Topological aspect of disclinations in two-dimensional crystals

By using topological current theory, this paper studies the inner topological structure of disclinations during the melting of two-dimensional systems. From two-dimensional elasticity theory, it finds that there are topological currents for topological defects in homogeneous equation. The evolution of disclinations is studied, and the branch conditions for generating, annihilating, crossing, splitting and merging of disclinations are given.     

Model of formation of broken dislocation boundaries at joint disclinations

We propose a physical model of formation of broken dislocation boundaries (partial disclinations of deformation origin) at the joints of large-angle grain boundaries. The model explains why and how rotational-type defects are necessarily formed in polycrystals in which plastic deformation at the microscopic level occurs exclusively via translational slips for strains ε > 0.2.     

Effective schrödinger equation for a classical massless dislocation plasma

Summary The statistical behaviour of classical massless excitations finds an increasing importance in the physics of low-dimensional condensed matter. Dislocation and disclination dipole-gases and plasmas play such a relevant role in the theory of 2D melting. Here the equilibrium statistical mechanics of a system of strongly interactingparticles of this type is faced searching for the approximate stationary solution of the multivariate associated Fokker-Planck equation corresponding to zero eigenvalue. The problems, encountered in a preceding paper, involved by the nonhermiticity of Fokker-Planck operator, are evaded, following Risken, building anequivalent many-body Schr?dinger equation. This last is solved self-consistently in a Hartree-like way starting with a free-particleproduct wavefunction in the case of a uniform background whosecharge is of sign opposite to that of theparticles. Unlike thetrue quantum case, here the integral part of the equivalent Hamiltonian operator is not simply Coulomb-like and defines a more difficult novel integrodifferential problem which is solved using a convergence in mean procedure. The author of this paper has agreed to not receive the proofs for correction.     

Power laws in dislocation plasticity

Starting from the idea that plastic flow produces dislocation structures in a state of self-organised criticality, it is shown that one expects power-law relationships between variables. If slip bands are modelled as avalanches of shear with an ellipsoidal shape, slightly tilted from the crystallographic slip plane, limited in size by interaction with secondary slip, the observed exponents of the power laws can be rationalised. In some cases, the constant of proportionality can also be estimated, and found to agree with experiment, even though the detailed mechanism of avalanche formation is not addressed. To analyse creep data and slip-band statistics, it is further assumed that the role of cross-slip is to eliminate screw dislocation dipoles, removing them entirely at stresses found in Stage III of work-hardening. If physical constants, such as the atomic vibration frequency, play a role, the dimensionless power-law relationships do not apply. One then finds creep rates linear in stress and absolute temperature, proportional to the logarithm of time, obeying an equation of state, as observed.     

Elastic constitutive laws for incompatible crystalline media: the contributions of dislocations,disclinations and G-disclinations

Linear higher-grade higher-order elastic constitutive laws for compatible (defect-free) and incompatible (containing crystal line defects) media are presented. In the proposed model, the free energy density of a body subjected to elastic deformation under the action of surface tractions, moments or hyper-traction tensors (second-order tensors whose anti-symmetric part corresponds to moments) has contributions coming from the first two gradients of displacements. Thermodynamic considerations reveal that only the symmetric component of the gradient of elastic displacement, i.e., compatible elastic strain tensor, and the anti-symmetric component of the second gradient of elastic displacement, i.e., compatible third-order elastic curvature tensor, contribute to the free energy density during compatible deformation of the body. The line crystal defect contributions are accounted for by incorporating the incompatible components of elastic strains, curvatures and symmetric 2-distortions as state variables of the free energy density. In particular, the presence of generalized disclinations (G-disclinations) is acknowledged when the medium is subjected to surface hyper-traction tensors having a non-zero symmetric component along with surface-tractions on its boundary. Mechanical dissipation analysis provides for the coupling between the Cauchy stresses and third-order symmetric hyper-stresses. The free energy density and elastic laws for a defect-free and line crystal defected medium are proposed in a linear setting. In the special case of isotropy, the cross terms between elastic strains and curvatures contribute to the free energy density through a single elastic constant. More interestingly, the Cauchy and couple stresses are found to have contributions coming from both, elastic strains and curvatures.     

Screening of the elastic field of disclinations by a dislocation ensemble

A self-consistent dynamics of a dislocation ensemble in the disclination field is analyzed within the kinetic approach. The effective Airy stress functions for the wedge disclination and the disclination dipole with due regard for the screening effect of the system of distributed dislocation charges are determined. The coordinate dependences of the stress tensor components and dislocation charge density for the screened disclination systems are found. The elastic energies of the screened disclination systems are calculated.     

Relaxation of a dislocation in a nanocrystallite

Experimental studies of dislocations in nanoparticles are just beginning. The corresponding theoretical models are still lacking. In this context, the author analyzes relaxation of a dislocation in a nanoparticle. Mechanistically, this process is considered to occur primarily via dislocation drift induced by the stress-related image forces. Elementary dislocation displacements include the formation of a kink at one of the sides of the dislocation line, its diffusion along this line, and annihilation at the opposite side. For this mechanism, the dependence of the time of dislocation disappearance on the nanoparticle size has been identified.     

Free energy change of a dislocation due to a Cottrell atmosphere

The free energy reduction of a dislocation due to a Cottrell atmosphere of solutes is computed using a continuum model. We show that the free energy change is composed of near-core and far-field components. The far-field component can be computed analytically using the linearized theory of solid solutions. Near the core the linearized theory is inaccurate, and the near-core component must be computed numerically. The influence of interactions between solutes in neighbouring lattice sites is also examined using the continuum model. We show that this model is able to reproduce atomistic calculations of the nickel–hydrogen system, predicting hydride formation on dislocations. The formation of these hydrides leads to dramatic reductions in the free energy. Finally, the influence of the free energy change on a dislocation’s line tension is examined by computing the equilibrium shape of a dislocation shear loop and the activation stress for a Frank–Read source using discrete dislocation dynamics.     

Experimental examination of displacement field in an edge dislocation core in aluminum

The displacement field of an edge dislocation in aluminum was experimentally investigated. Three typical theoretical models were discussed. High-resolution transmission electron microscopy (HRTEM) and geometric phase analysis (GPA) were used to map the displacement field of an edge dislocation. The displacement field near the dislocation core was determined. The experimental show that Peierls-Nabarro model is the most appropriate theoretical model for displacement field of dislocation in aluminum.     

A computational approach towards modelling dislocation transmission across phase boundaries

To study the nanoscopic interaction between edge dislocations and a phase boundary within a two-phase microstructure the effect of the phase contrast on the internal stress field due to the dislocations needs to be taken into account. For this purpose a 2D semi-discrete model is proposed in this paper. It consists of two distinct phases, each with its specific material properties, separated by a fully coherent and non-damaging phase boundary. Each phase is modelled as a continuum enriched with a Peierls–Nabarro (PN) dislocation region, confining dislocation motion to a discrete plane, the glide plane. In this paper, a single glide plane perpendicular to and continuous across the phase boundary is considered. Along the glide plane bulk induced shear tractions are balanced by glide plane shear tractions based on the classical PN model. The model's ability to capture dislocation obstruction at phase boundaries, dislocation pile-ups and dislocation transmission is studied. Results show that the phase contrast in material properties (e.g. elastic stiffness, glide plane properties) alone creates a barrier to the motion of dislocations from a soft to a hard phase. The proposed model accounts for the interplay between dislocations, external boundaries and phase boundary and thus represents a suitable tool for studying edge dislocation–phase boundary interaction in two-phase microstructures.     

The two-dimensional Peierls-Nabarro model for interfacial misfit dislocation networks of cubic lattice

A model is developed to investigate the two-dimensional interfacial misfit dislocation networks that follows the original Peierls-Nabarro idea. Structure and energies of heterophase interfaces are considered for the cubic lattice. To examine the energy contribution of misfit dislocations, where interactions between two dislocation arrays are concerned, a generalized stacking fault energy is proposed. Combined with first-principles calculations, we apply this model to a practical metal-ceramic example: the Ag/MgO(100) interface. An important correction to the adhesive energy is proposed in addition to its dislocation structure being confirmed.     

Spin polarization in carbon nanostructures with disclinations

We performed ab initio calculations, using density functional theory, to study spin polarization in carbon nanostructures with disclinations. The results indicate that compounds with positive and negative Gaussian curvature may exhibit a net magnetic moment in the ground state. Additionally, we can conclude that, carbon compounds that display an odd number of pentagons and heptagons, present polarization in the ground state.     

Interaction of an inclusion and (±1)-wedge disclination in smectic C free standing films

The interaction between an inclusion and (±1)-disclination situated near it in smectic C free standing films is investigated. The inclusion is modelled by the simplified model of an inclusion represented by (+1)-wedge disclination in the inclusion centre and two (−1/2)-wedge disclinations at opposite positions on the inclusion surface. New positions of (−1/2)-wedge disclinations due to the elastic interaction with (±1)-disclination were determined.     

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.     

Junction Line Disclinations: Characterisation and Observations

Experimental observations obtained using high resolution transmission electron microscopy (HREM) of junctions between two or more interfaces are analysed to determine whether they exhibit disclination character. The method of analysis used is circuit mapping, and the advantage of using rotation and mirror symmetry operations in the present context, rather than translation operations as is done conventionally, is demonstrated. This technique is applied to micrographs of junctions selected from published literature, and includes junctions in homophase and heterophase materials. It is concluded that few observations currently provide unequivocal evidence of junction line disclination character. This situation may be due in part to the special crystallographic constraints on the applicability of HREM to studies of junction lines. The examples which have been identified all arise at intersections where favourable interfaces, such as epitaxial and twin boundaries, intersect. Moreover, such juncions occur either in pairs, in the form of disclination dipoles, or in small particles.     

Electroelastic fields of moving dislocations and disclinations in piezoelectric crystals

Integral representations for the electroelastic fields of moving dislocations and disclinations in piezoelectric crystals have been obtained in terms of a four-dimensional formalism of dynamic Green’s functions. The cases of continuously distributed and single linear defects are considered. The correctness of the results is confirmed by the fact that they meet the necessary requirements as one goes to the purely elastic solution. Fiz. Tverd. Tela (St. Petersburg) 41, 1210–1213 (July 1999)