at the phase transition from the isotropic to the smectic A (SmA) phase. In different geometries, we studied and analysed creation of defects which can be explained by attractive elastic forces between nanoparticles in the SmA phase. On cooling the studied hybrid system, clusters grow up to the critical dimension, and the smectic texture is stabilised. The presented effects are theoretically described and explained if we consider the elastic interaction of two point defects and stabilisation of prismatic dislocation loops due to the presence of nanoparticles.

Domain growth in ferronematics: slaved coarsening, emergent morphologies and growth laws

Ferronematics (FNs) are suspensions of magnetic nanoparticles in nematic liquid crystals (NLCs). They have attracted much experimental attention, and are of great interest both scientifically and technologically. There are very few theoretical studies of FNs, even in equilibrium. In this paper, we study the non-equilibrium phenomenon of domain growth after a thermal quench (or coarsening) in this coupled system. Our modeling is based on coupled time-dependent Ginzburg-Landau (TDGL) equations for two order parameters: the LC tensor order parameter Q, and the magnetization M. We consider both shallow and deep quenches from a high-temperature disordered phase. The system coarsens by the collision and annihilation of topological defects. We focus on slaved coarsening, where a disordered Q (or M) field is driven to coarsen by an ordered M (or Q) field. We present detailed results for the morphologies and growth laws, which exhibit unusual features purely due to the magneto-nematic coupling. To the best of our knowledge, this is the first study of non-equilibrium phenomena in FNs.

Field-induced anti-nematic and biaxial ordering in binary mixtures of discotic mesogens and spherical magnetic nanoparticles

Using computer simulations we explore the equilibrium structure and response to external stimuli of complex magnetic hybrids consisting of magnetic particles in discotic liquid crystalline matrices. We show that the anisotropy of the liquid crystalline matrix (either in the nematic or in the columnar phase) promotes the collective orientational ordering of self-assembled magnetic particles. Upon applying an external homogeneous magnetic field in an otherwise isotropic state, the magnetic particles self-assemble into linear-rodlike-chains. At the same time structural changes occur in the matrix. The matrix transforms from an isotropic to a non-conventional anti-nematic state in which the symmetry axis of the discs is, on average, perpendicular to the magnetic field. In addition, a stable biaxial nematic state is found upon applying an external field to an otherwise uniaxial discotic nematic state. These observed morphologies constitute an appealing alternative to binary mixtures of rigid rod-disc system and indicate that non-trivial biaxial ordering can be obtained in the presence of a uniaxial external stimulus.

Structure and rheology of soft hybrid systems of magnetic nanoparticles in liquid-crystalline matrices: results from particle-resolved computer simulations

Hybrid mixtures composed of magnetic nanoparticles (MNP) in liquid crystalline (LC) matrices are a fascinating class of soft materials with intriguing physical properties and a wide range of potential applications, e.g., as stimuli-responsive and adaptive materials. Already in the absence of an external stimulus, these systems can display various types of orientationally disordered and ordered phases, which are enriched by self-assembled structures formed by the MNPs. In the presence of external fields, one typically observes highly nonlinear macroscopic behavior. However, an understanding of the structure and dynamics of such systems on the particle level has, so far, remained elusive. In the present paper we review recent computer simulation studies targeting the structure, equilibrium dynamics and rheology of LC-MNP systems, in which the particle sizes of the two components are comparable. As a numerically tractable model system we consider mixtures of soft spherical or elongated particles with a permanent magnetic dipole moment and ellipsoidal non-magnetic particles interacting via a Gay-Berne potential. We address, first, equilibrium aspects such as structural organization and self-assembly (cluster formation) of the MNPs in dependence of the orientational state of the matrix, the role of the size ratio, the impact of an external magnetic field, and the translational and orientational diffusion of the two components. Second, we discuss the non-equilibrium dynamics of LC-MNP mixtures under planar shear flow, considering both, spherical and non-spherical MNPs. Our results contribute to a detailed understanding of these intriguing hybrid materials, and they may serve as a guide for future experiments.

Steady-state rheology and structure of soft hybrid mixtures of liquid crystals and magnetic nanoparticles

Using non-equilibrium molecular dynamics simulations, we study the rheology of a model hybrid mixture of liquid crystals (LCs) and dipolar soft spheres (DSS) representing magnetic nanoparticles. The bulk isotropic LC-DSS mixture is sheared with different shear rates using Lees-Edwards periodic boundary conditions. The steady-state rheological properties and the effect of the shear on the microstructure of the mixture are studied for different strengths of the dipolar coupling, λ, among the DSS. We find that at large shear rates, the mixture shows a shear-thinning behavior for all considered values of λ. At low and intermediate values of λ, a crossover from Newtonian to non-Newtonian behavior is observed as the rate of applied shear is increased. In contrast, for large values of λ, such a crossover is not observed within the range of shear rates considered. Also, the extent of the non-Newtonian regime increases as λ is increased. These features can be understood via the shear-induced changes of the microstructure. In particular, the LCs display a shear-induced isotropic-to-nematic transition at large shear rates with a shear-rate dependent degree of nematic ordering. The DSS show a shear-induced nematic ordering only for large values of λ, where the particles self-assemble into chains. Moreover, at large λ and low shear rates, our simulations indicate that the DSS form ferromagnetic domains.

Efficient ferronematic coupling with polymer-brush particles

Switching of liquid crystal phases is of enormous technological importance and enables digital displays, thermometers and sensors.

Speeding up the study of diluted dipolar systems

We study the regimes of a diluted dipolar system through Monte Carlo numerical simulations in the NVT ensemble. To accelerate the dynamics, several approximations and speed-up algorithms are proposed and tested. In particular, it turns out that "cluster move Monte Carlo" algorithm speeds-up to two decades faster than the traditional Monte Carlo, depending on temperature and density. We find simple-fluid, chain-fluid, ring-fluid, gel, and antiparallel columnar regimes, which are studied and characterized through positional, orientational, and thermodynamical observables.

Influence of tetrahedral order on ferromagnetic gel phases

We investigate the macroscopic dynamics of gels with tetrahedral/octupolar symmetry, which possess in addition a spontaneous permanent magnetization. We derive the corresponding static and dynamic macroscopic equations for a phase, where the magnetization is parallel to one of the improper fourfold tetrahedral symmetry axes. Apart from elastic strains, we take into account relative rotations between the magnetization and the elastic network. The influence of tetrahedral order on these degrees of freedom is investigated and some experiments are proposed that are specific for such a material and allow to indirectly detect tetrahedral order. We also consider the case of a transient network and predict that stationary elastic shear stresses arise when a temperature gradient is applied.

Anomalous transport of magnetic colloids in a liquid crystal-magnetic colloid mixture

We report an extensive molecular dynamics study on the translational dynamics of a hybrid system composed of dipolar soft spheres (DSS), representing ferromagnetic particles, suspended in a liquid crystal (LC) matrix. We observe that the LC matrix strongly modifies the dynamics of the DSS. In the isotropic regime, the DSS show a crossover from subdiffusive to normal diffusive behavior at long times, with an increase of the subdiffusive regime as the dipolar coupling strength is increased. In the nematic regime, the LC matrix, due to the collective reorientation of LC particles, imposes a cylindrical confinement on the DSS chains. This leads to a diffusive dynamics of DSS along the nematic director and a subdiffusive dynamics (with an exponent of ∼0.5) in the perpendicular direction. The confinement provided by the LC matrix is also reflected by the oscillatory behavior of the components of the velocity autocorrelation function of the DSS in the nematic phase.

Nematohydrodynamics for colloidal self-assembly and transport phenomena

HYPOTHESIS

Colloidal particles in a nematic liquid crystal (NLC) exhibit very different behaviour to that observed in an isotropic medium. Such differences arise principally due to the nematic-induced elastic stresses exerted due to the interaction of NLC molecules with interfaces, which compete with traditional fluid viscous stresses on the particle.

THEORY

A systematic mathematical analysis of particles in an NLC microfluidic channel is performed using the continuum Beris-Edwards framework coupled to the Navier-Stokes equations. We impose strong homeotropic anchoring on the channel walls and weak homeotropic anchoring on the particle surfaces.

FINDINGS

The viscous and NLC forces act on an individual particle in opposing directions, resulting in a critical location in the channel where the particle experiences zero net force in the direction perpendicular to the flow. For multi-particle aggregation we show that the final arrangement is independent of the initial configuration, but the path towards achieving equilibrium is very different. These results uncover new mechanisms for particle separation and routes towards self-assembly.

Phase diagrams of mixtures of dipolar rods and discs

Self-assembly of binary mixtures that contain anisotropic, interacting colloidal particles have been proposed as a way to create new, multi-functional materials. We simulate binary mixtures of dipolar rods and dipolar discs in two-dimensions using discontinuous molecular dynamics to determine how the assembled structures of these mixtures differ from those seen in single component systems. Two different binary mixtures are investigated: a mixture of an equal number of dipolar rods and dipolar discs ("equal number"), and a mixture where the area fraction of dipolar rods is equal to the area fraction of dipolar discs ("equal area"). Phase boundaries between fluid, string-fluid, and "gel" phases are calculated and compared to the phase boundaries of the pure components. Looking deeper at the underlying structure of the mixture reveals a complex interplay between the rods and discs and the formation of states where the two components are in different phases. The mixtures exhibit phases where both rods and discs are in the fluid phase, where rods form a string-fluid while discs remain in the fluid phase, a rod string-fluid coexisting with a disc string-fluid, a "gel" that consists primarily of rods while the discs form either a fluid or string-fluid phase, and a "gel" that contains both rods and discs. Our results give insight into the general assembly pathway of binary mixtures, and how complex aggregates can be created by varying the mixture composition, strength of interaction between the two components, and the temperature. By manipulating the properties of one of the components it should be possible to fabricate bifunctional, thermally responsive self-assembled materials.

Effects of flow on the dynamics of a ferromagnetic nematic liquid crystal

We investigate the effects of flow on the dynamics of ferromagnetic nematic liquid crystals. As a model, we study the coupled dynamics of the magnetization, M, the director field, n, associated with the liquid crystalline orientational order, and the velocity field, v. We evaluate how simple shear flow in a ferromagnetic nematic is modified in the presence of small external magnetic fields, and we make experimentally testable predictions for the resulting effective shear viscosity: an increase by a factor of 2 in a magnetic field of about 20 mT. Flow alignment, a characteristic feature of classical uniaxial nematic liquid crystals, is analyzed for ferromagnetic nematics for the two cases of magnetization in or perpendicular to the shear plane. In the former case, we find that small in-plane magnetic fields are sufficient to suppress tumbling and thus that the boundary between flow alignment and tumbling can be controlled easily. In the latter case, we furthermore find a possibility of flow alignment in a regime for which one obtains tumbling for the pure nematic component. We derive the analogs of the three Miesowicz viscosities well-known from usual nematic liquid crystals, corresponding to nine different configurations. Combinations of these can be used to determine several dynamic coefficients experimentally.

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

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

Magneto-optic dynamics in a ferromagnetic nematic liquid crystal

We investigate dynamic magneto-optic effects in a ferromagnetic nematic liquid crystal experimentally and theoretically. Experimentally we measure the magnetization and the phase difference of the transmitted light when an external magnetic field is applied. As a model we study the coupled dynamics of the magnetization, M, and the director field, n, associated with the liquid crystalline orientational order. We demonstrate that the experimentally studied macroscopic dynamic behavior reveals the importance of a dynamic cross-coupling between M and n. The experimental data are used to extract the value of the dissipative cross-coupling coefficient. We also make concrete predictions about how reversible cross-coupling terms between the magnetization and the director could be detected experimentally by measurements of the transmitted light intensity as well as by analyzing the azimuthal angle of the magnetization and the director out of the plane spanned by the anchoring axis and the external magnetic field. We derive the eigenmodes of the coupled system and study their relaxation rates. We show that in the usual experimental setup used for measuring the relaxation rates of the splay-bend or twist-bend eigenmodes of a nematic liquid crystal one expects for a ferromagnetic nematic liquid crystal a mixture of at least two eigenmodes.

Dynamic Magneto-optic Coupling in a Ferromagnetic Nematic Liquid Crystal

Hydrodynamics of complex fluids with multiple order parameters is governed by a set of dynamic equations with many material constants, of which only some are easily measurable. We present a unique example of a dynamic magneto-optic coupling in a ferromagnetic nematic liquid, in which long-range orientational order of liquid crystalline molecules is accompanied by long-range magnetic order of magnetic nanoplatelets. We investigate the dynamics of the magneto-optic response experimentally and theoretically and find out that it is significantly affected by the dissipative dynamic cross-coupling between the nematic and magnetic order parameters. The cross-coupling coefficient determined by fitting the experimental results with a macroscopic theory is of the same order of magnitude as the dissipative coefficient (rotational viscosity) that governs the reorientation of pure liquid crystals.

Orientational order and translational dynamics of magnetic particle assemblies in liquid crystals

Implementing extensive molecular dynamics simulations we explore the organization of magnetic particle assemblies (clusters) in a uniaxial liquid crystalline matrix comprised of rodlike particles. The magnetic particles are modelled as soft dipolar spheres with diameter significantly smaller than the width of the rods. Depending on the dipolar strength coupling the magnetic particles arrange into head-to-tail configurations forming various types of clusters including rings (closed loops) and chains. In turn, the liquid crystalline matrix induces long range orientational ordering to these structures and promotes their diffusion along the director of the phase. Different translational dynamics are exhibited as the liquid crystalline matrix transforms either from isotropic to nematic or from nematic to smectic state. This is caused due to different collective motion of the magnetic particles into various clusters in the anisotropic environments. Our results offer a physical insight for understanding both the structure and dynamics of magnetic particle assemblies in liquid crystalline matrices.

Colloidal Suspensions of Rodlike Nanocrystals and Magnetic Spheres under an External Magnetic Stimulus: Experiment and Molecular Dynamics Simulation

Using experiments and molecular dynamics simulations, we explore magnetic field-induced phase transformations in suspensions of nonmagnetic rodlike and magnetic sphere-shaped particles. We experimentally demonstrate that an external uniform magnetic field causes the formation of small, stable clusters of magnetic particles that, in turn, induce and control the orientational order of the nonmagnetic subphase. Optical birefringence was studied as a function of the magnetic field and the volume fractions of each particle type. Steric transfer of the orientational order was investigated by molecular dynamics (MD) simulations; the results are in qualitative agreement with the experimental observations. By reproducing the general experimental trends, the MD simulation offers a cohesive bottom-up interpretation of the physical behavior of such systems, and it can also be regarded as a guide for further experimental research.

Superelastic stress–strain behavior in ferrogels with different types of magneto-elastic coupling

Anisotropic ferrogels and magnetorheological elastomers allow for reversible tunability of their markedly nonlinear stress–strain properties.