Session D25: Liquid Crystals I

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Common fluids cannot sustain static mechanical stresses at the macroscopic scale because they lack molecular order. Conversely, crystalline solids exhibit long-range order and mechanical strength at the macroscopic scale. Combining the properties of fluids and solids, liquid crystal films respond to mechanical confinement by both flowing and generating static forces. The elastic response, however, is very weak for film thicknesses exceeding 10 nm. In this study, the mechanical strength of a fluid film was enhanced by introducing topological defects in a cholesteric liquid crystal, producing unique viscoelastic and optomechanical properties. The cholesteric was confined under strong planar anchoring conditions between two curved surfaces with sphere-sphere contact geometry, similar to that of colloidal particles, creating concentric dislocation loops. During surface retraction, the loops shrank and periodically disappeared at the surface contact point, where the cholesteric helix underwent discontinuous twist transitions, producing weak oscillatory surface forces. On the other hand, new loop nucleation was frustrated by a topological barrier during fluid compression, creating a metastable state. This generated exceptionally large forces with a range exceeding 100 nm, as well as extended blue-shifts of the photonic band-gap. The metastable cholesteric helix eventually collapsed under a high compressive load, triggering a stick-slip-like cascade of defect nucleation and twist reconstruction events. These findings were explained using a simple theoretical model and suggest a general approach to enhance the mechanical strength of 1d-periodic materials, particularly cholesteric-colloid mixtures.   

Singularity tracking methods for a unified description of topology, geometry, and the dynamics of disclinations in 3D nematic liquid crystals

Disclinations are pervasive in both passive and active nematic liquid crystals. In the former, they can be used to align and propel colloidal particles, actuate surfaces, and transport biomaterials; while in the latter they spontaneously nucleate and recombine, facilitate density gradients, and promote extrusion of nematogens and layered growth of the material. In three dimensions, characterizing the structure and understanding the dynamic behavior of line defects poses theoretical challenges. We introduce a disclination density tensor, a function of the tensor order parameter, that characterizes disclinations in 3D nematics. The evolution of the disclination density tensor obeys conservation of topological charge and can be used to obtain the velocity of disclination lines in both passive and active nematics. We also report numerical results in 3D to validate the analytical predictions of disclination motion for a variety of configurations.   

Backflow effects in p-atic liquid crystals

We investigate the effects of hydrodynamics on the motion of topological defects in p-atic liquid crystals: i.e. two-dimensional liquid crystals characterised by p-fold rotational symmetry, of which nematics p=2 and hexatics p=6 are the best known examples. We find that the strong distortion originating from the defects sources a flow, which, in turn, can lead to the defects' propulsion, provided their topological charge is given by s = (p-1)/p. Furthermore, we find that hydrodynamics can accelerate the annihilation dynamics of pairs of ±1/p defects and that this effect increases with p before approaching a plateau.    

Fractional defect charges in p-atic liquid crystals on cones

We study two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on the surfaces of cones with free boundary conditions on the cone edge, and find both the ground state(s) and a ladder of quantized metastable states as a function of the cone angle and the liquid crystal symmetry p. We find that these states are characterized by a fractional defect charge at the apex and that the ground states are in general frustrated due to effects of parallel transport along the azimuthal direction of the cone. We check our predictions numerically for a set of commensurate cone angles, whose surfaces can be polygonized as a perfect triangular or square mesh, and find exact agreement.    

Behavior of disclination arrays across the nematic-smectic transition

Versatile control over liquid crystal alignment is key to the development of novel electro-optical devices. We use photo-alignment to create topological defects in nematic liquid crystals. The preferred alignment direction of the nematic director at parallel glass substrates is set using a surface layer of azo-dye that undergoes photo-reorientation upon exposure to blue light. Our projector-based optical system can pattern molecular orientations at two independently controlled confining surfaces. Altering the projected pattern and the system geometry allows us to produce topological defects arrays of both half-integer and integer disclinations. This capability to precisely synthesize 3D disclination networks offers a promising step in constructing and manipulating mechanical and optical metamaterials. We observe and discuss the behavior of disclination networks across the nematic-smectic phase transition.   

Critical points of two-dimensional sigma models and implications for liquid crystals and gases of intersecting loops.

We use the recently introduced scale invariant scattering theory to exactly determine the renormalization group fixed points of $RP^{N-1}$ and $CP^{N-1}$ models in two dimensions, which differ from vector models for an additional local symmetry: respectively the liquid crystal head-tail symmetry and a $U(1)$ symmetry. We show that, also due to subtle degeneracies at specific values of $N$, above a threshold value $N_c$ there is only a zero temperature critical point of $O(N(N+1)/2-1)$ type for $RP^{N-1}$ and of $O(N^2-1)$ type for $CP^{N-1}$. Below $N_c$ new branches of fixed points emerge which are relevant for criticality in gases of loops with crossings. For liquid crystals $N_c=2.24421..$, and a topological transition of Berezinskii-Kosterlitz-Thouless type exists only for $N=2$. For $CP^{N-1}$ $N_c=2$ and no topological transition occurs for $N$ integer.   

Pairs of solitary domain walls in ferroelectric nematic

Polar ordering in confined samples of a ferroelectric nematic N_F often produces coexisting domains of opposite polarization. We report on the structure of the defect walls separating these domains in the material abbreviated DIO. In sandwich cells with parallel rubbing assembly, polarization is either antiparallel to the rubbing direction or parallel to it. Each polar domain is bound by two π-walls, with clockwise or counterclockwise rotation of polarization. The walls are of a finite width, representing static solitons. Two neighboring walls of the same sense of rotation form a 2π soliton of a winding number 1, while the 2π walls with opposite sense of rotation are topologically trivial. The pairs of a zero topological charge are removable by the electric fields that are much lower than the field required to remove the topologically stable 2π walls.   

Explicit Demonstration of Geometric Frustration in Cholesteric Liquid Crystals

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 two-dimensional disk with free boundaries. When the disk is sufficiently small, the director field forms a double-twist configuration, which is the ideal local structure. However, when the disk 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 core radius.   

Cholesteric shells: two-dimensional blue fog and finite quasicrystals

We study the phase behaviour of a quasi-two-dimensional cholesteric liquid crystal shell. Using Lattice Boltzmann simulations we characterise the topological phases arising close to the isotropic-cholesteric transition and show that they differ in a fundamental way from those observed on a flat geometry. For spherical shells, we discover two types of quasi-two-dimensional topological phases: finite quasicrystals and amorphous structures, both made up of mixtures of polygonal tessellations of half-skyrmions. These structures generically emerge instead of regular double twist lattices because of geometric frustration, which disallows a regular hexagonal tiling of curved space. For toroidal shells, the variations in the local curvature of the surface stabilise heterogeneous phases where cholesteric patterns coexist with hexagonal lattices of half-skyrmions. Quasicrystals, amorphous and heterogeneous structures could be sought experimentally by self-assembling cholesteric shells on the surface of emulsion droplets.   

Spatial structure of topological defect lines in three-dimensional nematic liquid crystals

Topological defects in nematic liquid crystal systems often exhibit intricate spatial structures with nontrivial morphology. Using the the Landau-de Gennes framework, we theoretically and numerically study the structure and energetics of topological defect configurations that arise in various nematic liquid crystal systems. We investigate the role of different experimentally tunable parameters such as system size and boundary condition in dictating the minimum-energy form of the topological defects. In particular, motivated by experiment, we study the equilibrium structure of the defect patterns in the presence of non-matching surface disclinations at the two opposite boundaries of the system. Interestingly, we find that the morphology of the three-dimensional disclination lines that connect different surface defects crucially depends on the thickness of the system and other system parameters. These structures and transitions arise from the geometric conflict, inflicted by boundary conditions, between having short defect paths and having small elastic distortion around them. Some of our theoretical findings corroborate experimental observations. We point to the crucial effect that these nematic structures may have on the topology of defects that will emerge in such a system once cooled into a smectic phase.   

Scaling and Spontaneous Symmetry Restoring in Reconnecting Nematic Disclinations

Topological defects -locations of local mismatch of order- are a universal concept playing essential roles in various scientific domains. In nematic liquid crystals, they appear as disclinations, which move on an observable time scale and are amenable to optical observations. In the two-dimensional (2D) case, it is known that two point disclinations asymmetrically approach and annihilate. In three dimensions (3D), string-like disclinations are moving and interacting. However, due to the difficulty of imaging, it is not easy to fully resolve the motion of  3D disclinations. Here we report direct 3D measurement of dynamics of disclination lines by fluorescence confocal microscopy. We analyzed reconnections, a characteristic of disclination lines, and determined a scaling law for the distance between disclinations. Furthermore, we found that apparently asymmetric motion of the disclination pairs is symmetric in an appropriate co-moving frame, in contrast to the 2D counterpart. This "symmetry restoring" is considered to occur because any director profile around a 3D disclination is homeomorphic, and the symmetric twist configuration is energetically favorable. We expect this mechanism to have generality.   

Classifying Non-trivial Links of Biaxial Nematic Defect Lines

Unlike uniaxial nematics, biaxial nematic systems have a non-Abelian fundamental group. As a consequence, defects lines that form in these systems have non-trivial combination rules that obey the algebra of its fundamental group, the quaternions. This non-Abelian nature of the defect lines leads to non-trivial linking, in which two defects braided around each other may remain linked depending on the individual defect types, a feature that is not present in systems with Abelian properties. This allows one to form links of N defect lines that may not necessarily be equivalent to N unknots. We attempt to model this system as a colored braid theory subject to a set of equivalence relations on its generators in order to determine a classification system for the possible non-trivial links   

Focal conic array in patterned microchannels near the cholesteric-smectic phase transition

It is known that the liquid crystalline smectic-A phase has geometric defects, called focal conic domains, whose size and positions are heavily influenced by their boundaries. We have recently shown that chiral nematic phases have toron-like defects that can, for varying ranges of temperatures, reversibly transform into focal conic domains across the phase transition, displaying geometric memory [1]. We explore a weakly chiral system in which the positions of focal conic domains are controlled by confinement, and these defects are formed as the system relaxes near the phase transition. To study this, we have patterned microchannels with curved boundaries to induce arrays of focal conic domains and explore their formation for a different range of channel geometries. We show that certain defects are topologically protected, and we observe a variety of seemingly equilibrium configurations that are only observed as a result of conditions present at the phase transition.

[1] Hare, Sean M., Beatrice Lunsford-Poe, MinSu Kim, and Francesca Serra. "Chiral Liquid Crystal Lenses Confined in Microchannels." Materials 13, no. 17 (2020): 3761. https://doi.org/10.3390/ma13173761.   

Defect coarsening in three-dimensional active nematics

Coarsening of defect lines has a universal scaling in different physical systems; first described by Kibble for cosmic strings [1], predicted by Zurek in superfluid helium [2], and experimentally observed in nematic liquid crystals [3]. Here, we consider defect coarsening in three-dimensional active nematics [4,5], where following an initial quench the density of a defect line network decreases until a dynamic steady state is reached [6]. Activity-generated flows play an important part in the coarsening, which we demonstrate by constructing a phenomenological theory for defect density dynamics and validate it by a fully-resolved numerical simulation. First, we consider dynamics of a single loop, and then of a whole defect network following a quench or for a time-varying activity. Finally, our phenomenological equations are then compared to the time evolution of the defect line coarsening in other physical systems.

[1] T. W. B. Kibble, Topology of cosmic domains and strings, J. Phys. A: Math. Gen. 9, 1387 (1976).

[2] W. H. Zurek, Cosmological experiments in superfluid helium?, Nature 317, 505 (1985).

[3] I. Chuang, R. Durrer, N. Turok and B. Yurke, Cosmology in the laboratory: Defect dynamics in liquid crystals, Science 251, 1336 (1991).

[4] G. Duclos et al, Topological structure and dynamics of three-dimensional active nematics, Science 367, 1120 (2020).

[5] Ž. Krajnik, Ž. Kos and M. Ravnik, Spectral energy analysis of bulk three-dimensional active nematic turbulence, Soft Matter 16, 9059 (2020).

[6] N. Kralj, M. Ravnik, Ž. Kos, to be submitted   

Active Control of Periodic, Three-Dimensional Disclination Networks in Nematic Liquid Crystals

Disclinations are of fundamental interests to soft matter physics, particles physics, and mathematics. Topological defects are also of practical importance in sensing, photonics, and directed self-assembly of colloids and molecules. Disclinations in active nematic liquid crystals (LCs) are especially interesting due to their autonomous dynamics. However, these structures in three-dimensional (3D) active nematics are difficult to control, limiting their further applications. Here, we demonstrate a full control over the transformation of 3D disclinations that mimics their dynamics in active nematics. Specifically, using photo-patterned surfaces and continuum simulations, we show that during a mechanical or photo-induced transformation of the surface anchoring, periodic 3D disclinations can nucleate, deform, merge, and split in a nematic cell in a programmable manner. Continuum simulations based on Landau-de Gennes free energy functional demonstrate an excellent agreement with the experiment and reveal the change in their topologies during the transformation. These highly controlled defect transformations allow us to examine the topology and elastic properties of disclinations of varying morphologies, and to facilitate applications including novel LC-based photonic devices.