Macroscopic dynamics of the ferroelectric smectic [Formula: see text] phase with [Formula: see text] symmetry.

We present the macroscopic dynamics of ferroelectric smectic A, smectic [Formula: see text], liquid crystals reported recently experimentally by three groups. In this fluid and orthogonal smectic phase, the macroscopic polarization, [Formula: see text], is parallel to the layer normal thus giving rise to [Formula: see text] overall symmetry for this phase in the spatially homogeneous limit. A combination of linear irreversible thermodynamics and symmetry arguments is used to derive the resulting dynamic equations applicable at sufficiently low frequencies and sufficiently long wavelengths. Compared to non-polar smectic A phases, we find a static cross-coupling between compression of the layering and bending of the layers, which does not lead to elastic forces, but to elastic stresses. In addition, it turns out that a reversible cross-coupling between flow and the magnitude of the polarization modifies the velocities of both, first and second sound. At the same time, the relaxation of the polarization gives rise to dissipative effects for second sound at the same order of the wavevector as for the sound velocity. We also analyze reversible cross-coupling terms between elongational flow and electric fields as well as temperature and concentration gradients, which lend themselves to experimental detection. Apparently this type of terms has never been considered before for smectic phases. The question how the linear [Formula: see text] coupling in the energy alters the macroscopic response behavior when compared to usual non-polar smectic A phases is also addressed.

External concentration gradients can drive a propagating wave in a coherently layered system of nanosheets.

We present a model for the dynamics observed recently by Sano et al. [Nat. Commun. 12, 6771 (2021)] in a coherently layered system made up of sheetlike colloidal particles (nanosheets) subjected to an external concentration gradient. Adding a new macroscopic variable characteristic for the nonequilibrium situation encountered in the experiments to the hydrodynamics of smectic A liquid crystals, we show that all salient dynamic features observed in the experiments can be accounted for. For this nonequilibrium phenomenon, we identify the symmetry of the underlying ground state as undulating smectic A-like layering and the applied concentration gradient applied in the layer planes as the nonequilibrium driving force. As a result of our analysis, we find a coherent motion of undulating layers generated by a Helfrich-Hurault type instability propagating at a fixed velocity in accordance with the observations. If the coherence of the layering is lost, there is no longer any coherent propagation to be expected-as is also observed.

A two-fluid model for the macroscopic behavior of polar nematic fluids and gels in a nonchiral or a chiral solvent.

We present the macroscopic dynamics of polar nematic liquid crystals in a two-fluid context. We investigate the case of a nonchiral as well as of a chiral solvent. In addition, we analyze how the incorporation of a strain field for polar nematic gels and elastomers in a solvent modifies the macroscopic dynamics. It turns out that the relative velocity between the polar subsystem and the solvent gives rise to a number of cross-coupling terms, reversible as well as irreversible, unknown from the other two-fluid systems considered so far. Possible experiments to study those novel dynamic cross-coupling terms are suggested. As examples we just mention that gradients of the relative velocity lead, in polar nematics to heat currents and in polar cholesterics to temporal changes of the polarization. In polar cholesterics, shear flows give rise to a temporal variation in the velocity difference perpendicular to the shear plane, and in polar nematic gels uniaxial stresses or strains generate temporal variations of the velocity difference.

Ferromagnetic nematics: A macroscopic two-fluid description.

We present the macroscopic dynamic description of a ferromagnetic nematic, where the nematic part and the magnetic part can move relative to each other. The relative velocity that describes such movements can be a slowly relaxing variable. Its couplings to the nematic and the magnetic degrees of freedom are particularly interesting since the symmetry properties (behavior under spatial inversion and time reversal) of the three vectorial quantities involved are all different. As a consequence, a number of new crosscouplings involving the relative velocity exist. Some of them are discussed in more detail. First, we demonstrate that transverse temperature gradients generate transverse relative velocities and, vice versa, that transverse relative velocities give rise to temperature gradients. Second, we show that a simple shear flow in the relative velocity with the preferred direction in the shear plane can lead in a stationary situation to a tilt of the magnetization.

Two-fluid model for a fluid with tetrahedral-octupolar order.

We investigate the macroscopic dynamics of a two-fluid system with tetrahedral order. As all normal-fluid two-fluid systems one has-compared to a simple fluid-the velocity difference between the two subsystems and the concentration of one component as additional macroscopic variables. Depending on the type of system, the concentration can either be a conserved quantity or relax on a long, but finite timescale. Due to the existence of the tetrahedral order such a system breaks parity symmetry. Here we discuss physical systems without preferred direction in real space, meaning that our description applies to optically isotropic materials. We find a number of reversible as well as dissipative dynamic cross-coupling terms due to the additional octupolar order, when compared to a fluid mixture. As the most interesting cross-coupling term from an experimental point of view, we identify a dissipative cross-coupling between the relative velocity and the usual velocity gradients. Applying a shear flow in a plane, this dissipative cross-coupling leads to a velocity difference perpendicular to the shear plane. As a result one can obtain a spatially homogeneous oscillation of the relative velocity. In addition, this induced relative velocity can couple as a function of time and space to the concentration, which gives rise to an overdamped propagating soundlike mode, where the overdamping arises from the fact that velocity difference is a macroscopic variable and not strictly conserved. We also show that electric field gradients are connected with an analogous reversible cross-coupling and can lead in a planar shear geometry to an overdamped propagating mode as well.

Two-fluid model for the breakdown of flow alignment in nematic liquid crystals.

We present a macroscopic two-fluid model to explain the breakdown of flow alignment in nematic liquid crystals under shear flow due to smectic clusters. We find that the velocity difference of the two fluids plays a key role to mediate the time-dependent behavior as soon as a large enough amount of smectic order is induced by flow. For the minimal model it is sufficient to keep the nematic degrees of freedom, the mass density of the smectic clusters and the degree of smectic order, the density, and two velocities as macroscopic variables. While frequently a smectic A or C phase arises at lower temperatures, this is not required for the applicability of the present model. Indeed, as pointed out before by Gähwiller, there are compounds showing a breakdown of flow alignment over a large temperature range and no smectic phase, but a solid phase at lower temperatures. We also demonstrate that, using a one velocity model, there is no coupling under shear flow between induced smectic order and the director orientation in stationary situations thus rendering such a model to be unsuitable to describe the breakdown of flow alignment. In a two-fluid description, flow alignment breaks down and becomes unstable with regard to a space- and time-dependent state due to an induced finite velocity difference. In an Appendix we outline a mesoscopic model to account for the sign change in the anisotropy of the electric conductivity observed in nematics with smectic clusters.

Macroscopic two-fluid effects in magnetorheological fluids.

We investigate macroscopic two-fluid effects in magnetorheological fluids generalizing a one-fluid model studied before. In the bulk of the paper we use a model in which the carrier fluid, with density ρ_{1}, moves with velocity v_{1}, while the magnetic component (density ρ_{2}) and, therefore, the magnetization and the magnetic-field-induced relaxing strain field move with velocity v_{2}. In the framework of macroscopic dynamics we find, in particular, reversible dynamic and dissipative cross-coupling terms between the magnetization and the velocity difference. Experiments to detect some of these cross-coupling terms are suggested. We also compare the results of the two-fluid model presented here with two-fluid models available for electrorheological fluids. In two appendices we discuss the simplifying assumptions made to arrive at the model used in this paper and we also outline how to detect potential deviations from this model.

Cubic and tetragonal liquid crystal phases composed of non-chiral molecules: Chirality and macroscopic properties.

We discuss the symmetry properties as well as the macroscopic behavior of the cubic liquid crystal phases showing large chiral domains of either hand in some non-chiral compounds reported recently in the group of Tschierske. These phases are tricontinuous. While they have O or I432 symmetry in each chiral domain, the overall symmetry is [Formula: see text] as there is no net chirality for compounds composed of non-chiral molecules. It turns out that a rather similar type of phase has also been reported for triblock copolymers. Here we analyze in detail the macroscopic static and dynamic behavior of such phases and we predict, among other results, that they show the analog of static and dissipative Lehmann-type effects in their chiral domains. A description of a cubic liquid crystalline phase of Th symmetry, which has not yet been found experimentally, is also included. Suggestions for experiments are outlined to identify such a phase. In addition, we discuss tetragonal liquid crystalline phases of D4h and D4 (I422) symmetry as they have been reported last year experimentally in connection to the Q phase.

Continuum model of magnetic field induced viscoelasticity in magnetorheological fluids.

An effective macroscopic model of magnetorheological fluids in the viscoelastic regime is proposed. Under the application of an external magnetic field, columns of magnetizable particles are formed in these systems. The columns are responsible for solidlike properties, such as the existence of elastic shear modulus and yield stress, and are captured by the strain field, while magnetic properties are described by the magnetization. We investigate the interplay of these variables when static shear or normal pressure is imposed in the presence of the external magnetic field. By assuming a relaxing strain field, we calculate the flow curves, i.e., the shear stress as a function of the imposed shear rate, for different values of the applied magnetic field. Focusing on the small amplitude oscillatory shear, we study the complex shear modulus, i.e., the storage and the loss moduli, as a function of the frequency. We demonstrate that already such a minimal model is capable of furnishing many of the key physical features of these systems, such as yield stress, enhancement of the shear yield stress by pressure, threshold behavior in the spirit of the frequently employed Bingham law, and several features in the frequency dependence of storage and loss moduli.

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.

A dynamic preferred direction model for the self-organization dynamics of bacterial microfluidic pumping.

It is known that some flagellated bacteria like Serratia marcescens, when deposited and affixed onto a surface to form a "bacterial carpet", self-organize in a collective motion of the flagella that is capable of pumping fluid through microfluidic channels. We set up a continuum model comprising two macroscopic variables that is capable of describing this self-organization mechanism as well as quantifying it to the extent that an agreement with the experimentally observed channel width dependence of the pumping is reached. The activity is introduced through a collective angular velocity of the helical flagella rotation, which is an example of a dynamic macroscopic preferred direction. Our model supports and quantifies the view that the self-coordination is due to a positive feedback loop between the bacterial flagella and the local flow generated by their rotation. Moreover, our results indicate that this biological active system is operating close to the self-organization threshold.

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.

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.

Two-dimensional localized chaotic patterns in parametrically driven systems.

We study two-dimensional localized patterns in weakly dissipative systems that are driven parametrically. As a generic model for many different physical situations we use a generalized nonlinear Schrödinger equation that contains parametric forcing, damping, and spatial coupling. The latter allows for the existence of localized pattern states, where a finite-amplitude uniform state coexists with an inhomogeneous one. In particular, we study numerically two-dimensional patterns. Increasing the driving forces, first the localized pattern dynamics is regular, becomes chaotic for stronger driving, and finally extends in area to cover almost the whole system. In parallel, the spatial structure of the localized states becomes more and more irregular, ending up as a full spatiotemporal chaotic structure.

On the influence of a network on optically isotropic fluid phases with tetrahedral/octupolar order.

We investigate the influence of transient or permanent elasticity on liquid phases with octupolar (tetrahedral) order, a question that has never been addressed before. The focus will be on optically isotropic liquid phases with tetrahedral order including the T d phase and the chiral T phase introduced by Fel. It turns out that the presence of both, a network as well as tetrahedral order can lead to the formation of chiral domains of both hands in an optically isotropic fluid due to a completely novel mechanism, thus providing a possible macroscopic explanation for recent experimental observations. We study in detail how elasticity influences the macroscopic dynamics of both, the T d and the T phase. The simultaneous presence of a transient network as well as of octupolar order is shown to lead to completely new cross-coupling phenomena for optically isotropic systems including transient dissipative elastic strains due to temperature gradients.

Macroscopic behavior of polar nematic gels and elastomers.

We present the derivation of the macroscopic equations for uniaxial polar nematic gels and elastomers. We include the strain field as well as relative rotations as independent dynamic macroscopic degrees of freedom. As a consequence, special emphasis is laid on possible static and dynamic cross-couplings between these macroscopic degrees of freedom associated with the network, and the other macroscopic degrees of freedom including reorientations of the macroscopic polarization. In particular, we find static and dissipative dynamic cross-couplings between strain fields and relative rotations on one hand and the macroscopic polarization on the other that allow for new possibilities to manipulate polar nematics. To give one example each for the effects of a static and a dissipative cross-coupling: we find that a static electric field applied perpendicularly to the polar preferred direction leads to relative rotations while dynamically relative rotations can lead to transverse electric currents. In addition to a permanent network, we also consider the effect of a transient network, which is particularly important for the case of gels, melts and concentrated polymer solutions. A section on the influence of macroscopic chirality is included as well.

Transient elasticity and polymeric fluids: Small-amplitude deformations.

Transient elasticity (TE) is a concept useful for a systematic generalization of viscoelasticity. Due to its thermodynamic consistency, it naturally leads to a simple description of non-Newtonian effects displayed by polymeric fluids, granular media, and other soft matter. We employ a continuum-mechanical theory that is derived from TE and tailored to polymeric fluids, showing how it captures a surprisingly large number of phenomena in shear and elongational flows, including stationary, oscillatory, and transient ones, as well as the flow down an inclined channel. Even the Weissenberg effect is well accounted for. This theory is applicable for small- as well as large-amplitude deformations. We concentrate on the former in the present article, leaving the latter to a companion article.

Transient elasticity and the rheology of polymeric fluids with large amplitude deformations.

Transient elasticity is a systematic generalization of viscoelasticity. Its purpose is to give a coherent description of non-Newtonian effects displayed by soft-matter systems, especially polymer melts and solutions. Using the concept of transient elasticity we describe here a hydrodynamic model for polymeric fluids, which is applicable for large amplitude deformations. We present an energy density with only two independent parameters, which is compatible with all thermodynamic requirements and which reduces for small deformations to models studied previously. The expression discussed is simple enough to allow full analytic treatment and shows semiquantitative agreement with experimental data. This model is used to capture many of the interesting effects thought to be characteristic of polymer rheology for large deformations including viscosity overshoot near the onset of shear flow, the onset of elongational flows in situations for which there is no stationary solution as well as shear thinning and normal stress differences for a large range of shear rates. In addition, we analyze how well our model accounts for empirical relations including the Cox-Merz rule, the Yamamoto relation, and Gleißle's mirror relations.

Macroscopic behavior of ferrocholesteric liquid crystals and ferrocholesteric gels and elastomers.

We study the influence of macroscopic chirality on the macroscopic properties of superparamagnetic liquid crystals and gels. Specifically we derive macroscopic dynamic equations for ferrocholesteric low molecular weight (LMW) liquid crystals and for ferrocholesteric gels and elastomers in the local description using the director field as macroscopic variable. The magnetization is treated as a macroscopic dynamic degree of freedom and its coupling to all other macroscopic variables is examined in detail. We incorporate into our dynamic analysis terms that are linear in a magnetic field giving rise to a number of cross-coupling terms not possible otherwise. A number of properties that are unique to the class of systems studied arise. As an example for a static property we find a term in the generalized energy which is linear in the electric field and quadratic in the magnetic field. We find that applying a magnetic field to a ferrocholesteric can lead to reversible electric currents, heat currents and concentration currents, which change their sign with a sign change of macroscopic chirality. As an example of a rather intriguing dissipative dynamic contribution we point out that for ferrocholesterics and for ferrocholesteric gels and elastomers in a magnetic field extensional flow leads to electric and heat currents.