We demonstrate that a first order isotropic-to-nematic phase transition in liquid crystals can be succesfully modeled within the generalized Landau-de Gennes theory by selecting an appropriate combination of elastic constants. The numerical simulations of the model established in this paper qualitatively reproduce the experimentally observed configurations that include interfaces and topological defects in the nematic phase.
Tactoids are pointed, spindle-like droplets of nematic liquid crystal in an isotropic fluid. They have long been observed in inorganic and organic nematics, in thermotropic phases as well as lyotropic colloidal aggregates. The variational problem of determining the optimal shape of a nematic droplet is formidable and has only been attacked in selected classes of shapes and director fields. Here, by considering a novel class of admissible solutions for a bipolar droplet, we study the prevalence in the population of all equilibrium shapes of each of the three that may be optimal (tactoids primar…
The occurrence of new textures of liquid crystals is an important factor in tuning their optical and photonics properties. Here, we show, both experimentally and by numerical computation, that under an electric field chitin tactoids (i.e. nematic droplets) can stretch to aspect ratios of more than 15, leading to a transition from a spindle-like to a cigar-like shape. We argue that the large extensions occur because the elastic contribution to the free energy is dominated by the anchoring. We demonstrate that the elongation involves hydrodynamic flow and is reversible, the tactoids return to th…
Tactoids are spindle shaped-droplets of a uniaxial nematic phase suspended in the co-existing isotropic phase. They are found in dispersions of a wide variety of elongated colloidal particles, including actin, fd virus, carbon nanotubes, vanadium peroxide and chitin nanocrystals. Recent experiments on tactoids of chitin nanocrystals in water show that electric fields can very strongly elongate tactoids even though the dielectric properties of the co-existing isotropic and nematic phases differ only subtly. We develop a model for partially bipolar tactoids, where the degree of bipolarness of th…
We study two dimensional tactoids in nematic liquid crystals by using a $\mathbf{Q}$-tensor representation. A bulk free energy of the Maier-Saupe form with eigenvalue constraints on $\mathbf{Q}$, plus elastic terms up to cubic order in $\mathbf{Q}$ are used to understand the effects of anisotropic anchoring and Frank-Oseen elasticity on the morphology of nematic-isotropic domains. Further, a volume constraint is introduced to stabilize tactoids of any size at coexistence. We find that anisotropic anchoring results in differences in interface thickness depending on the relative orientation of t…
We theoretically investigate under what conditions the director field in a spindle-shaped nematic droplet or tactoid obtains a twisted, parity-broken structure. By minimizing the sum of the bulk elastic and surface energies, we show that a twisted director field is stable if the twist and bend elastic constants are small enough compared to the splay elastic constant, but only if the droplet volume is larger than some minimum value. We furthermore show that the transition from an untwisted to a twisted director-field structure is a sharp function of the various control parameters. We predict th…
Recent experiments have shown how nematically-ordered tactoid shaped actin droplets can be reorganized and divided by the action of myosin molecular motors. In this paper, we consider how similar morphological changes can potentially be achieved under equilibrium conditions. Using simulations, both atomistic and continuum, and a phenomenological model, we explore how the nucleation dynamics, shape changes, and the final steady state of a nematic tactoid droplet can be modified by interactions with model adhesive colloids that mimic a myosin motor cluster. Our results provide a prescription for…
Motivated by recent experiments, the isotropic-nematic phase transition in chromonic liquid crystals is studied. As temperature decreases, nematic nuclei nucleate, grow, and coalesce, giving rise to tactoid microstructures in an isotropic liquid. These tactoids produce topological defects at domain junctions (disclinations in the bulk or point defects on the surface). We simulate such tactoid equilibria and their coarsening dynamics with a model using degree of order, a variable length director, and an interfacial normal as state descriptors. We adopt Ericksen's work and introduce an augmented…
Disclination lines in nematic liquid crystals can exist in different geometric conformations, characterised by their director profile. In certain confined, colloidal and even more prominently in chiral nematics, the director profile may vary along the disclination line. We construct a robust geometric decomposition of director profile variations in closed disclination loops based on a quaternion description and use it to apply topological classification to linked loops with arbitrary variation of the profile. The description bridges the gap between the known abstract classification scheme deri…
In 3D nematic liquid crystals, disclination lines have a range of geometric structures. Locally, they may resemble $+1/2$ or $-1/2$ defects in 2D nematic phases, or they may have 3D twist. Here, we analyze the structure in terms of the director deformation modes around the disclination, as well as the nematic order tensor inside the disclination core. Based on this analysis, we construct a vector to represent the orientation of the disclination, as well as tensors to represent higher-order structure. We apply this method to simulations of a 3D disclination arch, and determine how the structure…
Topological defects play a pivotal role in the physics of liquid crystals and represent one of the most prominent and well studied aspects of mesophases. While in two-dimensional nematics, disclinations are traditionally treated as point-like objects, recent experimental studies on active nematics have suggested that half-strength disclinations might in fact possess a polar structure. In this article, we provide a precise definition of polarity for half-strength nematic disclinations, we introduce a simple and robust method to calculate this quantity from experimental and numerical data and we…
We show that the isotropic to nematic liquid crystalline phase transition of F-actin can be either continuous or discontinuous, depending critically on the filament length. For F-actin with average filament length > 3 um, we confirm that the transition is continuous in both filament alignment and local concentration. In contrast, for filament length < 2 um the F-actin solution undergoes a first order transition. Tactoidal droplets of co-existing isotropic and nematic domains were observed. Phenomena of nucleation-and-growth and spinodal decomposition both occur, depending sensitively on the ex…
We introduce a characterization of disclination lines in three dimensional nematic liquid crystals as a tensor quantity related to the so called rotation vector around the line. This quantity is expressed in terms of the nematic tensor order parameter $\mathbf{Q}$, and shown to decompose as a dyad involving the tangent vector to the disclination line and the rotation vector. Further, we derive a kinematic law for the velocity of disclination lines by connecting this tensor to a topological charge density as in the Halperin-Mazenko description of defects in vector models. Using this framework, …
We explore the structure of nuclei and topological defects in the first-order phase transition between the nematic (N) and isotropic (I) phases in lyotropic chromonic liquid crystals (LCLCs). The LCLCs are formed by self-assembled molecular aggregates of various lengths and show a broad biphasic region. The defects emerge as a result of two mechanisms. 1) Surface anisotropy mechanism that endows each N nucleus (tactoid) with topological defects thanks to preferential (tangential) orientation of the director at the closed I-N interface, and 2) Kibble mechanism with defects forming when differen…
Understanding and controlling the director field configuration, shape, and orientation in nematic and cholesteric liquid crystals is of fundamental importance in several branches of science. Liquid crystalline droplets, also known as tactoids, which spontaneously form by nucleation and growth within the biphasic region of the phase diagram where isotropic and nematic phases coexist, challenge our current understanding of liquid crystals under confinement, due to the influence of anisotropic surface boundaries at vanishingly small interfacial tension and are mostly studied under quiescent, quas…
Nematic liquid crystals exhibit configurations in which the underlying ordering changes markedly on macroscopic length scales. Such structures include topological defects in the nematic phase and tactoids within nematic-isotropic coexistence. We discuss a computational study of inhomogeneous configurations that is based on a field theory extension of the Maier-Saupe molecular model of a uniaxial, nematic liquid crystal. A tensor order parameter is defined as the second moment of an orientational probability distribution, leading to a free energy that is not convex within the isotropic-nematic …
The conventional topological description given by the fundamental group of nematic order parameter does not adequately explain the entangled defect line structures that have been observed in nematic colloids. We introduce a new topological invariant, the self-linking number, that enables a complete classification of entangled defect line structures in general nematics, even without particles, and demonstrate our formalism using colloidal dimers, for which entangled structures have been previously observed. We also unveil a simple rewiring scheme for the orthogonal crossing of two -1/2 disclina…
Aqueous dispersions of exfoliated, bile-salt stabilized single-wall carbon nanotubes exhibit a first order transition to a nematic liquid-crystalline phase. The nematic phase presents itself in the form of micron-sized nematic droplets also known as tactoids, freely floating in the isotropic host dispersion. The nematic droplets are spindle shaped and have an aspect ratio of about four, irrespective of their size. We attribute this to a director field that is uniform rather than bipolar, which is confirmed by polarization microscopy. It follows that the ratio of the anchoring strength and the …
Using Landau-de Gennes effective theory for nematic liquid crystals we analyse the structure of rectilinear n=1/2 smooth disclination line in the case of equal elastic constants. We find that at certain temperature there is an exact mathematical correspondence with a rectilinear vortex in superfluid $^4He$. With a help of polynomial approximation difference of free energies of the smooth and of a singular disclination lines is estimated. It turns out that the smooth disclination line is energetically preferred only if temperature is low enough. At higher temperatures a disordered core should b…
Using a geometric approach for the propagation of light in anisotropic media, we investigate what effect the director field of disclinations may have on the polarization state of light. Parallel transport around the defect, of the spinor describing the polarization, indicates the acquisition of a topological phase, in analogy with the Aharonov-Bohm effect.
The fluctuation eigenmode problem of the nematic topological disclination line with strength $\pm 1/2$ is solved for the complete nematic tensor order parameter. The line tension concept of a defect line is assessed, the line tension is properly defined. Exact relaxation rates and thermal amplitudes of the fluctuations are determined. It is shown that within the simple string model of the defect line the amplitude of its thermal fluctuations is significantly underestimated due to the neglect of higher radial modes. The extent of universality of the results concerning other systems possessing l…
Topological defects in elastic media may be described by a geometric field akin to three-dimensional gravity. From this point of view, disclinations are line defects of zero width corresponding to a singularity of the curvature in an otherwise flat background. On the other hand, in two dimensions, the Frank free energy of a nematic liquid crystal may be interpreted as an Abelian Higgs Lagrangian. In this work, we construct an Abelian Higgs model coupled to "gravity" for the nematic phase, with the perspective of finding more realistic disclinations. That is, a cylindrically symmetric line defe…
Nematic solids respond strongly to changes in ambient heat or light, significantly differently parallel and perpendicular to the director. This phenomenon is well characterized for uniform director fields, but not for defect textures. We analyze the elastic ground states of a nematic glass in the membrane approximation as a function of temperature for some disclination defects with an eye towards reversibly inducing three-dimensional shapes from flat sheets of material, at the nano-scale all the way to macroscopic objects, including non-developable surfaces. The latter offers a new paradigm to…
We use full nematohydrodynamic simulations to study the statics and dynamics of monolayers of cholesteric liquid crystals. Using chirality and temperature as control parameters we show that we can recover the two-dimensional blue phases recently observed in chiral nematics, where hexagonal lattices of half-skyrmion topological excitations are interleaved by lattices of trefoil topological defects. Furthermore, we characterise the transient dynamics during the quench from isotropic to blue phase. We then proceed by confining cholesteric stripes and blue phases within finite-sized tactoids and s…
Both uniaxial and biaxial nematic liquid crystals are defined by orientational ordering of their building blocks. While uniaxial nematics only orient the long molecular axis, biaxial order implies local order along three axes. As the natural degree of biaxiality and the associated frame, that can be extracted from the tensorial description of the nematic order, vanishes in the uniaxial phase, we extend the nematic director to a full biaxial frame by making use of a singular value decomposition of the gradient of the director field instead. New defects and degrees of freedom are unveiled and th…
Chiral additives in the nematic liquid crystal can alter the dynamics of point defects moving on a disclination line. They exert a constant force on defects, leading to the bimodal distribution of distances between them at long times. The evolution of the system of defects in the presence of chiral additives provides a very direct proof of the existence of repulsive forces between the defects at large distances. We find that addition of a sufficient amount of chiral compound removes all point defects from the system. The process is studied in the system of 8CB (4-n-octyl-4 '-cyanobiphenyl) dop…
We construct a phenomenological Landau-de Gennes theory for hard colloidal rods by performing an order parameter expansion of the chemical-potential dependent grand potential. By fitting the coefficients to known results of Onsager theory, we are not only able to describe the isotropic-nematic phase transition as function of density, including the well-known density jump, but also the isotropic-nematic planar interface. The resulting theory is applied in calculations of the isotropic core size in a radial hedgehog defect, the density dependence of linear defects of hard rods in square confinem…
Detection of chiral molecules requires amplification of chirality to measurable levels. Typically, amplification mechanisms are considered at the microscopic scales of individual molecules and their aggregates. Here we demonstrate chirality amplification and visualization of structural handedness in water solutions of organic molecules that extends over the scale of many micrometers. The mechanism is rooted in the long-range elastic nature of orientational order in lyotropic chromonic liquid crystals (LCLCs) formed in water solutions of achiral disc-like molecules. The nematic LCLC coexists wi…
Nonsingular disclination dynamics in a uniaxial nematic liquid crystal is modeled within a mathematical framework where the kinematics is a direct extension of the classical way of identifying these line defects with singularities of a unit vector field representing the nematic director. It is well known that the universally accepted Oseen-Frank energy is infinite for configurations that contain disclination line defects. We devise a natural augmentation of the Oseen-Frank energy to account for physical situations where, under certain conditions, infinite director gradients have zero associate…
We investigate the correlation between the point disclination energies and the surface curvature modulation of nematic liquid crystal membranes with a Gaussian bump geometry. Due to the correlation, disclinations feel an attractive force that confines them to an annulus region, resulting in a halo distribution around the top of the bump. The halo formation is a direct consequence of the nonzero Gaussian curvature of the bump that affects preferable configurations of liquid crystal molecules around the disclination core.
Disclination configurations of a nematic liquid crystal are studied within a self-consistent molecular field theory. The theory is based on a tensor order parameter, and can accommodate anisotropic elastic energies without the known divergences in the Landau-de Gennes formulation. Our results agree with the asymptotic results of Dzyaloshinsky for the Frank-Oseen energy and far from the defect core, but reveal biaxial order at intermediate distances from the core, crossing over to uniaxial, but isotropic configurations as the core is approached. The elastic terms considered in our energy allow …
We investigate the emergence of orientational order among +1/2 disclinations in active nematic liquid crystals. Using a combination of theoretical and experimental methods, we show that +1/2 disclinations have short-range antiferromagnetic alignment, as a consequence of the elastic torques originating from their polar structure. The presence of intermediate -1/2 disclinations, however, turns this interaction from anti-aligning to aligning at scales that are smaller than the typical distance between like-sign defects. No long-range orientational order is observed. Strikingly, these effects are …
We have studied the nucleation and the physical properties of a -1/2 wedge disclination line near the free surface of a confined nematic liquid crystal. The position of the disclination line has been related to the material parameters (elastic constants, anchoring energy and favored anchoring angle of the molecules at the free surface). The use of a planar model for the structure of the director field (whose predictions have been contrasted to those of a fully three-dimensional model) has allowed us to relate the experimentally observed position of the disclination line to the relevant propert…
We study the effect of the rotation of an external electric field on the dynamics of half-integer disclination networks in two dimensional nematic liquid crystals with a negative dielectric anisotropy using LICRA, a LIquid CRystal Algorithm developed by the authors. We show that a rotation of $π$ of the electric field around an axis of the liquid crystal plane continuously transforms all half-integer disclinations of the network into disclinations of opposite sign via twist disclinations. We also determine the evolution of the characteristic length scale, thus quantifying the impact of the ext…
The spontaneous emergence of collective flows is a generic property of active fluids and often leads to chaotic flow patterns characterised by swirls, jets, and topological disclinations in their orientation field. However, the ability to achieve structured flows and ordered disclinations is of particular importance in the design and control of active systems. By confining an active nematic fluid within a channel, we find a regular motion of disclinations, in conjunction with a well defined and dynamic vortex lattice. As pairs of moving disclinations travel through the channel, they continuall…
An individual defect in a nematic liquid crystal moves not only in response to its interaction with other defects but also in response to an external field. We analyze the motion of a wedge disclination in the presence of an applied field of strength $H$. We neglect backflow and seek steadily travelling patterns. The stationary picture yields a semi-infinite wall of strength $π$, bounded by the defect line. We find that the disclination advances into the region containing the wall at velocity $v(H)$, where $v$ scales as $H/|\log H|$ as long as the magnetic coherence length is greater than the …
Dipartimento Interateneo di Fisica, Università degli Studi di Bari and INFN, Sezione di Bari（巴里大学物理跨大学系与意大利国家核物理研究所巴里分部）；SUPA, School of Physics and Astronomy, University of Edinburgh（爱丁堡大学物理与天文学院）
We study two-dimensional equilibrium configurations of nematic liquid crystal (NLC) director in a cell confined between two parallel surfaces: a planar surface and a spatially modulated one. The relief of the modulated surface is described by a smooth periodic sine-like function. The director easy axis orientation is homeotropic at one of the bounding surfaces and is planar at the other one. Strong NLC anchoring with both surfaces is assumed. We consider the case where disclination lines occur in the bulk of NLC strictly above local extrema of the modulated surface. These disclination lines ru…
We investigate the behavior of disclination loops in the vicinity of the first order nematic-isotropic transition in the Lebwohl-Lasher and related models. We find that two independent measures of the transition temperature, the free energy and the distribution of disclination line segments, give essentially identical values. We also calculate the distribution function D(p) of disclination loops of perimeter p and fit it to a quasiexponential form. Below the transition, D(p) falls off exponentially, while in the neighborhood of the transition it decays with a power law exponent approximately e…
Various director configurations of a nematic liquid crystal, confined in a toroidal volume and subject to an external magnetic field, were evaluated numerically by performing finite element calculations in three dimensions. The equilibrium director field is found from a minimization of the elasto-magnetic energy of the nematic. The director at the inner surface of the torus is fixed, either along homeotropic or tangential direction. We consider both a homogeneous and an azimuthal magnetic field, with its direction being in conflict with the respective surface anchoring. The relative stability …
We present a novel framework for the study of disclinations in two-dimensional active nematic liquid crystals, and topological defects in general. The order tensor formalism is used to calculate exact multi-particle solutions of the linearized static equations inside a uniformly aligned state. Topological charge conservation requires a fixed difference between the number of half charges. Starting from a set of hydrodynamic equations, we derive a low-dimensional dynamical system for the parameters of the static solutions, which describes the motion of a half-disclination pair, or of several pai…
While studies of active nematics in two dimensions have shed light on various aspects of the flow regimes and topology of active matter, three-dimensional properties of topological defects and chaotic flows remain unexplored. By confining a film of active nematics between two parallel plates, we use continuum simulations and analytical arguments to demonstrate that the crossover from quasi-2D to 3D chaotic flows is controlled by the morphology of the disclination lines. For small plate separations, the active nematic behaves as a quasi-2D material, with straight topological disclination lines …
Defects in liquid crystals are of great practical importance and theoretical interest. Despite tremendous efforts, predicting the location and transition of defects under various topological constraint and external field remains to be a challenge. We investigate defect patterns of nematic liquid crystals confined in three-dimensional spherical droplet and two-dimensional disk under different boundary conditions, within the Landau-de Gennes model. We implement a spectral method that numerically solves the Landau-de Gennes model with high accuracy, which allows us to study the detailed static st…
We investigate energetically favorable configurations of point disclinations in nematic films having a bump geometry. Gradual expansion in the bump width Δ gives rise to a sudden shift in the stable position of the disclinations from the top to the skirt of the bump. The positional shift observed across a threshold Δth obeys a power law function of |Δ-Δth|, indicating a new class of continuous phase transition that governs the defect configuration in curved nematic films.
We study ionic and mass transport in a liquid crystalline fluid film in its nematic phase under an applied electrostatic field. Both analytic and numerical solutions are given for some prototypical configurations of interest in electrokinetics: Thin films with spatially nonuniform nematic director that are either periodic or comprise a set of isolated disclinations. We present a quantitative description of the mechanisms inducing spatial charge separation in the nematic, and of the structure and magnitude of the resulting flows. The fundamental solutions for the charge distribution and flow ve…
The structure of the half-integer rectilinear disclination line within the framework of the Landau-de Gennes effective theory of nematic liquid crystals is investigated. The consistent perturbative expansion is constructed for the case of $L_2\neq 0$. It turns out that such expansion can be performed around only a discrete subset of an infinite set of the degenerate zeroth order solutions. These solutions correspond to the positive and negative wedge disclination lines and to four configurations of the twist disclination line. The first order corrections to both the order parameter field as we…
Molecular simulations of a nematic liquid crystal confined in cylinder geometry with homeotropic anchoring have been carried out. The core structure of a disclination line defect of strength +1 has been examined, and comparison made with various theoretical treatments, which are presented in a unified way. It is found that excellent fits to the cylindrically-symmetrized order tensor profiles may be obtained with appropriate parameter choices; notwithstanding this, on the timescales of the simulation, the cylindrical symmetry of the core is broken and two defects of strength +1/2 may be resolve…
