Wedge Disclination Research Articles

Twist disclinations mediated transformations in confined nematic liquid crystals

We studied numerically structural transformations in different confined nematic liquid crystals (LC) mediated by chargeless (twist) disclinations. Firstly, we focus on the role of twist disclinations in Moiré-type geometrical setup where LC is confined in a plane-parallel cell. We impose an azimuthal rotation to initially parallel wedge disclinations where the total topological charge of the system equals to zero. Two qualitatively different scenarii are observed depending on the cell thickness. In thin enough cells the disclinations touch in the mid-plane which enables continuous transformation into a final structure consisting of two twist disclinations confined to the limiting substrates. On the other hand, in thinner cells an additional chargeless loop is formed in the mid-plane. In this case, on rotating the initially parallel disclinations periodic structural reentrant behavior is observed. In cylindrical geometry we considered transformations between the initial metastable escaped radial structure with point defects (ERPD) into escaped radial (ER) or planar polar structure with line defects (PPLD). In both cases two nearby ring-like point defects within the ERPD structure collide and temporarily form a chargeless twist disclination loop. The latter in the ERPD-ER transformation shrinks and enables continuous transformation into the final thermodynamically stable structure. On the contrary, in the ERPD-PPLD transformation the twist loop, whose local structure matches PPLD configuration, increases and converts the system continuously into the global PPLD configuration. Detailed understanding of chargeless defect mediated transformations is beneficial for development of future switching devices based on continuous nematic structural transformations between LC structures exhibiting significantly different optical features.

EQUILIBRIUM OF A NORMALLY LOADED ELASTIC SHALLOW CYLINDRICAL SHELL WITH DISLOCATIONS AND DISCLINATIONS

We consider the problem of the equilibrium of a normally loaded elastic shallow rectangular circular cylindrical shell, which contains sources of internal stress in the form of fields of continuously distributed edge dislocations and wedge disclinations or other sources, for example, thermal ones. Based on the modified system of Karman equations for the equilibrium of an elastic plate with dislocations and disclinations, a system of nonlinear equilibrium equations for the shell was constructed. The resulting system of equations differs from the Karman system of equilibrium equations for an elastic shallow cylindrical shell under the action of a normal load by the presence on the right side of the continuity equation of a nonzero function, called the incompatibility function, which is expressed through the densities of edge dislocations and wedge disclinations. The edges of the shell are freely clamped or hinged. In the absence of internal stress sources, this system transforms into a system of equilibrium equations for a shallow shell, and in the case of an infinitely large radius of curvature (and the presence of internal stress sources) into a modified system of Karman equilibrium equations for a flexible elastic plate with dislocations and disclinations. To solve the system of nonlinear shell equilibrium equations, the Newton – Kantorovich method in combination with the difference method is proposed. For the case of small values of the normal load and small values of the incompatibility function, linearization is carried out with respect to the deflection and stress function. As a result, a linear boundary value problem was obtained in the form of a system of differential equations for the deflection and the stress function, which is solved by the difference method. The solution of the linearized system is used as an initial approximation in the implementation of the Newton – Kantorovich method. Examples of numerical calculations of deflection and stress function for given values of normal load and incompatibility function are given, and corresponding graphs are constructed.

ON THE INFLUENCE OF NON-EQUILIBRIUM VACANCIES ON THE CHARACTERISTICS OF STRAIN INDUCED BROKEN DISLOCATION BOUNDARIES

The influence of material supersaturation with deformation vacancies on the characteristics of broken dislocation boundaries that appear near wedge disclinations as a result of accommodative plastic deformation is considered. An analysis is made for the elastic and osmotic forces acting on the dislocations of the broken boundary. Using discrete dislocation dynamics method, computer simulation of the nonconservative motion (climb) of dislocations in the boundary plane has been carried out. The broken boundary formed as a result of the splitting of the original disclination into two partial with equal strengths and the boundary with an inhomogeneous distribution of dislocations (with a linearly decreasing misorientation) were considered as initial ones. It is shown that in the case of broken dislocation boundary formed near a positive wedge disclination, dislocation climb leads to its unlimited growth through the model grain with a simultaneous decrease of misorientation. In the case of a boundary formed near a negative disclination, the osmotic forces acting on the dislocations of the broken boundary can, at sufficiently large super saturations of the material with nonequilibrium vacancies, lead to its significant compression with a simultaneous increase of the density of the Burgers vector. In this case, the choice of the initial configuration of the broken boundary has practically no effect on the characteristics of the equilibrium distribution of dislocations. The density distributions of the Burgers vector along the broken boundary are calculated for the case of small and large supersaturations of the material with nonequilibrium vacancies. The critical value of supersaturation above which the boundary compression occurs is calculated. The boundary compression leads to a significant increase in tensile stresses in its vicinity. In this case, the maximum tensile stresses in the plane coinciding with the plane of the boundary are reached at the boundary break point. The results obtained may be of interest in the analysis of possible mechanisms for the initiation of cracks and pores during ductile fracture of metals.

Pair interaction of intersecting dilatation and disclination defects

An elastic interaction of the intersecting dilatation and disclination defects located in an infinite linear isotropic media is investigated. The eigenstrain approach is employed to obtain the analytical expressions describing the pair interaction between intersecting dilatational lines and intersecting wedge disclinations. It is demonstrated that the interaction energy strongly depends on the intersection angle between the defects. The energy reaches the maximum value if the defect lines are coincided while the energy reaches the minimum value if the defect lines are orthogonal. Besides, it is shown that interaction energy of intersecting wedge disclinations strongly depends on the elastic properties of the media: the less thePoisson ratio, the less the energy. The obtained analytical results seem to be applicable for the theoretical analysis of the residual stress relaxation mechanisms in heterostructures with pentagonal symmetry such as icosahedral particles

Mechanics and Energetics of Kink Deformation Studied by Nonlinear Continuum Mechanics Based on Differential Geometry

This study conducts dislocation-based modeling and numerical analysis for kink deformation using the theory of nonlinear continuum mechanics based on differential geometry. The kinematics of the continuum is represented by the reference, intermediate, and current states, which are related to the multiplicative decomposition of the deformation gradient into plasticity and elasticity. The intermediate state is determined by the Cartan first structure equation, whereas the current state is obtained from the stress equilibrium equation. Kink deformation is modeled by a planar array of edge dislocations, and the governing equations are solved numerically using the finite element method. Quantitative verification of the model is conducted by comparing the bending angle at a kink interface and the theoretical prediction given for the tilt grain boundary. We construct two types of ortho-type kink models for several interface lengths to understand the energetics and mechanics of the kink growth process. The present numerical analysis demonstrates the significant stress concentration at the growth front due to the formation of wedge disclination. We also discuss the material strengthening mechanism from elastic strain energy, nonsingular stress fields, and nonlinear elastic interaction between the disclinations.

On the nucleation of cracks near stress sources with weak singularities

Analytical expressions are obtained for the configuration force and the value of elastic energy relaxation at the nucleation of a microcrack in a small neighborhood of an arbitrary singular stress source. When analyzing the conditions of crack nucleation at sources with weak divergences of the stress fields, ideas about the instantaneous nucleation of a crack of finite length are used. Simultaneous fulfillment of stress and energy conditions is considered as a criterion for the nucleation of such a crack. Within the framework of these representations, in the configuration space of the system parameters (geometric characteristics and strength of mesodefects, the value of external stress), the regions in which crack nucleation is possible in the case of a combined mesodefect, which is a superposition of the dipole of wedge disclinations and planar shear mesodefect, are determined. It is shown that the nucleation of the crack is significantly facilitated by the instability of the shear mesodefect.

Zero-energy states in graphene quantum dot with wedge disclination

We show that wedge disclination affects the density of states for fermions that are electrostatically confined in graphene quantum dots. By considering a magnetic flux and solving the Dirac equation, we determine the eigenspinors. These are used for large arguments to derive an approximate scattering matrix, which allows us to get the density of states. Under various conditions, it is found that the density of states exhibits several resonance peaks. In particular, it is shown that the wedge disclination is able to change the amplitude, width, and positions of resonance peaks.

Revisiting the electronic properties of disclinated graphene sheets

The interplay between topological defects, such as dislocations or disclinations, and the electronic degrees of freedom in graphene have been extensively studied. In the literature, for the study of this kind of problems, it is in general used either a gauge theory or a curved spatial Riemannian geometry approach, where, in the geometric case, the information about the defects is contained in the metric and the spin-connection. However, these topological defects can also be associated with a Riemann–Cartan geometry where curvature and torsion plays an important role. In this article, we study the interplay between a wedge disclination in a planar graphene sheet and the properties of its electronic degrees of freedom. Our approach relies on its relation with elasticity theory through the so called elastic-gauge, where their typical coefficients, as for example the Poisson’s ratio, appear directly in the metric, and consequently also in the electronic spectrum.

Reconfiguration of Nematic Disclinations in Plane-Parallel Confinements

We study numerically the reconfiguration process of colliding m=1/2 strength disclinations in an achiral nematic liquid crystal (NLC). A Landau–de Gennes approach in terms of tensor nematic-order parameters is used. Initially, different pairs m1,m2 of parallel wedge disclination lines connecting opposite substrates confining the NLC in a plane-parallel cell of a thickness h are imposed: {1/2,1/2}, {−1/2,−1/2} and {−1/2,1/2}. The collisions are imposed by the relative rotation of the azimuthal angle θ of the substrates that strongly pin the defect end points. Pairs {1/2,1/2} and {−1/2,−1/2} “rewire” at the critical angle θc1=3π4 in all cases studied. On the other hand, two qualitatively different scenarios are observed for {−1/2,1/2}. In the thinner film regime h<hc, the disclinations rewire at θc2=5π4. The rewiring process is mediated by an additional chargeless loop nucleated in the middle of the cell. In the regime h>hc, the colliding disclinations at θc2 reconfigure into boojum-like twist disclinations.

Wedge disclination description of emergent core-shifted grain boundaries at free surfaces

Emergent grain boundaries at free surface control material properties such as nanomaterial strength, catalysis, and corrosion. Recently the restructuring of emergent boundaries on copper (111) surfaces was discovered experimentally and atomic calculations point to its universality in fcc metal systems. Restructuring is due to a preference for boundaries to shift their tilt axis across the (11¯0) plane towards [112] and ultimately to form low energy [112] core shifted boundaries (CSBs). However, the observed geometry of these emergent boundaries is not reproduced by atomic calculations and the driving force is still controversial due to inconsistencies between the computational continuum analysis and atomic calculations. Here, using atomic calculations that involve a methodical shift of the dislocation core, we confirmed the geometry of emergent boundaries observed in experiment and reconciled the atomic calculations with the elastic analysis through the inclusion of a straight wedge disclination at the free surface.

Numerical Analysis of Disclinations in Connecting Kink Bands Formed by Multiple Basal Shear

We analyzed the kinematic (geometric) aspects of kink bands with multiple basal shear using rank-1 connection to investigate the type of disclination and annihilation of disclinations in kink microstructures. We found that wedge disclinations occur in connecting kink bands for realistic magnitude of shear that can occur, regardless of the shear direction. The normal vector of the junction plane between kink bands and Frank vector of the resulting wedge-disclination varied continuously with respect to the changes in the magnitude and direction of the basal shear. Annihilation of disclinations is possible even between the kinks which are formed by multiple basal shears. Kinematical model of the three-dimensional wavy kinks are proposed using the obtained results.

RAMIFICATION OF EQUILIBRIA OF A COMPRESSED ELASTIC ORTHOTROPIC PLATE WITH INTERNAL STRESSES

The problem of loss of stability and post-critical behavior of a compressed orthotropic rectangular plate on a nonlinearly elastic base containing continuously distributed fields of dislocations and disclinations and under the influence of a small normal load is considered. The compressive force components are evenly distributed along the edges and act parallel to the main directions of elasticity. The problem is formulated as an analogue of a system of nonlinear Karman equations for an orthotropic plate containing a function called the incompatibility measure, which is expressed in terms of the densities of edge dislocations and wedge disclinations. The system of equations takes into account the small transverse pressure and the reaction of the elastic base in the form of a polynomial of the third degree of deflection. The following boundary conditions are considered: the edges of the plate are freely pinched or movably pivotally supported; two opposite edges of the plate are freely pinched or movably pivotally supported, and the other two are free from loads. The stress function is sought in the form of two components: the stress function caused by the presence of internal sources, determined from the linear boundary value problem, and the stress function caused by the external impact of compressive loads and a nonlinear elastic base, which is determined from the nonlinear boundary value problem. The nonlinear boundary value problem is investigated by the Lyapunov – Schmidt method. To solve the linearized equation from which the critical value of the compressive load is determined, the variational method is used in combination with the difference method. A system of branching equations of the Lyapunov – Schmidt method is constructed, which is investigated by numerical methods. The post-critical behavior of the plate is investigated and asymptotic formulas for new equilibria in the vicinity of the critical load are derived. For various values of compressive loads, the orthotropy parameters of the plate and the intensity parameter of internal stresses, the relations between the values of the base parameters are established, at which its bearing capacity is preserved in the vicinity of the classical value of the critical load.

Quantum tunneling in graphene Corbino disk in a solenoid magnetic potential with wedge disclination

We investigate the wedge disclination effect on quantum tunneling of a Corbino disk in gapped-graphene of inner R1 and outer R2 radii in the presence of magnetic flux Φi. We solve Dirac equation for different regions and obtain the solutions of energy spectrum in terms of Hankel functions. The asymptotic behaviors for large arguments allow us to determine the transmission, Fano factor and conductance. We establish the case where the crystal symmetry is modified locally by replacing a hexagon by pentagon, square, heptagon or octagon. We show that the wedge disclination n modifies the amplitude of transmission oscillations. We find that the period of Fano factor oscillations is of the Aharonov–Bohm type, which strongly depends on n where intense peaks are observed. As another result, n changes the minimum and period of conductance oscillations of the Aharonov–Bohm type. We show that n minimizes the effect of resonance and decreases the amplitude of conductance magnitude ΔG oscillations.

Quantum Tunneling in Graphene Corbino Disk in a Solenoid Magnetic Potential with Wedge Disclination

Quantum Tunneling in Graphene Corbino Disk in a Solenoid Magnetic Potential with Wedge Disclination

Zero-Energy States in Graphene Quantum Dot with Wedge Disclination

Zero-Energy States in Graphene Quantum Dot with Wedge Disclination

Zero-Energy States in Graphene Quantum Dot with Wedge Disclination

Zero-Energy States in Graphene Quantum Dot with Wedge Disclination

Positive disclination in a thin elastic sheet with boundary.

An isolated positive wedge disclination deforms an initially flat elastic sheet into a perfect cone when the sheet is of infinite extent and is elastically inextensible. The latter requires the elastic stretching strains to be vanishingly small. In this paper, rigorous analytical and numerical results are obtained for the disclination-induced deformed shape and stress field of a bounded Föppl-von Kármán elastic sheet with finite extensibility, while emphasizing the deviations from the perfect cone solution. In particular, the Gaussian curvature field is no longer localized as a Dirac singularity at the defect location whenever elastic extensibility is allowed and is necessarily negative in large regions away from the defect. The stress field, similarly, has no Dirac singularity in the presence of elastic extensibility. However, with increasing Young's modulus of the sheet, while keeping the bending modulus and the domain size fixed, both of these fields tend to develop a Dirac singularity. Noticeably, in this limiting behavior, inextensibility eludes the bounded elastic sheet due to persisting regions of nontrivial Gaussian curvature away from the defect. Other results in the paper include studying the effect of specific boundary conditions (free, simply supported, or partially clamped) on the Gaussian curvature field away from the defect and on the buckling transition from the flat to a conical solution.

On the Effect of External Stress on the Stability of a Crack Located near a Wedge Disclination Dipole

On the Effect of External Stress on the Stability of a Crack Located near a Wedge Disclination Dipole

Ionically Charged Topological Defects in Nematic Fluids

Charge profiles in liquid electrolytes are of crucial importance for applications, such as supercapacitors, fuel cells, batteries, or the self-assembly of particles in colloidal or biological settings. However, creating localised (screened) charge profiles in the bulk of such electrolytes, generally requires the presence of surfaces -- for example, provided by colloidal particles or outer surfaces of the material -- which poses a fundamental constraint on the material design. Here, we show that topological defects in nematic electrolytes can perform as regions for local charge separation, forming charged defect cores and in some geometries even electric multilayers, as opposed to the electric double layers found in isotropic electrolytes. Using a Landau-de Gennes-Poisson-Boltzmann theoretical framework, we show that ions highly effectively couple with the topological defect cores via ion solvability, and with the local director-field distortions of the defects via flexoelectricity. The defect charging is shown for different defect types -- lines, points, and walls -- using geometries of ionically screened flat isotropic-nematic interfaces, radial hedgehog point defects and half-integer wedge disclinations in the bulk and as stabilised by (charged) colloidal particles. More generally, our findings are relevant for possible applications where topological defects act as diffuse ionic capacitors or as ionic charge carriers.

Elastic fields of a wedge disclination in a functionally graded cylinder

In this paper, the elastic fields of a wedge disclination in a functionally graded cylinder are obtained in closed-form by directly solving the equilibrium equations. The cylinder material is graded radially according to a power law characterized by a gradient index. The disclination has a small fixed core of nanometer dimension. The results show that: (1) the displacement, radial and transverse stresses vary with the radial coordinate according to a sum of power laws, (2) the gradient index changes the elastic fields through modification of the elastic parameters, and modification of the power law indices of the field dependence on (i) the radial coordinate and (ii) the cylinder dimensions, (3) the stresses at and near the disclination core are generally reduced by both positive and negative gradients, (4) the transverse stress at the disclination core can be reduced to zero and even compressive for small negative gradients, (5) a significant size effect exists for the disclination stresses, and the effect is influenced strongly by the gradient index, (6) the energy of the disclination is a strong, non-monotonic function of the gradient. These results suggest that various relaxation phenomena such as crack nucleation and disclination splitting may be controlled by selecting a suitable gradient for the material variation.