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Brand HR, Pleiner H. Macroscopic dynamics of the ferroelectric smectic [Formula: see text] phase with [Formula: see text] symmetry. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2024; 47:10. [PMID: 38305841 PMCID: PMC11226535 DOI: 10.1140/epje/s10189-024-00406-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/09/2023] [Accepted: 01/05/2024] [Indexed: 02/03/2024]
Abstract
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.
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Affiliation(s)
- Helmut R Brand
- Department of Physics, University of Bayreuth, 95440, Bayreuth, Germany
| | - Harald Pleiner
- Max Planck Institute for Polymer Research, 55021, Mainz, Germany.
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Brand HR, Pleiner H. A two-fluid model for the macroscopic behavior of polar nematic fluids and gels in a nonchiral or a chiral solvent. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2022; 45:17. [PMID: 35218411 PMCID: PMC8882122 DOI: 10.1140/epje/s10189-022-00172-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/15/2021] [Accepted: 02/14/2022] [Indexed: 06/14/2023]
Abstract
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.
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Affiliation(s)
- Helmut R. Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| | - Harald Pleiner
- Max Planck Institute for Polymer Research, 55021 Mainz, Germany
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Brand HR, Pleiner H, Svenšek D. Macroscopic behavior of polar nematic gels and elastomers. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2016; 39:105. [PMID: 27822626 DOI: 10.1140/epje/i2016-16105-7] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2016] [Accepted: 10/05/2016] [Indexed: 06/06/2023]
Abstract
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.
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Affiliation(s)
- Helmut R Brand
- Theoretische Physik III, Universität Bayreuth, 95440, Bayreuth, Germany
- Max-Planck-Institute for Polymer Research, POBox 3148, 55021, Mainz, Germany
| | - Harald Pleiner
- Max-Planck-Institute for Polymer Research, POBox 3148, 55021, Mainz, Germany.
| | - Daniel Svenšek
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, 1000, Ljubljana, Slovenia
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Pleiner H, Svenšek D, Brand HR. Active polar two-fluid macroscopic dynamics. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2013; 36:135. [PMID: 24287686 DOI: 10.1140/epje/i2013-13135-7] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2013] [Accepted: 11/12/2013] [Indexed: 06/02/2023]
Abstract
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.
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Affiliation(s)
- H Pleiner
- Max Planck Institute for Polymer Research, 55021, Mainz, Germany,
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Brand HR, Pleiner H, Svenšek D. Macroscopic behavior of systems with an axial dynamic preferred direction. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2011; 34:128. [PMID: 22120542 DOI: 10.1140/epje/i2011-11128-2] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2011] [Revised: 10/19/2011] [Accepted: 11/03/2011] [Indexed: 05/31/2023]
Abstract
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.
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Affiliation(s)
- H R Brand
- Theoretische Physik III, Universität Bayreuth, Bayreuth, Germany.
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Wittkowski R, Löwen H, Brand HR. Microscopic and macroscopic theories for the dynamics of polar liquid crystals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:041708. [PMID: 22181158 DOI: 10.1103/physreve.84.041708] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2011] [Indexed: 05/31/2023]
Abstract
We derive and analyze the dynamic equations for polar liquid crystals in two spatial dimensions in the framework of classical dynamical density functional theory (DDFT). Translational density variations, polarization, and quadrupolar order are used as order-parameter fields. The results are critically compared with those obtained using the macroscopic approach of time-dependent Ginzburg-Landau (GL) equations for the analogous order-parameter fields. We demonstrate that, for both the microscopic DDFT and the macroscopic GL approach, the resulting dissipative dynamics can be derived from a dissipation function. We obtain microscopic expressions for all diagonal contributions and for many of the cross-coupling terms emerging from a GL approach. Thus, we establish a bridge between molecular correlations and macroscopic modeling for the dissipative dynamics of polar liquid crystals.
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Affiliation(s)
- Raphael Wittkowski
- Institut für Theoretische Physik II, Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
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Wittkowski R, Löwen H, Brand HR. Polar liquid crystals in two spatial dimensions: the bridge from microscopic to macroscopic modeling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061706. [PMID: 21797386 DOI: 10.1103/physreve.83.061706] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2011] [Indexed: 05/31/2023]
Abstract
Two-dimensional polar liquid crystals have been discovered recently in monolayers of anisotropic molecules. Here, we provide a systematic theoretical description of liquid-crystalline phases for polar particles in two spatial dimensions. Starting from microscopic density functional theory, we derive a phase-field-crystal expression for the free-energy density that involves three local order-parameter fields, namely the translational density, the polarization, and the nematic order parameter. Various coupling terms between the order-parameter fields are obtained, which are in line with macroscopic considerations. Since the coupling constants are brought into connection with the molecular correlations, we establish a bridge from microscopic to macroscopic modeling. Our theory provides a starting point for further numerical calculations of the stability of polar liquid-crystalline phases and is also relevant for modeling of microswimmers, which are intrinsically polar.
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Affiliation(s)
- Raphael Wittkowski
- Institut für Theoretische Physik II, Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany
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Brand HR, Pleiner H. Macroscopic behavior of non-polar tetrahedratic nematic liquid crystals. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2010; 31:37-50. [PMID: 20101516 DOI: 10.1140/epje/i2010-10547-9] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2009] [Revised: 10/30/2009] [Accepted: 11/20/2009] [Indexed: 05/28/2023]
Abstract
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.
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Affiliation(s)
- H R Brand
- Max-Planck-Institute for Polymer Research, Mainz, Germany.
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