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Skyrmion lattice. Each unit cell possesses a single central s=2 umbilic, with the six outer s=1 umbilics each shared between three unit cells giving a total count of 2+6×1/3=4 umbilics =1 Skyrmion per unit cell. Top left: Unit vector field n in a Skyrmion lattice. Top right: Contour plot of |Δ| showing the umbilic lines forming a lattice of tubes. Bottom right: Eigenvector field of Π, illustrating the +1 winding around the central umbilics and −1/2 winding around the outer umbilics. Note that both are counted with the same sign as the orientation of n changes from one to another. Bottom left: Umbilics form lines in three dimensions.
Source publication
Three-dimensional orientational order in systems whose ground states possess
non-zero, chiral gradients typically exhibits line-like structures or defects:
$\lambda$ lines in cholesterics or Skyrmion tubes in ferromagnets for example.
Here we show that such lines can be identified as a set of natural geometric
singularities in a unit vector field,...
Contexts in source publication
Context 1
... Int(U i , Σ) is the signed number of intersections of U i with Σ and s i ∈ N is the absolute strength of the um- bilic U i . Examples illustrating this are shown in Figs. 1 and 4; we describe only the latter (the former is essen- tially identical). Fig. 4 shows a Skyrmion lattice, a well- known structure seen in a variety of systems [13,[15][16][17]23,29,[84][85][86]. The umbilics form a lattice, with s = 2 umbilics each surrounded by six s = 1 umbilics. The eigenvectors of Π show the central umbilic having a +1 profile, and the six outer umbilics a −1/2 profile. All the umbilics have ...
Context 2
... the sum of the indices of the umbilics piercing Σ is equal to four times the Skyrmion number. This estab- lishes a general relationship between Skyrmions and um- bilics. This is illustrated in Fig. 4 where each unit cell contains one s = 2 umbilic and six s = 1 umbilics, each of which are shared between three unit cells, giving a total count of 2 + 6 × 1/3 = 4, i.e. there is one Skyrmion in each unit cell. This approach can also be extended to the charcterisation of merons, or half-Skyrmions, which can be thought of as a single s = ...
Citations
... The same effect was described contemporaneously by Cladis & Kleman [2]. These escaped structures are commonly observed in experiments on materials confined in cylindrical capillaries [3,4], and they also occur in torodial droplets of liquid crystal [5][6][7], at the centre of a Skyrmion [8,9], and as part of three-dimensional knotted solitons [10,11]. In this paper we consider the chiral version of Meyer's escape in which the director maintains a preferred handedness of twist. ...
... Depending on the structure being studied the material domain is a cylindrical capillary of radius R, a spherical droplet of radius R, or a cube of side length R. At the boundary we impose either homeotropic or planar degenerate anchoring, depending on the structure being studied, and in our simulations we implement this as a hard constraint, corresponding to an infinite energy penalty for deviating from the desired anchoring. A close similarity exists between the textures of cholesteric liquid crystals and those of chiral ferromagnets [8,40,41]; indeed (5) is exactly the energy of a ferromagnet with the Dzyaloshinskii-Moriya interaction. Thus, our results in this paper may also be applied to chiral ferromagnets. ...
... A simulation of the resulting structure is shown in figure 7(b). The λ +1 -lines that result from the escape can be extracted from the tensor ∆, the anisotropic part of the director gradients [8,50,51]. This tensor is defined by breaking the gradient tensor ∇n into the gradients ∇ n n along the director, and the gradients ∇ ⊥ n along directions orthogonal to the director. ...
Integer winding disclinations are unstable in a nematic and are removed by an ‘escape into the third dimension’, resulting in a non-singular texture. This process is frustrated in a cholesteric material due to the requirement of maintaining a uniform handedness and instead results in the formation of strings of point defects, as well as complex three-dimensional solitons such as heliknotons that consist of linked dislocations. We give a complete description of this frustration using methods of contact topology. Furthermore, we describe how this frustration can be exploited to stabilise regions of the material where the handedness differs from the preferred handedness. These ‘twist solitons’ are stable in numerical simulation and are a new form of topological defect in cholesteric materials that have not previously been studied.
... NLC fluctuation modes are important generators of different structural deformations [46,[66][67][68][69][70][71][72]. Recent theoretical analysis by Selinger et al. [69,70] revealed that four independent modes exist in the NLC if its structure is described solely by . ...
... NLC fluctuation modes are important generators of different structural deformations [46,[66][67][68][69][70][71][72]. Recent theoretical analysis by Selinger et al. [69,70] revealed that four independent modes exist in the NLC if its structure is described solely byn. ...
We consider the influence of different nanoparticles or micrometre-scale colloidal objects, which we commonly refer to as particles, on liquid crystalline (LC) orientational order in essentially spatially homogeneous particle–LC mixtures. We first illustrate the effects of coupling a single particle with the surrounding nematic molecular field. A particle could either act as a “dilution”, i.e., weakly distorting local effective orientational field, or as a source of strong distortions. In the strong anchoring limit, particles could effectively act as topological point defects, whose topological charge q depends on particle topology. The most common particles exhibit spherical topology and consequently act as q = 1 monopoles. Depending on the particle’s geometry, these effective monopoles could locally induce either point-like or line-like defects in the surrounding LC host so that the total topological charge of the system equals zero. The resulting system’s configuration is topologically equivalent to a crystal-like array of monopole defects with alternating topological charges. Such configurations could be trapped in metastable or stable configurations, where the history of the sample determines a configuration selection.
... Some general references include [38][39][40][41]. Our style of presentation is similar to analogous analysis of director gradients in liquid crystals [70][71][72]. ...
... The eigenvectors of J · D are the directions E ± = 1 √ 2 (E 1 ± E 2 ), making an angle ± π 4 with the principal curvature directions. They may be called directions of "principal torsion" [70] as for curves in the surface along these directions the Darboux torsion is extremal. ...
Living systems are chiral on multiple scales, from constituent biopolymers to large scale morphology, and their active mechanics is both driven by chiral components and serves to generate chiral morphologies. We describe the mechanics of active fluid membranes in coordinate-free form, with focus on chiral contributions to the stress. These generate geometric “odd elastic” forces in response to mean curvature gradients but directed perpendicularly. As a result, they induce tangential membrane flows that circulate around maxima and minima of membrane curvature. When the normal viscous force amplifies perturbations the membrane shape can become linearly unstable giving rise to shape instabilities controlled by an active Scriven-Love number. We describe examples for spheroids, membranes tubes, and helicoids, discussing the relevance and predictions such examples make for a variety of biological systems from the subcellular to tissue level.
... Selinger [101], extending earlier work [71], suggested a new interpretation of the elastic modes for nematic liquid crystals described by the Oseen-Frank elastic free energy, which penalizes in a quadratic fashion the distortions of n away from any uniform state. The Oseen-Frank energy density W OF is defined as (see, e.g., [31,Chap. ...
In its most restrictive definition, an octupolar tensor is a fully symmetric traceless third-rank tensor in three space dimensions. So great a body of works have been devoted to this specific class of tensors and their physical applications that a review would perhaps be welcome by a number of students. Here, we endeavour to place octupolar tensors into a broader perspective, considering non-vanishing traces and non-fully symmetric tensors as well. A number of general concepts are recalled and applied to either octupolar and higher-rank tensors. As a tool to navigate the diversity of scenarios we envision, we introduce the octupolar potential, a scalar-valued function which can easily be given an instructive geometrical representation. Physical applications are plenty; those to liquid crystal science play a major role here, as they were the original motivation for our interest in the topic of this review.
... Topological phenomena in physics have attracted persistent attention since Lord Kelvin's vortex-atom theory in 1869 [1], which hypothesized atoms to be knotted vortices in the ether that permeates throughout the cosmos. Since then, interesting topological structures have been discovered in various other contexts: the topologically non-trivial gauge field theory of the Dirac monopole [2], topological defects [3, 4] that appear in various ordered systems such as superfluids [5], Bose-Einstein condensates (BEC) [6,7], liquid crystals [8][9][10][11], and the vacuum structures of quantum field theories [12][13][14], as well as topological phases of matter such as topological superconductors [15] and topological insulators [16]. Interestingly, such phenomena are amenable to investigation by numerical [17,18], and experimental means [19][20][21][22], resulting in a discipline in which practical experiments and numerical simulations meet methods of abstract topology. ...
Ordered media often support vortex structures with intriguing topological properties. Here, we investigate non-Abelian vortices in tetrahedral order, which appear in the cyclic phase of spin-2 Bose--Einstein condensates and in the tetrahedratic phase of bent-core nematic liquid crystals. Using these vortices, we construct topologically protected knots in the sense that they cannot decay into unlinked simple loop defects through vortex crossings and reconnections without destroying the phase. The discovered structures are the first examples of knots bearing such topological protection in a known experimentally realizable system.
... 7 In particular, K 22 − K 24 < 0 would promote a double twist deformation, as in blue phases. 9 In a recent reformulation of nematic elasticity, 10 based on a mathematical construction proposed in Ref. 11, the director gradients are decomposed into four modes: in addition to splay, twist, and bend, there is a fourth mode, denoted as Δ, which is directly related to K 24 . Thus, a form of the free energy density is proposed, with the saddle-splay contribution explicitly regarded as a bulk term. ...
The elastic behavior of nematics is commonly described in terms of the three so-called bulk deformation modes, i.e., splay, twist, and bend. However, the elastic free energy contains also other terms, often denoted as saddle-splay and splay-bend, which contribute, for instance, in confined systems. The role of such terms is controversial, partly because of the difficulty of their experimental determination. The saddle-splay (K24) and splay-bend (K13) elastic constants remain elusive also for theories; indeed, even the possibility of obtaining unambiguous microscopic expressions for these quantities has been questioned. Here, within the framework of Onsager theory with Parsons-Lee correction, we obtain microscopic estimates of the deformation free energy density of hard rod nematics in the presence of different director deformations. In the limit of a slowly changing director, these are directly compared with the macroscopic elastic free energy density. Within the same framework, we derive also closed microscopic expressions for all elastic coefficients of rodlike nematics. We find that the saddle-splay constant K24 is larger than both K11 and K22 over a wide range of particle lengths and densities. Moreover, the K13 contribution comes out to be crucial for the consistency of the results obtained from the analysis of the microscopic deformation free energy density calculated for variants of the splay deformation.
... Some general references include [36][37][38][39]. Our style of presentation is similar to analogous analysis of director gradients in liquid crystals [66][67][68]. ...
... The eigenvectors of J·D are the directions E ± = 1 √ 2 (E 1 ±E 2 ), making an angle ± π 4 with the principal curvature directions. They may be called directions of 'principal torsion' [66] as for curves in the surface along these directions the Darboux torsion is extremal. ...
Living systems are chiral on multiple scales, from constituent biopolymers to large scale morphology, and their active mechanics is both driven by chiral components and serves to generate chiral morphologies. We describe the mechanics of active fluid membranes in coordinate-free form, with focus on chiral contributions to the stress. These generate geometric `odd elastic' forces in response to mean curvature gradients but directed perpendicularly. As a result, they induce tangential membrane flows that circulate around maxima and minima of membrane curvature. When the normal viscous force amplifies perturbations the membrane shape can become linearly unstable giving rise to shape instabilities controlled by an active Scriven-Love number. We describe examples for spheroids, membranes tubes and helicoids, discussing the relevance and predictions such examples make for a variety of biological systems from the sub-cellular to tissue level.
... To appreciate this better, we recall that according to the decomposition of ∇n proposed by [16] and reprised and reinterpreted by [17], ...
Chromonic nematics are lyotropic liquid crystals that have already been known for half a century, but have only recently raised interest for their potential applications in life sciences. Determining elastic constants and anchoring strengths for rigid substrates has thus become a priority in the characterization of these materials. Here, we present a method to determine chromonics' planar anchoring strength. We call it geometric as it is based on recognition and fitting of the stable equilibrium shapes of droplets surrounded by the isotropic phase in a thin cell with plates enforcing parallel alignments of the nematic director. We apply our method to shapes observed in experiments; they resemble elongated rods with round ends, which are called batonnets. Our theory also predicts other droplets' equilibrium shapes, which are either slender and round, called discoids, or slender and pointed, called tactoids. In particular, sufficiently small droplets are expected to display shape bistability, with two equilibrium shapes, one tactoid and one discoid, exchanging roles as stable and metastable shapes upon varying their common area.
... Here this goal is achieved by means of a decomposition of ∇n first proposed in [7] and then reprised and reinterpreted in [8], where the main players are the splay scalar S := div n, the twist pseudoscalar T := n · curl n and the bend vector b := n × curl n: ...
... The solution of (24) is uniquely defined only at those points in the plane belonging precisely to a single characteristic line. Since the slope of a characteristic line (28) depends on b * (which is constant) and φ 0 (which propagates unchanged on the entire line), the unique solution filling the whole plane must have φ constant, and so it generates thorough (20) a constant director field n. 7 To construct a genuine planar quasi-uniform distortion, we must therefore prescribe a nonconstant φ 0 on the straight line y = 0, so that there are no intersections of characteristic lines in the half-plane y > 0. It follows from direct inspection of (28) that the only way to achieve this goal is to ensure that the slope along the line y = 0, ...
... 6 A classical reference is chapter 2 of [12]; for the related Lagrange-Charpit method, the reader is also referred to [13]. 7 In three space dimensions, general compatibility conditions for the function f were given in [4] for a regular quasiuniform distortion to fill the whole space (see their equation (48)). Examples were also given to show that this class is not empty. ...
Frustration in nematic-ordered media (endowed with a director field) is treated in a purely geometric fashion in a flat, two-dimensional space. We recall the definition of quasi-uniform distortions and envision these as viable ways to relieve director fields prescribed on either a straight line or the unit circle. We prove that using a planar spiral is the only way to fill the whole plane with a quasi-uniform distortion. Apart from that, all relieving quasi-uniform distortions can at most be defined in a half-plane; however, in a generic sense, they are all asymptotically spirals.
... More subtly, it does not build in a consistent handedness (sense of twist). More recent geometric approaches take the pitch axis to be derived from the director gradients [28][29][30] and constrain the handedness by adopting methods of contact geometry [19,[21][22][23][24] and it is this approach that we employ here. We give a complete classification of the local structure of disclination lines in cholesterics. ...
We give a complete topological classification of defect lines in cholesteric liquid crystals using methods from contact topology. By focusing on the role played by the chirality of the material, we demonstrate a fundamental distinction between “tight” and “overtwisted” disclination lines not detected by standard homotopy theory arguments. The classification of overtwisted lines is the same as nematics, however, we show that tight disclinations possess a topological layer number that is conserved as long as the twist is nonvanishing. Finally, we observe that chirality frustrates the escape of removable defect lines, and explain how this frustration underlies the formation of several structures observed in experiments.
