Intrinsic geometry and director reconstruction for three-dimensional liquid crystals
01 Jun 2021-New Journal of Physics (IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft)-Vol. 23, Iss: 6, pp 063006
TL;DR: In this article, the intrinsic geometry of elastic distortions in three-dimensional liquid crystals is described and necessary and sufficient conditions for a set of functions to represent these distortions by describing how they couple to the curvature tensor.
Abstract: We give a description of the intrinsic geometry of elastic distortions in three-dimensional nematic
liquid crystals and establish necessary and sufficient conditions for a set of functions to represent
these distortions by describing how they couple to the curvature tensor. We demonstrate that, in
contrast to the situation in two dimensions, the first-order gradients of the director alone are not
sufficient for full reconstruction of the director field from its intrinsic geometry: it is necessary to
provide additional information about the second-order director gradients. We describe several
different methods by which the director field may be reconstructed from its intrinsic geometry.
Finally, we discuss the coupling between individual distortions and curvature from the perspective
of Lie algebras and groups and describe homogeneous spaces on which pure modes of distortion
can be realised.
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TL;DR: In this article , the authors analyzed modulated phases in liquid crystals, from the long-established cholesteric and blue phases to the recently discovered twist-bend, splay-bent, and splay nematic phases, as well as the twist-grain-boundary and helical nanofilament variations on smectic phases.
Abstract: This article analyzes modulated phases in liquid crystals, from the long-established cholesteric and blue phases to the recently discovered twist-bend, splay-bend, and splay nematic phases, as well as the twist-grain-boundary (TGB) and helical nanofilament variations on smectic phases. The analysis uses the concept of four fundamental modes of director deformation: twist, bend, splay, and a fourth mode related to saddle-splay. Each mode is coupled to a specific type of molecular order: chirality, polarization perpendicular and parallel to the director, and octupolar order. When the liquid crystal develops one type of spontaneous order, the ideal local structure becomes nonuniform, with the corresponding director deformation. In general, the ideal local structure is frustrated; it cannot fill space. As a result, the liquid crystal must form a complex global phase, which may have a combination of deformation modes, and may have a periodic array of defects. Thus, the concept of an ideal local structure under geometric frustration provides a unified framework to understand the wide variety of modulated phases.
21 citations
TL;DR: In this paper , the authors distinguish three regimes of elastic constants for nematic liquid crystals: the regime where the Ericksen inequalities are satisfied, the ground state of the director field is uniform, and certain necessary inequalities are violated, and the free energy is thermodynamically unstable.
Abstract: By analyzing elastic theory for nematic liquid crystals, we distinguish three regimes of elastic constants. In one regime, the Ericksen inequalities are satisfied, and the ground state of the director field is uniform. In a second regime, certain necessary inequalities are violated, and the free energy is thermodynamically unstable. Between those possibilities, there is an intermediate regime, where the Ericksen inequalities are violated but the system is still stable. Remarkably, lyotropic chromonic liquid crystals are in the intermediate regime. We investigate the nonuniform structure of the director field in that regime, show that it depends sensitively on system geometry, and discuss the implications for lyotropic chromonic liquid crystals.
7 citations
TL;DR: In this article, it was shown that a 3D director field can be determined by five local fields, which are related to each other and to the curvature of the embedding space through six differential relations.
Abstract: The geometry and topology of the region in which a director field is embedded impose limitations on the kind of supported orientational order. These limitations manifest as compatibility conditions that relate the quantities describing the director field to the geometry of the embedding space. For example, in two dimensions (2D) the splay and bend fields suffice to determine a director uniquely (up to rigid motions) and must comply with one relation linear in the Gaussian curvature of the embedding manifold. In 3D there are additional local fields describing the director, i.e. fields available to a local observer residing within the material, and a number of distinct ways to yield geometric frustration. So far it was unknown how many such local fields are required to uniquely describe a 3D director field, nor what are the compatibility relations they must satisfy. In this work, we address these questions directly. We employ the method of moving frames to show that a director field is fully determined by five local fields. These fields are shown to be related to each other and to the curvature of the embedding space through six differential relations. As an application of our method, we characterize all uniform distortion director fields, i.e., directors for which all the local characterizing fields are constant in space, in manifolds of constant curvature. The classification of such phases has been recently provided for directors in Euclidean space, where the textures correspond to foliations of space by parallel congruent helices. For non-vanishing curvature, we show that the pure twist phase is the only solution in positively curved space, while in the hyperbolic space uniform distortion fields correspond to foliations of space by (non-necessarily parallel) congruent helices. Further analysis is expected to allow to also construct of new non-uniform director fields.
6 citations
TL;DR: In this paper , the authors consider a chiral liquid crystal confined in a long cylinder with free boundaries and show how the relative free energies depend on the system size, the natural twist, and the disclination energy.
Abstract: Many solid materials and liquid crystals exhibit geometric frustration, meaning that they have an ideal local structure that cannot fill up space. For that reason, the global phase must be a compromise between the ideal local structure and geometric constraints. As an explicit example of geometric frustration, we consider a chiral liquid crystal confined in a long cylinder with free boundaries. When the radius of the tube is sufficiently small, the director field forms a double-twist configuration, which is the ideal local structure. However, when the radius becomes larger (compared with the natural twist of the liquid crystal), the double-twist structure cannot fill space, and hence the director field must transform into some other chiral structure that can fill space. This space-filling structure may be either (1) a cholesteric phase with single twist, or (2) a set of double-twist regions separated by a disclination, which can be regarded as the beginning of a blue phase. We investigate these structures using theory and simulations, and show how the relative free energies depend on the system size, the natural twist, and the disclination energy. As another example, we also study a cholesteric liquid crystal confined between two infinite parallel plates with free boundaries.
4 citations
TL;DR: In this article, the authors exploit the intrinsic approach to provide a framework for directly addressing the frustration in physical assemblies, highlighting the role of the compatibility conditions associated with the intrinsic fields describing the physical assembly.
Abstract: Geometric frustration arises whenever the constituents of a physical assembly locally favor an arrangement that cannot be realized globally. Recently, such frustrated assemblies were shown to exhibit filamentation, size limitation, large morphological variations and other exotic response properties. While these unique characteristics can be shown to be a direct outcome of the geometric frustration, some geometrically frustrated systems do not exhibit any of the above phenomena. In this work we exploit the intrinsic approach to provide a framework for directly addressing the frustration in physical assemblies. The framework highlights the role of the compatibility conditions associated with the intrinsic fields describing the physical assembly. We show that the structure of the compatibility conditions determines the behavior of small assemblies and in particular predicts their superextensive energy growth exponent. We illustrate the use of this framework to several well-known frustrated assemblies.
1 citations
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TL;DR: In this paper, a Landau theory for bend flexoelectricity in liquid crystals of bent-core molecules was developed, which predicts a second-order transition from a high-temperature uniform nematic phase to a non-uniform polar phase composed of twist bend or splay bend deformations.
Abstract: We develop a Landau theory for bend flexoelectricity in liquid crystals of bent-core molecules. In the nematic phase of the model, the bend flexoelectric coefficient increases as we reduce the temperature toward the nematic to polar phase transition. At this critical point, there is a second-order transition from high-temperature uniform nematic phase to low-temperature nonuniform polar phase composed of twist-bend or splay-bend deformations. To test the predictions of Landau theory, we perform Monte Carlo simulations to find the director and polarization configurations as functions of temperature, applied electric field, and interaction parameters.
152 citations
TL;DR: The nematic elastic energy promotes the alignment of the flux lines of the nematic director towards geodesics and/or lines of curvature of the surface, which enables the manipulation of nematic alignment for the design of new materials and technological devices.
Abstract: When nematic liquid crystals are constrained to a curved surface, the geometry induces distortions in the molecular orientation. The mechanisms of the geometrical frustration involve the intrinsic as well as the extrinsic geometry of the underlying substrate. We show that the nematic elastic energy promotes the alignment of the flux lines of the nematic director towards geodesics and/or lines of curvature of the surface. As a consequence, the influence of the curvature can be tuned through the Frank elastic moduli. To illustrate this effect, we consider the simple case of nematics lying on a cylindrical shell. By combining the curvature effects with external magnetic fields, the molecular alignment can be reoriented or switched between two stable configurations. This enables the manipulation of nematic alignment for the design of new materials and technological devices.
126 citations