Image forces on edge dislocations: a revisit of the fundamental concept with special regard to nanocrystals

Two conditions under which image forces become significant are when a dislocation is close to a surface (or interface) or when the dislocation is in a nanocrystal. This investigation pertains to the calculation of image forces under these circumstances. A simple edge dislocation is simulated using finite element method (FEM) by feeding-in the appropriate stress-free strains in idealised domains, corresponding to the introduction of an extra half-plane of atoms. Following basic validation of the new model, the energy of the system as a function of the position of the simulated dislocation is plotted and the gradient of the curve gives the image force. The reduction in energy of the system arises from two aspects: firstly, due to the position of the dislocation in the domain and, secondly, due to deformations to the domain (/surfaces). The second aspect becomes important when the dislocation is positioned near a free-surface or in nanocrystals and can be calculated using the current methodology without constructing fictitious images. It is to be noted that domain deformations have been ignored in the standard theories for the calculation of image forces and, hence, they give erroneous results (magnitude and/or direction) whenever image forces play an important role. An important point to be noted is that, under certain circumstances, where domain deformations occur in the presence of an edge dislocation, the ‘image' can be negative (attractive), zero or even positive (repulsive). The current model is extended to calculate image forces based on the usual concept of an ‘image dislocation’.     

On the elastic field and image force of dislocations in anisotropic solids and its application to GaN nanostructures

The determination of the elastic field and image force of dislocation in anisotropic media is a nontrivial problem. This work revisits Eshelby’s sextic anisotropic elasticity theory to obtain the stress field of a screw dislocation in an infinite anisotropic solid. The image force of a dislocation in an anisotropic nanowire is then derived by using the concept of ‘image dislocation’. Moreover, this work proposes to study the image force of nanorods by approximating the three-dimensional shape effect as a height-dependent shape function, which could be obtained through curve fitting of the finite element data. The analytical solution is applied to analyse image forces on different dislocations in GaN nanorods oriented along polar (c-axis) and nonpolar (a, m-axis) directions. The result shows the dislocation dissipation could be more effective in a-GaN but less in m-GaN by comparing with the standard growth of c-GaN. The approach developed in this work is applicable to other material systems. Therefore, it could contribute to a wide range of nanostructure design and fabrication for dislocation-free devices.     

Screw dislocation in semi-infinite non-local hexagonal medium

Abstract

Nonlocal stresses of a screw dislocation near a free surface in a semi-infinite hexagonal medium are investigated by a surface dislocation model. The nonlocal image force on the screw dislocation due to the existing free surface is also obtained. All classical singularities for the stress and image force are eliminated. The maxima of the stress and image force are evaluated. A zero point of the stress is found, which predicts that different states of the shear stress exist simultaneously near the dislocation. The appearance of a zero value at the free surface and a maximum of the dislocation image force can be used to explain the existence of the dislocation free zone.     

Elastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack

The interaction between an edge dislocation and a crack emanating from a semi-elliptic hole is dealt with. Utilizing the complex variable method, closed form solutions are derived for complex potentials and stress fields. The stress intensity factor at the tip of the crack and the image force acting on the edge dislocation are also calculated. The influence of the morphology of the blunt crack and the position of the edge dislocation on the shielding effect to the crack and the image force is examined in detail. The results indicate that the shielding or anti-shielding effect to the stress intensity factor increases acutely when the dislocation approaches the tip of the crack. The effect of the morphology of the blunt crack on the stress intensity factor of the crack and the image force is very significant.     

Dynamic blocking of the influence of surface point defects on the glide of edge dislocations

The influence of image forces on an edge dislocation gliding parallel to the free crystal surface having point defects is studied theoretically. It is shown that image forces block the drag mechanism that involves the excitation of dislocation vibrations by surface defects.     

磁电弹性体中螺型位错与唇口裂纹的相互作用

研究磁电弹性体中螺型位错与唇口裂纹的相互作用。结合Muskhelishvili方法和干扰技术, 在假定裂纹面具有不可渗透条件下得到磁电弹性体中由位错和唇口裂纹所诱导的应力场、电场和磁场的解析解。应用广义Peach-Koehler公式,得到作用在位错上的影像力。通过数值算例,得到场强度因子的变化规律及影像力和广义力随位错位置的变化规律。     

Edge dislocations near phase boundaries in the gradient theory of elasticity

A solution of the boundary-value problem in the gradient theory of elasticity concerning a rectilinear edge dislocation parallel to the interface between phases with different elastic moduli and gradient coefficients is obtained. The interaction between the dislocation and the interface is considered on a nanoscopic level. It is shown that the stress field has no singularities on the dislocation line and remains continuous at the interface, unlike the classical solution, which is singular at the dislocation line and allows a discontinuity of two stress components at the interface. The gradient solution also removes the classical singularity of the image force for the dislocation on the interface. An additional elastic image force associated with the difference in the gradient coefficients of contacting phases is also determined. It is found that this force, which has a short range and a maximum at the interface, expels the edge dislocation into the material with a smaller gradient coefficient.     

Asymptotic behaviors of the stress fields in the vicinity of dislocations and dislocation segments

We obtain the singular asymptotic behavior of the stress field in the vicinity of a non-planar dislocation in three dimensions and the nearly singular behavior of the full self-force of the dislocation including both glide and climb forces, using asymptotic analysis. We also derive asymptotic formulas for the stress field in the vicinity of a curved dislocation segment. Numerical examples are presented to examine the asymptotic formulas. The obtained formulas can be used for qualitative understanding of the stress tensor associated with dislocations and efficient and accurate calculation of the stress tensor in dislocation dynamics simulations.     

Dislocations in two-dimensional liquid foams

In this analysis we make the analogy between the dislocations in metals and the behaviour of 2D liquid foams in the presence of external forces. It is shown that the edge defect which appears in metals can also be introduced in a similar manner in 2D liquid foams and that the motion of this defect through the foam can occur by the process of T1 transformations. The analogy leads to the motion of local cells as opposed to the motion of a row of cells. That is, the T1 process occurs ‘locally’ and the ‘defect’, primarily introduced here as an edge dislocation, can move through the whole 2D liquid foam through the local motion of these edge dislocations. These defects as in the theory of metals lead to the reduction of the critical stress required for the ‘slipping’ of the 2D liquid foam. Furthermore, since it is these defects which are responsible for the behaviour of the 2D liquid foam to external forces and since it will be the motion of these defects which is related to the motion of the cells, it is possible, again through the analogy with ‘slipping’ in metals to determine the strain rate. The result is given as strain rate = ϱ b 〈V〉.     

Edge misfit dislocation formation at the interface of a nanopore and infinite substrate with surface/interface effects

The model of an edge misfit dislocation at the interface of the hollow nanopore and the infinite substrate with surface/interface stress is investigated. Using the complex variable method, analytical solutions for complex potentials of a film due to an edge misfit dislocation located in the film with surface/interface effect are derived, and the stress fields of the film and the edge misfit dislocation formation energy can be obtained. The critical conditions for edge misfit dislocation formation are given at which the generation of an edge misfit dislocation is energetically favourable. The influence of the ratio of the shear modulus between the film and the infinite substrate, the misfit strain, the radius of the nanopore and the surface/interface stress on the critical thickness of the film is discussed.     

基于点缺陷扩散理论与离散位错动力学耦合的位错攀移模型研究

位错的攀移运动对高温下晶体材料的塑性行为有重要影响,为了能够有效揭示攀移的物理本质及其对塑性行为的作用,本文基于点缺陷扩散理论,通过将体扩散和管扩散机理的共同作用与三维离散位错动力学耦合,建立了适用条件更广的位错攀移模型. 利用此模型我们模拟了单个及多个棱柱型位错环的收缩变形过程,发现影响位错攀移速率的决定因素不是传统理论认为的机械攀移力,而是位错周围(体扩散)及位错段上(管扩散)的空位浓度梯度. 该模型也能够完全重现棱柱型位错环群的粗化过程中不同位错环半径及晶体内平均空位浓度随时间变化的三个阶段. 关键词： 位错攀移 点缺陷扩散理论 位错动力学 棱柱位错环     

Elastic behavior of a screw dislocation in the wall of a hollow nanotube

The elastic behavior of a screw dislocation lying in the wall of a hollow cylindrical nanotube is studied theoretically. The corresponding boundary-value problem of the classical theory of elasticity is formulated and solved (for stresses) for a linear elastically isotropic or transversely isotropic body. The elastic energy of the nanotube with the dislocation and the image force exerted on this dislocation by the inner and outer nanotube surfaces are calculated. The internal space of the nanotube is shown to cause the following qualitative differences in the dislocation stress-field distribution: a change in the sign of stress-tensor components near the inner nanotube surface, a high stress concentration at this surface, and a strong stress gradient at this surface. The dislocation has only one position of unstable equilibrium in the nanotube wall, which is always shifted toward the inner nanotube surface. As the internal-space radius increases, the dislocation energy decreases and the position of its equilibrium shifts toward the outer nanotube surface; in the limiting case of a flat plate, it reaches the center of the plate. Near the nanotube free surfaces, the image force on the dislocation is large and can substantially affect the dislocation behavior.     

Special features of the stress field induced by the edge of the plastic shear band near the crystal surface

The characteristics of the stress field produced by the edge of the plastic shear band near the surface are examined on the basis of numerical calculations. A finite dislocation pileup is chosen as a shear band model. The stress intensity is shown to increase markedly in the vicinity of the band edge due to “image” force effects as it approaches the surface. The peculiarity of propagation of localized shear bands in coated materials is discussed. Institute of Strength Physics and Material Science. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 65–71, September, 1999.     

Stress field and interaction forces of dislocations in anisotropic multilayer thin films

Utilizing Fourier transforms, the elastic field of three-dimensional dislocation loops in anisotropic multilayer materials is developed. Green's functions and their derivatives, obtained first in the Fourier domain and then in the real domain by numerical inversion, are used in integrals to determine the elastic field of dislocation loops. The interaction forces between dislocations and free surfaces or interfaces in multilayer thin films are then investigated. The developed method is based on rigorous elasticity solutions for dislocations approaching to within one to two atomic planes from the interface. For a dislocation in one layer, the interface image force is determined mainly by the elastic moduli and thicknesses of neighbouring layers. When a dislocation approaches an interface between two layers, within 10–20 atomic planes, the image force changes rapidly. Interaction forces are then kept constant up to the interface. The model shows that, when a dislocation crosses an interface from a soft to a hard layer, additional external forces must be applied to overcome an elastic mismatch barrier. The developed method extends the concept of the Kohler barrier in 2D, and shows that the interface force barrier not only depends on the relative ratio of the elastic moduli of neighbouring layers, but also on the 3D shape of the dislocation, the number of interacting adjacent layers, and on layer thicknesses.     

Post-yielding dislocation retraction of nano-lamellar TiAl single crystals

The perfect single crystal has ultra-high strength but is often accompanied by catastrophic failures after yielding. This study reveals that nano-lamellar TiAl single crystals alleviate the catastrophic failure due to a post-yielding dislocation retraction through atomistic simulations and theoretical analyses. This dislocation retraction leads to a retained post-yielding strength of1.03 to 2.33 GPa(about 50% of the yielding strength). It is shown that this dislocation retraction is caused by local stress relaxation and interface-mediated image force. The local stress relaxation is due to successive dislocation nucleation in different slip systems, and the interface-mediated image force is caused by the heterogeneous interface. Based on dislocation theory, this study demonstrates that the size effect also plays a vital role in dislocation retraction. Theoretical modeling shows that the dislocation retraction occurs when the lamellar thickness is less than approximately 12 nm. Additionally, the post-yielding dislocation retraction is more pronounced at higher temperatures, making it more effective in alleviating catastrophic failures.These findings demonstrate a viable option for avoiding catastrophic failure of single crystals through nanoscale-lamellar design.     

Disconnections in simple and complex structures

Disconnections are interfacial defects with dislocation and step character; for example, twinning dislocations in homo-phase and transformation dislocations in hetero-phase materials. They play important structural roles and are classified as either ‘perfect’, separating energetically degenerate regions of interface, or ‘partial’, bounding a faulted region. In kinetic mechanisms, disconnection motion can be glissile, by conservative climb or climb. In the present paper, disconnection mobility is analyzed in terms of shear and shuffle-type atomic displacements and diffusional flux. This is applied to a sequence of hetero-phase materials with increasing structural complexity, namely the β to α transformation in Ti, θ′ precipitation in Al(Cu) and the orthorhombic to monoclinic martensitic transformation in ZrO₂. A disconnection source mechanism in Ti, involving conservative climb, is also described.     

非对称倾侧亚晶界的晶体相场法研究

采用晶体相场法研究非对称倾侧亚晶界结构及其在应力作用下的微观运动机制.分别从温度、倾斜度角以及应力施加方向等方面对其结构及迁移过程进行分析和讨论.结果表明,非对称倾侧亚晶界由符号相同的一排刃型位错等距排列,部分出现由两个相互垂直排列的刃型位错构成的位错组;在应力作用下,非对称倾侧亚晶界迁移的微观机制包括位错的滑移和攀移、位错组分解、单个位错与位错组反应、单个位错分解以及位错湮灭;温度降低和倾斜度增大都会阻碍亚晶界的迁移过程;应力方向改变导致位错运动方向改变,从而改变晶界迁移形式.     

Dislocation emission at the onset of plasticity during nanoindentation in gold

Nanoindentations and the subsequent plastic damage in the form of dislocation configurations have been both generated and imaged with scanning tunnelling microscopy on a reconstructed Au(001) surface, the resulting observations being interpreted in terms of the elastic theory of dislocations in a continuum. The rearranged pileup material around the nanoindentation is described in terms of dislocation emission and glide involving, in particular, multiple cross-slip. ‘Mesas’, shallow protusions stemming from a special dislocation configuration consisting of Schockley partial dislocations encompassing two stacking faults, are shown to glide parallel to the surface under the stress generated by further nanoindentations. The spatial distribution of ‘mesas’ around the nanoindentation traces is shown to be controlled by a balance between the interactions between the different ‘mesas’ and the stresses arising from the nanoindentation itself.     

On the structure of a water molecule chemisorbed on a metal surface

A model using the electrostatic image force and the harmonic restoring force has been adapted to find the equilibrium structure of a chemisorbed water molecule on a metal surface. An estimate of the maximum possible increase in the HOH interbond angle upon chemisorption due to image forces is given.     

Molecular Dynamics Study of Mechanical Behaviour of Screw Dislocation during Cutting with Diamond Tip on Silicon

By means of Tersoff and Morse potentials, a three-dimensional molecular dynamics simulation is performed to study atomic force microscopy cutting on silicon monocrystal surface. The interatomic forces between the workpiece and the pin tool and the atoms of workpiece themselves are calculated. A screw dislocation is introduced into workpiece Si. It is found that motion of dislocations does not occur during the atomic force microscopy cutting processing. Simulation results show that the shear stress acting on dislocation is far below the yield strength of Si.