The peierls energy and kink energy in fcc metals

In fcc crystals, dislocations are dissociated on the {111} glide plane into pairs of partial dislocations. Since each partial interacts individually with the Peierls potential and is coupled to its neighbour by a stacking fault, periodic variations in the separation distance d of the partials occur when dislocations running along closed packed lattice directions are displaced. This can drastically reduce the effective Peierls stress. By using the Peierls model the structure of 0°, 30°, 60° and 90° dislocations in a typical fcc metal with the elastic properties of Cu and a stacking-fault energy γ₀ in the interval 0.04?≤?γ₀?≤?0.05?J/m² was studied, and the magnitude of the Peierls energy ΔE _P and the resulting kink energies E _K were determined. Since the energies involved are of the order of 10^?3?eV/b or less, their magnitude cannot be asserted with high confidence, considering the simplifying assumptions in the model. The difference in the changes of the core configuration during displacement of dislocations of different orientations should, however, be of physical significance. It is found that a dissociated 60° dislocation generally has a higher effective Peierls energy than a screw dislocation, but the reverse is true for the kink energy, at least in Cu.     

A generalized Peierls–Nabarro model for kinked dislocations

We present a generalized Peierls–Nabarro model for curved dislocations incorporating directly the Peierls energies for both straight dislocations and dislocation kinks. In our model, the anisotropic elastic energy is calculated efficiently using the discrete Fourier transform on the discrete lattice sites of the slip plane, and the discreteness in both the elastic energy and the misfit energy is included. We have used our model to calculate the kink migration and nucleation energies of the 30° dislocations in silicon. The results agree well with those obtained using atomistic potentials and first principles calculations, and the experimental results.     

The Peierls model for dislocation rings

The energy of a shear dislocation ring is calculated in the framework of the Peierls model in which the displacement is represented by a density of infinitesimal dislocations in the glide plane. This avoids the introduction of an uncertain core cut-off radiusr ₀ to prevent divergence in the usual treatment. The atomic misfit energy in the glide plane is accounted for explicitly and the influence of the interplanar potential on the ring energy and the core structure is studied. Whereas spontaneous formation of shear rings in a homogenous stress field can be ruled out, the emission of dislocation rings from crack tips in glide planes not containing the crack front is feasible.The paper is dedicated to Dr. Frantiek Kroupa in honour of his 70^th birthday.Stimulating discussions with miss Petra Fiala are gratefully acknowledged.     

Numerical Simulation of the Movement of Frenkel–Kontorova Dislocations in Aluminum Single Crystals at Low Temperatures

The motion of Frenkel–Kontorova dislocations in the single crystals of aluminum at low temperatures has been studied, by means of the computer simulation. It is shown that the dislocation movement is realized by the quantum tunneling of the kinks of dislocations through the Peierls barriers. It is shown that the action of the Peierls high barrier is analogous to the action of low temperatures, and if the Peierls barrier overcome, the dislocation moves unevenly, accelerating under the action of the Peierls barrier and slowing down after overcoming the Peierls barrier. Based on the numerical experiment, the mean free path of dislocation, the distance between the Peierls potential barriers and the width of the Peierls barrier are calculated. The computed values correspond to the real values.     

Which stresses affect the glide of screw dislocations in bcc metals?

By direct application of stress in molecular statics calculations we identify the stress components that affect the glide of 1/2?111? screw dislocations in bcc tungsten. These results prove that the hydrostatic stress and the normal stress parallel to the dislocation line do not play any role in the dislocation glide. Therefore, the Peierls stress of the dislocation cannot depend directly on the remaining two normal stresses that are perpendicular to the dislocation but, instead, on their combination that causes an equibiaxial tension-compression (and thus shear) in the plane perpendicular to the dislocation line. The Peierls stress of 1/2?111? screw dislocations then depends only on the orientation of the plane in which the shear stress parallel to the Burgers vector is applied and on the magnitude and orientation of the shear stress perpendicular to the slip direction.     

Structure factors that influence the stability of plastic strain of bcc metals under tensile load

The effect of the Peierls stress on the ultimate tensile stress and uniform strain prior to the formation of a neck during stretching of metals and alloys with bcc structure is theoretically analyzed. The analysis is based on the equation for the variation of the dislocation density with deformation; this equation determines the shape of the work-hardening curve for a bcc material and the effect of the Peierls stress on the parameters of this equation (the annihilation coefficient for screw dislocations). Using the Considére condition for plastic instability of the neck type, the ultimate tensile stress and the magnitude of uniform strain are found theoretically as a function of the Peierls stress at different temperatures below 0.15T _m , where T _m is the melting temperature of the bcc metal. Theoretical results are illustrated with experimental data on the temperature dependences of the annihilation coefficients for screw dislocations and of the magnitude of uniform strain in molybdenum and Armco iron.     

剪切应变下刃型位错的滑移机理的晶体相场模拟

针对刃型位错的滑移运动, 构建包含外力场与晶格原子密度耦合作用的体系自由能密度函数, 建立剪切应变作用体系的晶体相场模型. 模拟了双相双晶体系的位错攀移和滑移运动, 计算了位错滑移的Peierls势垒和滑移速度. 结果表明: 施加较大的剪切应变率作用, 体系能量变化为单调光滑曲线, 位错以恒定速度做连续运动, 具有刚性运动特征; 剪切应变率较小时, 体系能量变化出现周期波动特征, 位错运动是处于低速不连续运动状态, 运动出现周期“颠簸”式滑移运动, 具有黏滞运动特征; 位错启动运动, 存在临界的势垒. 位错启动攀移运动的Peierls势垒要比启动滑移Peierls势垒大几倍. 位错攀移和滑移运动特征与实验结果相符合.     

On the core width and Peierls stress of bubble rafts dislocations within the framework of modified Peierls-Nabarro model

Using Foreman’s method, the core structure and Peierls stress of dislocations in bubble rafts have been investigated within the framework of the modified Peierls-Nabarro (P-N) model in which the discrete lattice effect is taken into account. The core width obtained from the modified P-N model is much wider than that from the P-N model owing to the discrete lattice effect. It is found that the core width of dislocation increases with a decrease of the bubble radius. The elastic strain energy associated with the discrete effect is considered while calculating the Peierls stress. The new expression of the Peierls stress obtained in this paper is not explicitly dependent on the particular form of the restoring force law, which is only related to the core structure parameter and can be used expediently to predict the Peierls stress of dislocations. The Peierls stress decreases rapidly with the decrease of the bubble radius.     

Non-singular dislocation continuum theories: strain gradient elasticity vs. Peierls–Nabarro model

Abstract

Non-singular dislocation continuum theories are studied. A comparison between Peierls–Nabarro dislocations and straight dislocations in strain gradient elasticity is given. The non-singular displacement fields, non-singular stresses, plastic distortions and dislocation core shapes are analysed and compared for the two models. The main conclusion of this study is that due to their characteristic properties, the non-singular displacement fields, non-singular stresses and dislocation core shape of screw and edge dislocations obtained in the framework of strain gradient elasticity are more realistic and physical than the corresponding fields of the Peierls–Nabarro model. Strain gradient elasticity of dislocations is a continuum dislocation theory including a weak non-locality within the dislocation core and predicting the size and shape of the dislocation core. The dislocation core is narrower in the strain gradient elasticity dislocation model than in the Peierls–Nabarro model and more evenly distributed in two dimensions. The present analysis shows that for the modelling of the dislocation core structure the non-singular dislocation fields of strain gradient elasticity are the suitable ones.     

Periodic image effects in dislocation modelling

The use of periodic boundary conditions for modelling crystal dislocations is predicated on one's ability to handle the inevitable image effects. This communication deals with an often overlooked mathematical subtlety involved in dealing with the periodic dislocation arrays, that is conditional convergence of the lattice sums of image fields. By analysing the origin of conditional convergence and the numerical artefacts associated with it, we establish a mathematically consistent and numerically efficient procedure for regularization of the lattice sums and the corresponding image fields. The regularized solutions are free from the artefacts caused by conditional convergence and regain periodicity and translational invariance of the periodic supercells. Unlike the other existing methods, our approach is applicable to general anisotropic elasticity and arbitrary dislocation arrangements. The capabilities of this general methodology are demonstrated by application to a variety of situations encountered in atomistic and continuum modelling of crystal dislocations. The applications include introduction of dislocations in the periodic supercell for subsequent atomistic simulations, atomistic calculations of the core energies and the Peierls stress and continuum dislocation dynamics simulations in three dimensions.     

Dislocation transmission across the Cu/Ni interface: a hybrid atomistic–continuum study

The strengthening mechanisms in bimetallic Cu/Ni thin layers are investigated using a hybrid approach that links the parametric dislocation dynamics method with ab initio calculations. The hybrid approach is an extension of the Peierls–Nabarro (PN) model to bimaterials, where the dislocation spreading over the interface is explicitly accounted for. The model takes into account all three components of atomic displacements of the dislocation and utilizes the entire generalized stacking fault energy surface (GSFS) to capture the essential features of dislocation core structure. Both coherent and incoherent interfaces are considered and the lattice resistance of dislocation motion is estimated through the ab initio-determined GSFS. The effects of the mismatch in the elastic properties, GSFS and lattice parameters on the spreading of the dislocation onto the interface and the transmission across the interface are studied in detail. The hybrid model shows that the dislocation dissociates into partials in both Cu and Ni, and the dislocation core is squeezed near the interface facilitating the spreading process, and leaving an interfacial ledge. The competition of dislocation spreading and transmission depends on the characteristics of the GSFS of the interface. The strength of the bimaterial can be greatly enhanced by the spreading of the glide dislocation, and also increased by the pre-existence of misfit dislocations. In contrast to other available PN models, dislocation core spreading in the two dissimilar materials and on their common interface must be simultaneously considered because of the significant effects on the transmission stress.     

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.     

Flexible Ab initio boundary conditions: simulating isolated dislocations in bcc Mo and Ta

We report the first ab initio density-functional study of the strain field and Peierls stress of isolated <111> screw dislocations in bcc Mo and Ta. The local dislocation strain field is self-consistently coupled to the long-range elastic field using a flexible boundary condition method. This reduces the mesoscopic atomistic calculation to one involving only degrees of freedom near the dislocation core. The predicted equilibrium core for Mo is significantly different from previous atomistic results and the Peierls stress shows significant non-Schmid behavior as expected for the bcc metals.     

Non-singular descriptions of dislocation cores: a hybrid ab initio continuum approach

The core structure of straight and curved dislocations is studied by developing a hybrid approach that links the parametric dislocation dynamics method with ab initio calculations. The approach is an extension of the Peierls–Nabarro (PN) model, with the following features: (1) all three components of the displacement vector for atoms within the dislocation core are included; (2) the entire generalized stacking fault energy surface (GSFS) obtained from ab initio calculations is utilized; and (3) the method is generalized to treat curved dislocations. We combine the parametric dislocation dynamics (DD) approach for the interaction and motion of dislocations with ab initio calculations of lattice restoring forces. These forces, which are extracted from the GSFS (γ-surface), are calculated from both first-principles density functional theory (DFT) and the embedded-atom method (EAM). Dislocation core structures in aluminium and silver are determined. For straight dislocations, the results from the model are shown to be in excellent agreement with experiments for both Al and Ag. In contrast to undissociated dislocation loops in Al, it is found that the core width and the separations between partials in Ag vary along the angular direction measured with respect to the Burgers vector. It is also shown that the core-cutoff radius, which is usually employed in DD calculations to avoid singularities, must be adjusted as a function of loop size to account for the correct dislocation core energy.     

The elastic potential energy of a thin foil deformed by an in-plane 60° dislocation

An explicit expression of the elastic potential energy, W ^p, stored in a thin foil by a dissociated dislocation running parallel to the free surfaces is obtained in isotropic elasticity. It is used to discuss the metastable elastic equilibrium of a 60° dislocation in an ultrathin silicon foil when the fault plane is slightly inclined with respect to the free surfaces. The energy W ^p depends in particular on the positions of the two partials in the foil and on the thickness h, a situation not fully considered previously. For such ultrathin foils, propitious to the observation of partial dislocations cores at near atomic resolution, the theory predicts rapid changes with h of the separation distance S. This result is in accordance with previous experimental observations of S realized with the so-called “forbidden-reflection lattice imaging” technique.     

Full-scale atomistic simulations of dislocations in Ni crystal by embedded-atom method

Full-scale atomistic simulations by the nudged elastic band method are performed to determine the energetics and core structures of dislocations in a Ni lattice using an embedded-atom method potential. We find that for an edge dislocation, the potential yields very weak coupling between the partials which move almost individually. For a screw dislocation, the coupling between the partials is somewhat stronger and the partials move with some dependence. As expected, the results indicate that stacking fault energy has a controlling influence on the coupling behaviour of the partials. The effective Peierls energies and stresses are 1.30?×?10^?6?eV/Å and 2.79?×?10^?6?μ for the edge dislocation, and 1.62?×?10^?4?eV/Å and 2.02?×?10^?4?μ for the screw dislocation.     

On the nature of ultrahigh plastic (Megaplastic) strain

The features of structural transformations occurring at ultrahigh plastic strains, which are proposed to refer to as gigascopic, have been analyzed. It is shown that, during such deformation, additional channels of elastic energy dissipation must be effectively implemented. It is concluded that structural changes are characterized by certain cyclicity. A specific route of structural rearrangements is determined by temperature, the height of the Peierls barrier for dislocations and their ability of diffusive rearranging, and the difference in the free energies of the crystalline and amorphous states.     

Hydrogen diffusion in the elastic fields of dislocations in iron

The effect of dislocation stress fields on the sink efficiency thereof is studied for hydrogen interstitial atoms at temperatures of 293 and 600 K and at a dislocation density of 3 × 10¹⁴ m^–2 in bcc iron crystal. Rectilinear full screw and edge dislocations in basic slip systems 〈111〉{110}, 〈111〉{112}, 〈100〉{100}, and 〈100〉{110} are considered. Diffusion of defects is simulated by means of the object kinetic Monte Carlo method. The energy of interaction between defects and dislocations is calculated using the anisotropic theory of elasticity. The elastic fields of dislocations result in a less than 25% change of the sink efficiency as compared to the noninteracting linear sink efficiency at a room temperature. The elastic fields of edge dislocations increase the dislocation sink efficiency, whereas the elastic fields of screw dislocations either decrease this parameter (in the case of dislocations with the Burgers vector being 1/2〈111〉) or do not affect it (in the case of dislocations with the Burgers vector being 〈100〉). At temperatures above 600 K, the dislocations affect the behavior of hydrogen in bcc iron mainly owing to a high binding energy between the hydrogen atom and dislocation cores.     

Plastic deformation of filamentary silicon crystals. Relation to formation of dislocations at the surface and evolution of ensemble of these in volume

Direct methods, as well as high-sensitivity indirect methods, have shown that in initially dislocation-free filamentary crystals with high Peierls barriers dislocations originate at the surface and become abruptly localized at sites of stress concentration. Formation of the first dislocations does not lead to large-scale multiplication of these and to irreversible plastic deformation; rather, a gradient of the dislocation density appears, reaching 10¹⁷ m^–3. After penetration of the shear front to about half of the radius of the filamentary crystals, a change in the mechanism of plastic deformation occurs.Voronezh Polytechnical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 60–68, May, 1993.     

The dislocations in graphene with the correction from lattice effect

The dislocation widths and Peierls stresses of glide dislocations and shuffle dislocations in graphene have been studied by the improved Peierls-Nabarro (P-N) equation which contains the discrete correction. The discrete parameter is obtained from a simple dynamic model in which the interaction attributed to the variation of bond length and angle was considered. The restoring force in the improved P-N equation is given by the gradient of the generalized stacking fault energy surface (γ-surface). Our calculation shows that the widths of the shuffle dislocation and the glide dislocation are narrow and the width of the shuffle dislocation is about twice wider than the glide dislocation. The Peierls stress of a shuffle dislocation is one order of magnitude smaller than that of a glide dislocation. As a consequence, the shuffle dislocation moves more easily than the glide dislocation.