Contribution of Disclination Lines to Free Energy of Liquid Crystals in Single-Elastic Constant Approximation

In the light of C-mapping method and topological current theory, the contribution of disclination lines to free energy density of liquid crystals is studied in the single-elastic constant approximation. It is pointed out that the total free energy density can be divided into two parts. One is the usual distorted energy density of director field around the disclination lines. The other is the free energy density of disclination lines themselves, which is shown to be centralized at the disclination lines and to be topologically quantized in the unit of kn/2. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director l~eld at the disclination lines, i.e. the disclination strengths. From the Lagrange‘s method of multipliers, the equilibrium equation and the molecular field ofliquid crystals are also obtained. The physical meaning of the Lagrangian multiplier is just the distorted energy density.     

Effect of Disclination Lines on Free Energy of Nematic Liquid Crystals

In the light of φ-mapping method-and topological current theory, the effect of disclination lines on the free energy density of nematic liquid crystals is studied. It is pointed out that the total Frank free energy density can be divided into two parts. One is the distorted energy density of director field around the disclination lines. The other is the saddle-splay energy density, which is shown to be centralized at the disclination lines and to he topologically quantized in the unit of kπ/2 when the Jacobian determinant of the director field does not vanish at the singularities of the director field. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, i.e., the disclination strengthes. When the Jacobian determinant vanishes, the generation, annihilation, intersection, splitting and merging processes of the saddle-splay energy density are detailed in the neighborhoods of the limit points and bifurcation points, respectively. It is shown that the disclination line with high topological quantum number is unstable and will evolve to the low topological quantum number states through the splitting process.     

Line defect dynamics around a colloidal particle

We study experimentally the dynamics of a topological defect located around a colloidal particle suspended in a thermotropic nematic liquid crystal. The considered defect consists of a disclination loop encircling the particle at the equator. Under specific conditions, it is shown that this disclination continuously shrinks to a hedgehog defect located in the immediate vicinity of the particle. This phenomenon corresponds to a transition between an elastic quadrupolar configuration and an elastic dipolar configuration. We performed a basic numerical calculation to get an estimate of the dissipated energy during the transition; we compare the results with theoretical predictions that describe the elastic energy of particles surrounded by defects. Received 21 December 2001     

Misfit dislocations in thin films on plastically deformed substrates

A theoretical model describing the nucleation of misfit dislocations (MD) in interfaces between films and plastically deformed substrates with disclinations is proposed. The ranges of the parameters (disclination strength, density of the disclination ensemble, film thickness, and degree of misfit) within which MD nucleation is energetically favorable are found. It is shown that at certain strengths of disclinations and densities of their ensemble the critical thickness of the film on a plastically deformed substrate with disclinations can exceed that on an undeformed defect-free substrate by a few times.     

Arrays of Charged (±2π)-Twist Disclinations in Ferroelectric Liquid Crystals

The notion of an electrostatic charge of (±2)-twist disclinations is used to approximate the evaluation of the electrostatic interaction energy among disclinations forming arrays in finite samples of ferroelectric chiral smectic C liquid crystals. Screening effects of free charges in a material surrounding the disclination are taken into account by introducing a phenomenological depolarisation factor.The electrostatic interaction energy is important in chiral smectic C materials with high values of the spontaneous polarisation when screening effects of free charges are small. Then the electrostatic interaction leads to elimination of disclinations from the sample. When there is a high concentration of free charges in the sample (smaller value of depolarisation factor), the electrostatic interaction energy is of the order of the elastic interaction energy of disclinations what influences the equilibrium of disclination arrays in the sample. Two disclination configurations are considered. In the Brunet-Williams configuration the disclinations of opposite topological charge have also the opposite electrostatic charge so their attraction is augmented. This attraction can be balanced by the helical structure in the central part of the sample when the sample thickness is rather high.On the contrary, in the Glogarová-Pavel configuration the disclinations of opposite topological charge have the electrostatic charge of the same sign. The equilibrium in this configuration is either a balance of elastic attraction and electrostatic repulsion if elastic and Coulomb forces are of the same order or it is governed by the value of the anchoring energy when electrostatic interaction prevails over the elastic one.     

Integer quantum Hall effect on an interface with disclinations

The Hall conductivity of an electron gas on an interface showing topological defects as disclinations in the presence of an orthogonal constant magnetic field is investigated. This kind of defect induces either positive or negative singular curvature in the medium. It is shown that the positive curvature decreases the quantum Hall plateau widths and shifts the steps in the Hall conductivity to lower magnetic fields. In contrast, the negative one leaves to the existence of two types of plateaus, one with higher widths and the other one with lower widths in comparison to the flat case. In this case, the shift in the steps of the Hall conductivity goes to higher magnetic fields. We also investigate the Hall conductivity for electrons around a cylindrically symmetric distribution of disclinations and it turns out that it is the same as that corresponding to a single effective disclination.     

Fields of stress and electric displacement produced by dislocations and disclinations in three-dimensional,anisotropic, elastic,and piezoelectric media. II. Infinite straight dislocation and Frank disclination

The fields of stress and electric displacement caused by infinitely extended straight dislocations and Frank disclinations are deduced from the author's statements for the fields caused by a continuous distribution of dislocations and disclinations (S. Minagawa, Phil. Mag. 84 2229 (2004)). The multiple integrals in the original statements are converted into functions of space coordinates. Cauchy's theorem plays an important part. The improper integral that appears in computations of the fields around a Frank disclination is interpreted as its finite part by Hadamard. Examples are the fields around an infinite straight defect in caesium copper chloride, as well as those in gallium arsenide. The contours and zero lines are plotted to illustrate the fields caused by a dislocation and a disclination dipole.     

A disclination-based approach to describing the structure of fullerenes

The possibility of using a disclination approach to describe the structure and properties of fullerenes is discussed. It is shown that the conversion of a planar carbon monolayer into a spherical macromolecule can be viewed as the result of introducing 12 disclinations with power π/3 into the original layer. Fiz. Tverd. Tela (St. Petersburg) 40, 1178–1180 (June 1998)     

Misfit disclinations at crystal/crystal and crystal/glass interfaces

Theoretical concepts have been developed for a new type of misfit defects, misfit disclinations, at crystal/crystal and crystal/glass interfaces. It is shown, in particular, that the formation of misfit disclinations is an efficient physical micromechanism of misfit stress relaxation at crystal/crystal interfaces. A model describing misfit disclinations at crystal/glass interfaces has been constructed. The energy characteristics of phase boundaries with misfit disclination ensembles are estimated. Fiz. Tverd. Tela (St. Petersburg) 41, 1637–1643 (September 1999)     

Fields of stress and electric displacement produced by dislocations and disclinations in three-dimensional,anisotropic, elastic and piezoelectric media: elliptical Frank disclination

A statement is made on the theory of continuous distributions of dislocations and disclinations in anisotropic elastopiezoelectric media. The basic field equations governing the fields of stress functions, electric vector potential and incompatibility are presented and solved to give the fields of stress and electric displacement caused by a distribution of dislocations and disclinations. They are expressed in terms of the dislocation- and disclination-density tensors by means of the convolution integrals, extended throughout the medium, and the Fourier integrals. To treat the fields around discrete defects, that is dislocation and/or Frank disclination, the convolution integrals are replaced by the line integrals belonging to the loop of the defect. The fields of stress and electric displacement are given in terms of three quadruple integrals, which are converted into single integrals of explicitly given functions, in the case where the loop of the defect is elliptical. Numerical computations are carried out to estimate the fields in gallium arsenide. The values of those fields at a certain point of the body are presented. The contours and zero lines of the fields of dilatational stress and electric displacement in the plane placed parallel to and at a certain distance from the loop are illustrated.     

Brownian motion in crystals with topological defects

The diffusion behaviour of a Brownian particle in a crystal with randomly distributed topological defects is analyzed by means of the continuum theory of defects combined with the theory of diffusion on manifolds. A path-integral representation of the diffusion process is used for the calculation of cumulants of the particle position averaged over a defect ensemble. For a random distribution of disclinations the problem of Brownian motion reduces to a known random-drift problem. Depending on the properties of the disclination ensemble, this yields various types of subdiffusional behaviour. In a random array of parallel screw dislocations one finds a normal, but anisotropic, diffusion behaviour of the mean-square displacement. Moreover, the process turns out to be non-Gaussian, and reveals long-time tails in the higher-order cumulants.Dedicated to Professor Herbert Wagner on the occasion of his 60th birthday     

Topology and geometry of nematic braids

Topological analysis of disclinations in nematic liquid crystals is an interesting and diverse topic that goes from strict mathematical theorems to applications in elaborate systems found in experiments and numerical simulations. The theory of nematic disclinations is shown from both the geometric and topological perspectives. Entangled disclination line networks are analyzed based on their shape and the behavior of their cross section. Methods of differential geometry are applied to derive topological results from reduced geometric information. For nematic braids, systems of −1/2 disclination loops, created by inclusion of homeotropic colloidal particles, a formalism of rewiring is constructed, allowing comparison and construction of an entire set of different conformations. The disclination lines are described as ribbons and a new topological invariant, the self-linking number, is introduced. The analysis is generalized from a constant −1/2 profile to general profile variations, while retaining the geometric treatment. The workings of presented topological statements are demonstrated on simple models of entangled nematic colloids, estimating the margins of theoretical assumptions made in the formal derivations, and reviewing the behavior of the disclinations not only under topological, but also under free-energy driven constraints.     

Point defects and ring defects in a nematic liquid crystal in a cylindrical capillary

The bidirectional escape into the third dimension of a linear disclination of strength m=1 (L ₊₁ ^p in a cylindrical capillary with normal boundary conditions is investigated. It is shown that in this case two types of defects arise in the capillary: point defects and ring defects, each of which can be of the radial or hyperbolic type. Exact solutions are obtained for the equation of equilibrium of the elastic field. The free energy of the point and ring defects is calculated approximately in a narrow, long capillary. New scenarios are proposed for the escape of the disclination L ₊₁ ^p . Zh. Tekh. Fiz. 69, 29–32 (July 1999)     

Novel defect structures in nematic liquid crystal shells

We use double-emulsion drops to experimentally investigate the defect structures of spherical shells of nematic liquid crystals. We uncover a rich scenario of coexisting defect structures dictated by the unavoidable finite thickness of even the thinnest shell and by the thickness variation around the sphere. These structures are characterized by a varying number of disclination lines and pairs of surface point defects on the inner and outer surfaces of the nematic shell. In the limit of very thick shells the defect structure ultimately merges with that of a bulk nematic liquid crystal drop.     

Electronic structure of carbon nanoparticles

The electronic structure of graphitic nanoparticles is investigated within a gauge field-theory model. The local and total densities of states (DOS) near the pentagonal defects (disclinations) are calculated for three geometries: sphere, cone, and hyperboloid. It is found that the low-energy electron states have a rather specific dependence on both the energy and the distance from a disclination line. In particular, the low-energy total DOS has a cusp that drops to zero at the Fermi energy for disclinations with the Frank index v<1/2, while a region of a nonzero DOS across the Fermi level is formed for v=1/2. The true zero-mode fermion state is found for the graphitic hyperboloid. The appearance of an enhanced charge density near the Fermi level for nanocones with a 60° opening angle (180° disclination) is predicted.     

Dislocation and disclination loops in the virtual-defect method

A method of virtual circular defect loops is developed for determining the elastic fields produced by defects in a bounded medium in the case of an axially symmetric geometry. In this method, continuously distributed virtual circular Volterra and Somigliana dislocation loops are adjusted in such a way as to satisfy the boundary conditions imposed at free surfaces and interfaces. Original calculations of the elastic fields of circular defect loops of different types are carried out. The elastic fields are found for the case of straight dislocations and disclinations in a plate that are perpendicular to the plate plane and for the case of circular disclination loops parallel to the plate plane or to an interface.     

Grain size refinement due to relaxation of disclination junction configurations in the course of plastic deformation of polycrystals

A model is proposed for the formation of the substructure in polycrystals during plastic deformation. According to this model, fragmentation of a grain occurs through the formation of a system of diagonal low-angle boundaries, which originate at the edges of a rectangular grain. Misorientation boundaries form through relaxation of a nonsymmetric junction quadrupole disclination configuration accumulated at the grain corners under severe deformation when the disclination strength reaches a certain critical value. The energetics of this process is analyzed. A general case is considered where the disclinations at the junctions of the chosen grain differ in strength. The energetic approach used makes it possible to determine the misorientation angle ω_x of the resulting boundaries corresponding to the maximum energy gain and to find the dependence of this angle on the degree of asymmetry of the quadrupole configuration of junction disclinations. According to the proposed model, the splitting of a grain with a short edge greater than 0.5 μm is energetically favorable and decreases the latent energy of the grain for any ratio between the junction disclination strengths if the grain length-to-width ratio is less than 30. It is shown that the minimum possible grain size in the proposed model does not exceed 0.1 μm.     

Defect microstructure in titanium nitride submicrocrystals

Transmission electron microscopy has been applied to the defect structure of titanium nitride in the submicrocrystalline (SMC) state produced under conditions of deviation from equilibrium in ion-plasma synthesis. The submicrocrystals contain a new type of defect substructure having a continuum disclination density up to 2.5 rad/μm². Direct structure methods give evidence for a high density of partial disclinations at the SMC grain boundaries in the nitride phase. A novel method has been used to examine substructures having a high defect density, which has been used to estimate the partial disclination density at the submicrocrystal boundaries. The origin of this highly defective state and the effects of it on the properties of SMC materials is discussed. Kuznetsov Siberian Technical Physics Institute, Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–12, July, 1998.     

Disclinations in nematics in the presence of flexoelectricity

The stability problems of wedge disclinations (with pure splay and bend distortions) in nematic liquid crystals (NLC) are considered in the presence of flexoelectricity. We write NLC director equations in the cylindrical coordinate system, taking into account flexoelectricity. These equations allow us to solve many interesting problems on the influence of flexoelectricity on the orientational structures of NLC. In particular, it is shown that the line of disclination with radial distribution of director can be stabilized in the presence of flexoelectricity even far from nematicsmectic phase transition point. For the disclination with azimuthal distribution of director for all physical sizes the line is not stable and escapes in the third dimension.