Reversible and dissipative macroscopic contributions to the stress tensor: active or passive?

The issue of dynamic contributions to the macroscopic stress tensor has been of high interest in the field of bio-inspired active systems over the last few years. Of particular interest is a direct coupling ("active term") of the stress tensor with the order parameter, the latter describing orientational order induced by active processes. Here we analyze more generally possible reversible and irreversible dynamic contributions to the stress tensor for various passive and active macroscopic systems. This includes systems with tetrahedral/octupolar order, polar and non-polar (chiral) nematic and smectic liquid crystals, as well as active fluids with a dynamic preferred (polar or non-polar) direction. We show that it cannot a priori be seen, neither from the symmetry properties of the macroscopic variables involved, nor from the structure of the cross-coupling contributions to the stress tensor, whether the system studied is active or passive. Rather, that depends on whether the variables that give rise to those cross-couplings in the stress tensor are driven or not. We demonstrate that several simplified descriptions of active systems in the literature that neglect the necessary counter term to the active term violate linear irreversible thermodynamics and lead to an unphysical contribution to the entropy production.

Active polar two-fluid macroscopic dynamics.

We study the dynamics of systems with a polar dynamic preferred direction. Examples include the pattern-forming growth of bacteria as well as shoals of fish, flocks of birds and migrating insects. Due to the fact that the preferred direction only exists dynamically, but not statically, the macroscopic variable of choice is the macroscopic velocity associated with the motion of the active units, which are typically biological in nature. We derive the macroscopic equations for such a system and discuss novel static, reversible and irreversible cross-couplings connected to a second velocity as a variable. We analyze in detail how the macroscopic behavior of an active system with a polar dynamic preferred direction compares to other systems with two velocities including immiscible liquids and electrically neutral quantum liquids such as superfluid (4)He and (3)He . We critically discuss changes in the normal mode spectrum when comparing uncharged superfluids, immiscible liquids and active system with a polar dynamic preferred direction. We investigate the influence of a macroscopic hand (collective effects of chirality) on the macroscopic behavior of such active media.

Macroscopic behavior of systems with an axial dynamic preferred direction.

We present the derivation of the macroscopic equations for systems with an axial dynamic preferred direction. In addition to the usual hydrodynamic variables, we introduce the time derivative of the local preferred direction as a new variable and discuss its macroscopic consequences including new cross-coupling terms. Such an approach is expected to be useful for a number of systems for which orientational degrees of freedom are important including, for example, the formation of dynamic macroscopic patterns shown by certain bacteria such a Proteus mirabilis. We point out similarities in symmetry between the additional macroscopic variable discussed here, and the magnetization density in magnetic systems as well as the so-called Î vector in superfluid (3)He-A. Furthermore we investigate the coupling to a gel-like system for which one has the strain tensor and relative rotations between the new variable and the network as additional macroscopic variables.

Macroscopic behavior of non-polar tetrahedratic nematic liquid crystals.

We discuss the symmetry properties and the macroscopic behavior of a nematic liquid crystal phase with D(2d) symmetry. Such a phase is a prime candidate for nematic phases made from banana-shaped molecules where the usual quadrupolar order coexists with octupolar (tetrahedratic) order. The resulting nematic phase is nonpolar. While this phase could resemble the classic D (infinityh) nematic in the polarizing microscope, it has many static as well as reversible and irreversible properties unknown to nonpolar nematics without octupolar order. In particular, there is a linear gradient term in the free energy that selects parity leading to ambidextrously helical ground states when the molecules are achiral. In addition, there are static and irreversible coupling terms of a type only met otherwise in macroscopically chiral liquid crystals, e.g. the ambidextrous analogues of Lehmann-type effects known from cholesteric liquid crystals. We also discuss the role of hydrodynamic rotations about the nematic director. For example, we show how strong external fields could alter the D(2d) symmetry, and describe the non-hydrodynamic aspects of the dynamics, if the two order structures, the nematic and the tetrahedratic one, rotate relative to each other. Finally, we discuss certain nonlinear aspects of the dynamics related to the non-commutativity of three-dimensional finite rotations as well as other structural nonlinear hydrodynamic effects.

Response of prestretched nematic elastomers to external fields.

We investigate the response of prestretched nematic side-chain liquid single-crystal elastomers to superimposed external shear, electric, and magnetic fields of small amplitude. The prestretching direction is oriented perpendicular to the initial nematic director orientation, which enforces director reorientation. Furthermore, the shear plane contains the direction of prestretch. In this case, we obtain a strongly decreased effective shear modulus in the vicinity of the onset and the completion of the enforced director rotation. For the same regions, we find that it becomes comparatively easy to reorient the director by external electric and magnetic fields. These results were derived using conventional elasticity theory and its coupling to relative director-network rotations.

Rosensweig instability of ferrogel thin films or membranes.

We derive the dispersion relation of surface waves for magnetic gel membranes or thin films at the interface between two fluids in the presence of an external magnetic field normal to the free surface. Above a critical field strength surface waves become linearly unstable with respect to a stationary pattern of surface protuberances. This linear stability criterion generalizes that of the Rosensweig instability for ferrofluid and ferrogel free surfaces to take into account bending elasticity and intrinsic elastic and magnetic surface properties of the film or membrane, additionally. The latter is of interest for uniaxial ferrogel film or membranes, which show a locked-in permanent magnetization.

Instabilities in nematic elastomers in external electric and magnetic fields.

In this paper we study the macroscopic behavior of nematic side-chain liquid single crystal elastomers exposed to an external electric or magnetic field. For this purpose we use the framework of a continuum model. The geometries investigated comprise the bend and the twist geometry known from the classical Frederiks transition in low molecular weight liquid crystals. For the bend geometry we find a laterally homogeneous and a two-dimensional undulatory instability, which may compete at onset. In the case of the twist geometry three instabilities can occur at onset, two of which are two dimensional and clearly show undulations. As a major result we propose how the values of the twist coefficient K(2) and the values of the material parameters D(1) and D(2) connected to relative rotations between the director field and the polymer network can be determined from experimental observations. In addition, we explain why a twist experiment is probably the most suitable set-up in order to measure the material parameter D(1).

Splay-bend textures involving tetrahedratic order.

We show that defect-free splay-bend textures are less energetic compared to uniform states in liquid-crystalline phases that possess both quadrupolar (nematic) and octupolar (tetrahedratic) order. This is because, in such systems, there is a symmetry-allowed linear gradient term in the energy. Another unusual feature of these splay-bend textures is the fact that they have a non-homogeneous, space-dependent free-energy density. These results may help clarify some mysterious features noted for the B7 liquid-crystal phase formed by achiral banana-shaped molecules.

A phenomenological theory of the isotropic to chiral smectic-C phase transition.

In this paper we discuss the direct isotropic to chiral smectic-C phase transition on the basis of a phenomenological theory. The model free energy is written in terms of the coupled order parameters including the spontaneous polarization. We present a detailed analysis of the different phases that can occur and analyze the question under which conditions a direct isotropic to chiral smectic-C phase transition is possible when compared to other phase transitions. On the basis of this model the isotropic-smectic-C* transition is always of first order. The theoretical predictions are compared with the available experimental results.

Undulation versus Frederiks instability in nematic elastomers in an external electric field.

We study the behaviour of a nematic side-chain elastomer under the influence of an external static electric field for a specific geometry. For this investigation, the nematic elastomer is considered to be a perfect insulator. On the basis of a macroscopic description we generalize the classical Frederiks transition in a low-molecular-weight (LMW) nematic liquid crystal to the elastomeric case. We predict, using a linear stability analysis, that the onset of the instability can be qualitatively different from the LMW case: in liquid crystalline elastomers an undulation instability can arise at onset. Whether the analogue of a Frederiks instability or an undulation instability occurs first depends on the sample thickness as well as on the material parameters. It turns out that the parameter which describes the coupling between the deformations of the elastomer and the relative rotations between the elastomer and the director field of the nematic phase is most important for the predicted response of the system. Furthermore, we find that the magnitude of the critical electric field is much higher in the elastomeric than in the low-molecular-weight case.

Mechanical properties of mono-domain side chain nematic elastomers.

We investigate the behavior of the shear rigidity modulus G = G' + iG" of three mono-domain side chain liquid-crystalline elastomers composed of side chain polysiloxanes cross-linked by either flexible or rigid cross-linkers. The measurements were taken in a frequency domain ranging from approximately 0.02 Hz to approximately 10(4) Hz applying the shear in a plane perpendicular to or containing the director. The measurements as a function of temperature show an anisotropy of G' which appears around T(NI), when decreasing the temperature, and which is due to the expected lowering of G'(//) coming from the coupling between the shear and the director. The measurements as a function of frequency show that G has two components for both geometries, in both the isotropic phase and in the nematic phase around the phase transition. One reflects the network behavior in its hydrodynamic regime ( G' is constant and G'' is approximately f, where f is the frequency), the other which appears at higher frequencies is characterized by a scaling law behavior (G' is approximately G" is approximately f(0.5)) of the Rouse type. We discuss the results in the framework of available theories and show that the three elastomers present a non-soft behavior, even for the elastomer for which the contrary was claimed, and that there is no separation of time scales between the director and the network. We also present data on a poly-domain sample and a non-mesomorphic one which complement these results.

Macroscopic dynamics of uniaxial magnetic gels.

We present the derivation of the macroscopic equations for uniaxial ferrogels. In addition to the usual hydrodynamic variables for gels we introduce the magnetization and the relative rotations between the magnetization and the network as macroscopic variables. The relative rotations introduced here for a system with magnetic degrees of freedom are the analog of the relative rotations introduced by de Gennes in nematic elastomers for rotations of the director with respect to the elastomeric network. These variables give rise to a large number of static as well as dynamic effects due to their coupling to the magnetization, the strain field, and the density of linear momentum. A few of them are discussed for specific geometries, for example, the case of a shear-induced magnetization perpendicular to the preferred direction.

Hydrodynamics of isotropic ferrogels.

We derive the complete set of macroscopic dynamic equations for ferrogels under an external magnetic field, including the magnetization as an independent dynamic degree of freedom. The magnetoelasticity comes in the form of magnetostriction and through the magnetic part of the Maxwell stress. Various dynamic couplings of the elastic degree of freedom with the magnetization and the magnetic field are found. We discuss static elongation, shear deformations, and the modified sound spectrum in the presence of an external magnetic field.

Deformable tetrahedratic phases: the effects of external fields and flows.

We discuss changes in the symmetry and physical properties of an isotropic phase which has initially tetrahedral symmetry characterized by four unit vectors. In its undeformed state, these four vectors are at the tetrahedral angle (109.47 degrees) to each other. We find that this optically isotropic phase becomes uniaxial under the influence of an external electric field, E, resulting in a phase with C3v symmetry. For an applied simple shear flow, the system becomes biaxial and a time-dependent state with C1 symmetry arises. We discuss to what extent deformations induced by external forces and flows on this optically isotropic phase, which we call a "deformable tetrahedratic phase", are consistent with observations at the isotropic-B7 transition found recently in compounds composed of banana-shaped molecules and suggest a number of experiments to test the conclusions of this model.

Macroscopic dynamics near the isotropic-smectic-A phase transition.

The hydrodynamic theory for the smectic-A phase and the isotropic phase is generalized to the macroscopic dynamics in the vicinity of the isotropic-smectic-A phase transition. The macroscopic dynamic equations are presented on the isotropic side as well as on the smectic-A side of the phase transition, incorporating the effect of an external electric field. Specific experiments to test some of the effects contained in the macroscopic dynamic equations are suggested.