Peierls-Nabarro Dislocation Model Research Articles

First-principles investigations of the interaction between alloying atom and dislocation and its implication to the rafting of Ni-based superalloys

The rafting of the microstructure of Ni-based superalloys significantly deteriorates the mechanical properties. According to the plastic rafting model based on the dislocation mechanism, the directional atomic diffusion induced by the interaction between the alloying atom and the interfacial dislocation is critical to the evolution of the rafting microstructure. In the present work, the interactions between the alloying atom and dislocation core in both matrix γ phase and the precipitate γ′ phase are investigated by using a first principles method in combination with the semi-discrete variational Peierls-Nabarro dislocation model. We show that the alloying atoms Al, Cr, Co, Mo, Ru, Ta, W, Re lower the (edge for Cr and Co) dislocation core energy in the γ phase, indicating that the alloying atoms are attractive to the dislocation core. For the γ′ phase, Cr, Mo, Ta, W, Re raise whereas Co and Ru lower the dislocation core energy, i.e., Cr, Mo, Ta, W, Re are repulsive but Co and Ru are attractive to the dislocation core. With the calculated interaction between the alloying atoms and the dislocation core, the influence of these alloying elements on the rafting of the superalloy is discussed.

Application of digital phase shifting moiré method in interface and dislocation location recognition and real strain characterization from HRTEM images.

In high-resolution transmission electron microscopy (HRTEM) images of heterostructures, it is always difficult to accurately determine the interface position and identify dislocations in a large field of view at tens to hundreds of nanometers due to the small lattice differences. However, in the heterostructure, the determination of the interface position is the key to obtain the true mismatch stress/strain field of the interface. Due to the magnifying effect of the digital moiré method on small differences, digital moiré technology was applied to determine Ge/Si heterostructure interfaces and large-area identification interface dislocations in HRTEM lattice diagrams in this study. By optimizing the frequency and angle of the reference lattice, the interface and dislocation position are clearly and intuitively displayed. How to accurately determine the position of the heterostructure interface and the dislocation of the large-area recognition interface from HRTEM images are studied through simulation experiments. The results show that when the frequency of the reference lattice and the specimen lattice are close, and the angle between them is within 10°, the position of the heterostructure interface can be accurately and intuitively determined by the naked eye according to the distortion characteristics of the moiré fringe. When the frequency of the reference lattice is 0.7 to 0.9 times of the specimen lattice, and the rotation angle is within 8°, the visually clear crossover phenomenon of the moiré fringes is used for large-area identification of interface dislocations. Using the phase measurement interface position sensitivity can reach the Å level. Using the phase-shifting digital moiré method the strain field on the dislocation core at the Ge/Si heterostructure interface and the interface stress distribution were quantitatively analyzed. Compared with the Peierls-Nabarro dislocation model and the Foreman dislocation model, Foreman's variable factor α = 4 is more suitable for describing the strain field of misfit dislocations on the Ge/Si heterostructure interface.

Long time behavior of dynamic solution to Peierls–Nabarro dislocation model

In this paper we study the relaxation process of Peierls-Nabarro dislocation model, which is a gradient flow with singular nonlocal energy and double well potential describing how the materials relax to its equilibrium with the presence of a dislocation. We prove the dynamic solution to Peierls-Nabarro model will converge exponentially to a shifted steady profile which is uniquely determined.

Comparing modeling predictions of aluminum edge dislocations: semidiscrete variational Peierls-Nabarro versus atomistics.

Multiple computational methods for modeling dislocations are implemented within a high-throughput calculation framework allowing for rigorous investigations comparing the methodologies. Focusing on aluminum edge dislocations, twenty-one classical aluminum interatomic potentials are used to directly model dislocation core structures using molecular dynamics, as well as provide input data for solving the semidiscrete variational Peierls-Nabarro dislocation model. The predicted dislocation core spreading obtained from both computational methods show similar trends across the potentials. Additionally, tests are done to rigorously determine if a recent correction to the Peierls-Nabarro model results in better agreement with the atomistic calculations.

The influence of anisotropy on the core structure of Shockley partial dislocations within FCC materials

Both theoretical and numerical models of dislocations often necessitate the assumption of elastic isotropy to retain analytical tractability in addition to reducing computational load. As dislocation based models evolve towards physically realistic material descriptions, the assumption of elastic isotropy becomes increasingly worthy of examination. We present an analytical dislocation model for calculating the full dissociated core structure of dislocations within anisotropic face centered cubic (FCC) crystals as a function of the degree of material elastic anisotropy, two misfit energy densities on the γ-surface (, ) and the remaining elastic constants. Our solution is independent of any additional features of the γ-surface. Towards this pursuit, we first demonstrate that the dependence of the anisotropic elasticity tensor on the orientation of the dislocation line within the FCC crystalline lattice is small and may be reasonably neglected for typical materials. With this approximation, explicit analytic solutions for the anisotropic elasticity tensor for both nominally edge and screw dislocations within an FCC crystalline lattice are devised, and employed towards defining a set of effective isotropic elastic constants which reproduce fully anisotropic results, however do not retain the bulk modulus. Conversely, Hill averaged elastic constants which both retain the bulk modulus and reasonably approximate the dislocation core structure are employed within subsequent numerical calculations. We examine a wide range of materials within this study, and the features of each partial dislocation core are sufficiently localized that application of discrete linear elasticity accurately describes the separation of each partial dislocation core. In addition, the local features (the partial dislocation core distribution) are well described by a Peierls-Nabarro dislocation model. We develop a model for the displacement profile which depends upon two disparate dislocation length scales which describe the core structure; (i) the equilibrium stacking fault width between two Shockley partial dislocations, Req and (ii) the maximum slip gradient, χ, of each Shockley partial dislocation. We demonstrate excellent agreement between our own analytic predictions, numerical calculations, and Req computed directly by both ab-initio and molecular statics methods found elsewhere within the literature. The results suggest that understanding of various plastic mechanisms, e.g., cross-slip and nucleation may be augmented with the inclusion of elastic anisotropy.

Local equilibrium configurations and minimum energy path of carbon nanotubes with Stone-Wales defects and their related pentagon-heptagon lattice defects

The energy landscape over configurational change of defects in carbon nanotube (CNT) is clarified for a variety of distributions of Stone–Wales defects and pentagon–heptagon (5–7) defects using the Peierls–Nabarro dislocation model. Based on the theoretical model, the configurational energy is estimated by relative displacement without fatal loss of accuracy, but the calculation of energy during the configurational change of defect is simplified with reduced degrees of freedom in the displacement field. The equilibrium configurations are obtained using the conjugate-gradient method, and the saddle point on the minimum-energy path between two equilibrium configurations for evolution of defects, such as the nucleation/annihilation and movement of defects, is estimated using the nudged elastic-band method. Many equilibrium configurations stably exist because of the lattice trapping of the 5–7 defects due to the relatively large Peierls–Nabarro barrier. The total energy does not only depend on the mean value of the relative displacement, but also on the complexity of the configurations of the defects, and we draw a directed graph which shows the possible paths of the configuration change of the defects. The results provide fundamental information related to the stability of the thermal activation of defects in CNT.

Multiscale simulation of incipient plasticity and dislocation nucleation on nickel film during tilted flat-ended nanoindentation

Multiscale simulations of the tilted flat-ended nanoindentation with different tilted angles (from 5° ∼ 30°) on the (−110) surface of nickel crystal were studied using the QC method. The model of the indentation is a flat-end indenter inclined by an angle ɛ driven into a half-plane vertically. Load-displacement responses, initial stages of the plasticity deformations and dislocation emissions for nickel film at different inclined angles were obtained and analyzed as well. An energy criterion was successfully proposed to analyze the critical load for the first dislocation emission beneath the edge of the indenter. Simulation results agree well with analytical ones. An elastic model based on the contact theory and the Peierls-Nabarro dislocation model were combined to analyze when and where the dislocation will be emitted beneath the lower surface of an inclined indenter. Results indicate that the key parameter is the ratio of the contact half-width to the position of the slip plane. This parameter shows the range in which a dislocation will probably be emitted. This mechanism explains the simulation results well. This work is of value for understanding the mechanism of dislocation emissions of FCC crystals under tilted flat-ended nanoindentation while providing approaches to predicting when the first dislocation will be emitted and where subsequent dislocations will probably be emitted.

Strain analysis of misfit dislocations in α-Fe2O3/α-Al2O3 heterostructure interface by geometric phase analysis

The α-Fe2O3/α-Al2O3 heterostructure interfaces have been studied using transmission electron microscopy (TEM). The interface exhibited coherent regions separated by equally spaced misfit dislocations. The misfit dislocations were demonstrated to be edge dislocations with dislocation spacing of ∼4nm. The strain fields around the misfit dislocation core were mapped using a combination of geometric phase analysis and high-resolution transmission electron microscopy images. The strain measurement results were compared with the Peierls–Nabarro dislocation model and the Foreman dislocation model. These comparisons show that the Foreman model (a=2) is the most appropriate theoretical model to describe the strain fields of the dislocation core.

NANOSCALE DEFORMATION ANALYSIS OF TWIN BOUNDARY DISLOCATIONS IN NANOCRYSTALLINE ALUMINUM

The nanoscale deformation field of twin boundary dislocations in nanocrystalline aluminum was experimentally investigated using a combination of high-resolution transmission electron microscopy and geometric phase analysis. The entire strain field of the twin boundary dislocations was mapped and then compared with those of the Peierls–Nabarro and elastic theory dislocation models. The comparison results demonstrated that the Peierls–Nabarro dislocation model best describes the strain field of the twin boundary dislocations in nanocrystalline aluminum.

Multiscale simulations of wedged nanoindentation on nickel

Mutiscale simulations of nanoindentation with different wedged indenters (with vertex angle from 120–170°) on the (−110) surface of nickel crystal were studied using the quasicontinuum method. Load–displacement responses, initial stages of the plasticity deformations and dislocation emissions for nickel film with each kind of indenter were obtained and analyzed as well. An elastic model based on the Hertz contact theory was proposed to analyze the critical vertex angle to emit dislocations. A critical vertex angle at the range of 156–162° was obtained. The Peierls–Nabarro dislocation model, as well as the elastic model and an energetic criterion were used to analyze the position of dislocations. A key parameter and its threshold values were obtained to characterize where the dislocation will be emitted. Simulation results matched well with analytical ones. This work is useful to research and understand the mechanism of dislocation emissions of nickel under wedged nanoindentation, and furthermore provides abilities to predict whether and where the dislocation will be emitted.

Crack tip dislocation emission and nanoscale deformation fields in silicon

A micro-crack in silicon was experimentally investigated by using a combination of transmission electron microscopy and geometric phase analysis. The strain fields of the crack tip, with scales of a few tens of nanometers, were mapped. The crack tip dislocation emission and stress relief by dislocation generation around a crack tip can be proved. And, the strain field of an edge dislocation was compared with the Peierls–Nabarro dislocation model at the scale of a dislocation width. We show that the Peierls–Nabarro model is the appropriate theoretical model to describe the deformation fields of the dislocation core.

Measurement of strain fields in an edge dislocation

Abstract We present a strain analysis of an edge dislocation core in gold. The strain fields around an edge dislocation were mapped with a geometric phase analysis of high-resolution transmission electron microscopy images. The strain measurement results were compared with the gradient elasticity dislocation model, the field theory of elastoplasticity dislocation model and the Peierls–Nabarro dislocation model. We then concluded that for the field theory of elastoplasticity, the non-dimensional constant e 0 is more suitable when chosen as 0.25 than with 0.399. However, the Peierls–Nabarro model is the most appropriate theoretical model to describe the deformation fields of the dislocation core.

Experimental examination of displacement and strain fields in an edge dislocation core

We present a displacement and strain analysis of edge dislocation core in gold. The displacement and strain field around the edge dislocation were mapped by geometric phase analysis of high-resolution transmission electron microscopy images. The displacement and strain measurement results were compared with the isotropic elastic theory dislocation model, the anisotropic elastic theory dislocation model, the Peierls–Nabarro dislocation model and the Foreman dislocation model ( a = 4). These comparisons show that the Peierls–Nabarro model is the most appropriate theoretical model to describe the deformation fields of the dislocation core.

位移和应变测量的定量数值云纹法

The numerical moire method with sensitivity as high as 0.03 nm has been presented. A quantitative displacement and strain analysis program has been proposed by using this method. It is applied to an edge dislocation and a stacking fault in aluminum. The measured strain of edge dislocation is compared with theoretical prediction given by Peierls-Nabarro dislocation model. The displacement of stacking fault is also obtained.

Quantitative measurement of deformation field around low-angle grain boundaries by electron microscopy

The strain field of low-angle grain boundaries in gold was experimentally investigated. The grain boundaries consist of the arrangement of discrete dislocations. High-resolution transmission electron microscopy (HRTEM) and geometric phase analysis (GPA) were employed to map the strain field of grain boundaries. The numerical moiré method was used to visualize the dislocations. The strain components ε xx , ε yy , ε xy and rigid rotation ω xy were mapped. The dislocation core regions are convergence regions of strain. The largest values of strain occur in the immediate dislocation core region. The strain field around an edge dislocation was compared with Peierls–Nabarro dislocation model. The comparison result has demonstrated that the Peierls–Nabarro model can describe the strain field around edge dislocation.

Geometrical effect on dislocation nucleation at crystal surface nanostructures

Crystal surface nanostructures such as steps and trenches in microelectronic layer structures have been considered as stress concentrators that may facilitate dislocation nucleation. Quantitative characterization of the critical condition of this atomic scale process is of considerable interest for the development of high performance electronic devices. This paper addresses this issue using a multiscale approach based on the variational boundary integral formulation of the Peierls–Nabarro dislocation model. By representing the profiles of embryonic dislocations as the relative displacements between the two adjacent atomic layers along the slip planes, the critical conditions for dislocation nucleation are obtained by solving the stress dependent activation energies required to activate embryonic dislocations from their stable to unstable saddle point configurations. The geometrical effect of surface nanostructures such as steps and trenches on dislocation nucleation is ascertained quantitatively. Our results show that the atomic scale surface nanostructures can reduce the critical stress for dislocation nucleation by nearly an order of magnitude and the trench configurations are more prone to dislocation nucleation than the step configurations. Nucleation of versatile dislocations in multiple slip systems at crystal surfaces may be attributed surface nanostructures of a variety of geometries.

Quasicontinuum simulation of nanoindentation of Nickel film

A large-scale atom simulation of nanoindentation into a thin nickel film using the quasicontinuum method was performed. The initial stages of the plasticity deformation of nickel were studied. Several useful results were obtained as follows: (1) The response of the load versus indentation depth — on the load versus indentation depth curve, besides the straight parts corresponding to the elastic property of nickel, the sudden drop of the load occurred several times; (2) The phenomena of dislocation nucleation — the dislocation nucleation took place when the load descended, which makes it clear that dislocation nucleation causes the drop of the load; (3) The mechanism of the dislocation emission — the Peierls-Nabarro dislocation model and a powerful criterion were used to analyze the dislocation emission. And the computational value was in good agreement with the predict value; (4) The density of geometrically necessary dislocations. A simple model was used to obtain the density of geometrically necessary dislocations beneath the indenter. Furthermore, the influence of the boundary conditions on the simulation results was discussed.

Critical conditions for dislocation nucleation at surface steps

Dislocation nucleation at a surface step is analyzed based on a general variational boundary integral formulation of the Peierls–Nabarro dislocation model. By modelling the surface step as part of the surface of a three-dimensional crack, the free surface effect is taken into account by transferring the half space problem into an equivalent one in the infinite medium. The profiles of embryonic dislocations, corresponding to the relative displacements between the two adjacent atomic layers along the slip planes, are then rigorously solved through the variational boundary integral method. The critical conditions for dislocation nucleation are determined by solving the stress-dependent activation energies required to activate the embryonic dislocations from their stable to unstable saddle-point configurations. In particular, the effect of the step geometry, such as the height of the step, the dip angle of the slip plane and the inclined angle of the step surface on dislocation nucleation, is quantitatively ascertained. The results show that the atomic-scale surface step can rapidly reduce the critical stress required for dislocation nucleation from the surface by nearly an order of magnitude. The decrease in critical stress as a function of the height of the step is more significant for slip planes with smaller dip angles and surface steps with smaller inclined angles.

Energetics of dislocation nucleation under a nanoindenter

We present an analysis of dislocation nucleation under an idealized nanoindenter based on the variational boundary integral formulation of the Peierls–Nabarro dislocation model. By solving the embryonic dislocation profiles, corresponding to the relative displacements between the two adjacent atomic layers along the slip plane, we have determined the critical conditions for athermal dislocation nucleation as well as the activation energies required to thermally activate embryonic dislocations from their stable to unstable saddle point configurations. The effect of the size of the indenter on the energetics of dislocation nucleation is quantitatively characterized. The result is compared with a simplified analysis based on the application of the Rice model for dislocation nucleation at a crack tip.

Critical Conditions for Dislocation Nucleation from Surface Steps

AbstractThe critical conditions for dislocation nucleation from the surface steps of various geometries are analyzed based on the Peierls-Nabarro dislocation model. By modeling a surface step as part of a three- dimensional crack surface, the half space problem is transferred into an equivalent three dimensional crack problem in an infinite medium. The profiles of embryonic dislocations, corresponding to the relative displacements between the two adjacent atomic layers along slip planes, are then rigorously solved through the variational boundary integral method. The critical conditions for dislocation nucleation are determined by solving the stress dependent activation energies required to activate embryonic dislocations from their stable to unstable saddle point configurations. For a given slip plane, the effects of step geometry such as the step height and inclined angle on dislocation nucleation are analyzed in detail. The results show that the atomic scale steps may reduce the critical stress required for dislocation nucleation from the surface by several factors. Compared to previous analyses of this type of problem based on continuum elastic dislocation theory, the presented analysis eliminates the uncertain core cutoff parameter by allowing for the existence of an extended dislocation core as the embryonic dislocation evolves. Because of the serious limitation of direct atomic simulation for this type of problem, the presented methodology of incorporating atomic information into continuum approach appears to be particularly noteworthy for providing insights of energetics of the atomic processes involved in dislocation nucleation.