Semi-discrete Variational Peierls-Nabarro 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.

Drastic oscillation of peierls stress from peierls-nabarro model calculation and its remedy

Peierls stress (τp) of face-centered-cubic (FCC) structures from the Peierls-Nabarro (PN) model calculation is extremely sensitive to the input parameters including the shear modulus (μ), Poisson's ratio (ν) and generalized stacking fault energy (GSFE), which makes the model not a very reliable tool for the prediction of τp because uncertainties occur inevitably in the theoretical calculation or experimental measurements of these input parameters. In the present work, we scrutinized systematically the sensitivity of τps of the FCC metals to the input parameters within the framework of semi-discrete variational PN (SVPN) model. We showed that τp oscillates drastically with varying input parameters. The period of the oscillation is associated with the dependence of the distance between the two partials of the extended dislocation (D) on μ, ν and GSFE while the amplitude of the oscillation is modulated by the dependence of the half-width of the partial dislocation (ζ) on these input parameters. Based on the explored origin of the oscillation, a modified SVPN model is proposed to relieve the sensitivity of τp on the input parameters. The Peierls stresses evaluated with the modified SVPN model are in general agreement with the available experimental values for all the FCC metals.

Insights into plastic deformation mechanisms of austenitic steels by coupling generalized stacking fault energy and semi-discrete variational Peierls-Nabarro model

The generalized stacking fault energy (GSFE) is a key parameter to determine the plastic deformation mechanisms of austenitic steels. However, the underlying physics why the GSFE can affect the plastic deformation behaviors remains unclear. In this paper, the plastic deformation mechanisms of austenitic steels with different carbon (C) additions were investigated by coupling the GSFE with the semi-discrete variational Peierls-Nabarro (P–N) model. The internal mechanisms behind the P–N stress and plastic deformation were explained at atomic scale. It is found that the positions and contents of C atoms affect the GSFE of austenite, and thus regulate plastic deformation behaviors of austenitic steels by influencing dislocation core structure. As exemplified that with 4 ​at.%C in austenite, the intrinsic stacking fault energy increases from −433 to −264 mJ/m2, and the stacking fault width increases to 6.62b from 4.72b of FCC-Fe with b being the Burgers vector. This corresponds to the plastic deformation mechanism dominated by the ε martensitic transformation with the lattice changing from FCC to HCP. With increasing C contents to 8 ​at.%, the intrinsic stacking fault energy of austenite increases to −9.01 mJ/m2, while the stacking fault width decreases to 6.03b. The plastic deformation tends to proceed via the mechanical twinning mode. The present investigation establishes a solid foundation for clarifying the plastic deformation mechanisms of austenitic steels from the perspective of the dislocation core structure.

Study on lattice discreteness effect on superdislocation core properties of Ni3Al by improved semi-discrete variational Peierls-Nabarro model

Nickel-based superalloys (NBSAs) serve as hot-end component materials of aero engines. NBSAs generally consist of γ and γ′ phases, the γ′ phase (i.e. Ni3Al) presents anomalous yield strength and non-Schmid effect, influencing the mechanical behaviors of NBSAs significantly. These phenomena are closely related to the ⟨110⟩{111} superdislocations. In the present work, based on the quasiharmonic density functional theory (DFT), the improved semi-discrete variational Peierls-Nabarro (SVPN) model is employed to systematically investigate the properties of such superdislocations, in terms of dislocation density distribution, splitting distance (i.e., anti-phase boundary APB, super-lattice intrinsic stacking fault SISF and complex stacking fault width CSF), total energy, Peierls stress and so on. It is found that the predicted superdislocation characters, especially the Peierls stress, are in good agreement with previous experimental results, considering the lattice discreteness effect. This implies that the lattice discreteness plays an important role in the total dislocation energy and Peierls stress, which has however been ignored in previous studies. Although such superdislocation may have several possible core configurations, the dissociation of four 1/6⟨112⟩ Shockley partials separated by two CSFs and an APB may be more energetically favorable at both 300 K and 1200 K. In addition, the superdislocation properties with such the core structure may be easily affected by applied stress. The present work provides an improved SVPN model with consideration of lattice discreteness to describe the superdislocation properties in Ni3Al, which may be helpful for understanding the plastic deformation behavior of the γ′ phase and NBSAs.

The effect of impurities on stacking fault energy and dislocation properties in γ-TiAl

The dislocation properties of γ-TiAl are systematically investigated under the effect of impurities with semi-discrete variational Peierls-Nabarro (SDVPN) model. The impurities prefer to locating at the first layer below the stacking fault plane along 1/2<112‾] which is attributed to the local effect between stacking fault and impurities by local energy scheme analysis. The dislocation dissociation width of γ-TiAl is 1.58 Å, which shows a compact core character and is well consistent with that by atomic simulation and experimental observation. Negligible changes occur in the dislocation dissociation width as C or O is doped, while the dislocation does not even dissociate with N or H addition. The Peierls stress is reduced and the dislocation mobility is improved by N, H, and O, while C takes an opposite effect. The improved dislocation mobility with H is consistent with the enhanced dislocation mobility by H in experiments, while the improved dislocation mobility with O in calculation excludes the dislocation pinning effect of oxygen in solution.

Stress-dependence of dislocation dissociation, nucleation and annihilation in elastically anisotropic Cu

Dislocation dynamics under external stresses are an important manifestation of plastic behaviors of materials. In this study, based on the work of Shen et al. (2014), an improved microscopic phase field (MPF) model incorporating gradient energy term into the energy functional is developed to thoroughly investigate the stress dependence of whole dislocation evolution processes, including dislocation dissociation, nucleation, and annihilation in Cu with prominent elastic anisotropy. Necessary physical parameters such as elastic constants, lattice energy and gradient energy coefficients are all determined by atomistic molecular statics (MS) simulations. The residue theorem is utilized to accurately solve the anisotropic stress kernels as pre-factors characterizing the long-range elastic interaction produced by edge and screw components of mixed dislocations in anisotropic Cu. The Peierls stress of screw dislocations and stacking fault width under applied stress are evaluated by the MPF model and compared with those assessed by the semi-discrete variational Peierls-Nabarro (SVPN) model and MS simulations. The MPF model is then applied to screw dislocation dipoles and dislocation loops to examine possible dislocation evolution paths under different stress states. Through systematic simulations, we find the dislocation evolution regimes are closely related with the applied stress, and can be distinguished by three important critical stresses, i.e., critical stress, singular stress, and nucleation stress. Specifically, when the applied stress is less than the critical stress, the dislocation loops always collapse in the end, and do not depend on the final states of Shockley partials, which may exist or have annihilated. Furthermore, when the applied stress exceeds the nucleation stress, partial dislocations will spontaneously nucleate within stacking fault (SF) area, followed by a series of dislocation reactions until partial dislocations bρ1 and bρ3 are left in the end. Besides, the shear stress components perpendicular to the Escaig stress in slip plane are found to suppress the nucleation of partial dislocations in SF area.

Generalized stacking fault energies and critical resolved shear stresses of random α-Ti-Al alloys from first-principles calculations

The critical resolved shear stress (CRSS) and its related plastic deformation of titanium with hexagonal close packed structure are highly anisotropic, leading to low ductility of the material. Understanding the alloying effect on the CRSS is crucial for the improvement of the mechanical properties through rational composition design. Accurate prediction of the CRSS is not straightforward due to the atomic randomness of the alloy. The generalized stacking fault energies (GSFEs) for the basal and prismatic plane 〈a〉 slips of random α-Ti1−xAlx alloys (0≤x≤0.1875) are calculated by using first-principles methods including exact muffin-tin orbitals (EMTO) and plane-wave psuedopotential (VASP) methods in this work. The random distribution of Al in the alloy is treated by using both coherent potential approximation (CPA) for EMTO and special quasirandom structure (SQS) techniques for VASP. The CRSSs are then evaluated within the frame work of semi-discrete variational Peierls-Nabarro model. The VASP-SQS calculations with atomic relaxation generate reasonably good GSFE and CRSS compared with the EMTO-CPA and VASP-SQS calculations without atomic relaxation. For pure Ti, the unstable stacking fault energy (γusf) and CRSS (τa) for the basal 〈a〉 slip are higher than those for the prismatic 〈a〉 slip. With increasing Al concentration x, γusfb for the basal 〈a〉 slip decreases whereas γusfp for the prismatic 〈a〉 increases. The CRSSs for the basal 〈a〉 slip (τab) and the prismatic 〈a〉 slip (τap) both increase with x while τap increases faster than τab such that τab approaches to τap. The calculated CRSSs explain successfully our recently measured mechanical properties (yield strength, fracture toughness, ultimate strength, and elongation) of the α-Ti1−xAlx alloy against x.

Unraveling the dislocation core structure at a van der Waals gap in bismuth telluride

Tetradymite-structured chalcogenides such as bismuth telluride (Bi2Te3) are of significant interest for thermoelectric energy conversion and as topological insulators. Dislocations play a critical role during synthesis and processing of such materials and can strongly affect their functional properties. The dislocations between quintuple layers present special interest since their core structure is controlled by the van der Waals interactions between the layers. In this work, using atomic-resolution electron microscopy, we resolve the basal dislocation core structure in Bi2Te3, quantifying the disregistry of the atomic planes across the core. We show that, despite the existence of a stable stacking fault in the basal plane gamma surface, the dislocation core spreading is mainly due to the weak bonding between the layers, which leads to a small energy penalty for layer sliding parallel to the van der Waals gap. Calculations within a semidiscrete variational Peierls-Nabarro model informed by first-principles calculations support our experimental findings.

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.

First-principles investigation of strain effects on the stacking fault energies, dislocation core structure, and Peierls stress of magnesium and its alloys

Taking pure Mg, Mg-Al, and Mg-Zn as prototypes, the effects of strain on the stacking fault energies (SFEs), dislocation core structure, and Peierls stress were systematically investigated by means of density functional theory and the semidiscrete variational Peierls-Nabarro model. Our results suggest that volumetric strain may significantly influence the values of SFEs of both pure Mg and its alloys, which will eventually modify the dislocation core structure, Peierls stress, and preferred slip system, in agreement with recent experimental results. The so-called ``strain factor'' that was previously proposed for the solute strengthening could be justified as a major contribution to the strain effect on SFEs. Based on multivariate regression analysis, we proposed universal exponential relationships between the dislocation core structure, the Peierls stress, and the stable or unstable SFEs. Electronic structure calculations suggest that the variations of these critical parameters controlling strength and ductility under strain can be attributed to the strain-induced electronic polarization and redistribution of valence charge density at hollow sites. These findings provide a fundamental basis for tuning the strain effect to design novel Mg alloys with both high strength and ductility.

Atomically informed nonlocal semi-discrete variational Peierls-Nabarro model for planar core dislocations

Prediction of Peierls stress associated with dislocation glide is of fundamental concern in understanding and designing the plasticity and mechanical properties of crystalline materials. Here, we develop a nonlocal semi-discrete variational Peierls-Nabarro (SVPN) model by incorporating the nonlocal atomic interactions into the semi-discrete variational Peierls framework. The nonlocal kernel is simplified by limiting the nonlocal atomic interaction in the nearest neighbor region, and the nonlocal coefficient is directly computed from the dislocation core structure. Our model is capable of accurately predicting the displacement profile, and the Peierls stress, of planar-extended core dislocations in face-centered cubic structures. Our model could be extended to study more complicated planar-extended core dislocations, such as <110> {111} dislocations in Al-based and Ti-based intermetallic compounds.

Improvement of nonlocal Peierls-Nabarro models

We review the major efforts that improve the accuracy of Peierls-Nabarro (PN) model in predicting core structure and Peierls stress, and recognize that the nonlocal atomic interactions in the core region should be accounted for in calculating the dislocation energy. Although some efforts have been devoted to taking the nonlocal interaction into account, further improvement is needed to simplify the computational complexity and resolve the inconsistency between the continuum model and the discrete nature of the lattice. Here we developed a two-dimensional (2D) and a three-dimensional (3D) nonlocal semi-discrete variational Peierls-Nabarro (SVPN) models by incorporating the nonlocal atomic interactions into the semi-discrete variational Peierls framework. The nonlocal SVPN models are applied to dislocations with extended core in copper and compact core in iron. Molecular dynamics simulations are performed to validate the model predictions. We found that the nonlocal SVPN model (both 2D and 3D) significantly improves the prediction accuracy for dislocation core structure and Peierls stress. Moreover, the results show that the 2D and 3D models give similar predictions of the Peierls stress and dislocation core structure, when the atomic relaxation in the normal direction is allowed in the 2D model to describe the interatomic interactions in the slip plane.

Dislocation core properties of β-tin: a first-principles study

Dislocation core properties of tin (β-Sn) were investigated using the semi-discrete variational Peierls–Nabarro (SVPN) model. The SVPN model, which connects the continuum elasticity treatment of the long-range strain field around a dislocation with an approximate treatment of the dislocation core, was employed to calculate various core properties, including the core energetics, widths, and Peierls stresses for different dislocation structures. The role of core energetics and properties on dislocation character and subsequent slip behavior in β-Sn was investigated. For instance, this work shows that a widely spread dislocation core on the {110} plane as compared to dislocations on the {100} and {101} planes. Physically, the narrowing or widening of the core will significantly affect the mobility of dislocations as the Peierls stress is exponentially related to the dislocation core width in β-Sn. In general, the Peierls stress for the screw dislocation was found to be orders of magnitude higher than the edge dislocation, i.e., the more the edge component of a mixed dislocation, the greater the dislocation mobility (lower the Peierls stress). The largest Peierls stress observed was 365 MPa for the dislocation on the {101} plane. Furthermore, from the density plot, we see a double peak for the 0° (screw) and 30° dislocations which suggests the dissociation of dislocations along these planes. Thus, for the {101} slip system, we observed dislocation dissociation into three partials with metastable states. Overall, this work provides qualitative insights that aid in understanding the plastic deformation in β-Sn.

Quasi-periodic variation of Peierls stress of dislocations in face-centered-cubic metals

The Escaig stress, i.e. the shear stress perpendicular to the Burgers vector, modulates the stacking fault area between two partials of a full dislocation, in turn, affects the mobility of the dislocation. In this paper, using the newly improved semi-discrete variational Peierls-Nabarro (SVPN) model we studied the variation of Peierls stress (τp) of dislocations in face-centered-cubic crystals with respect to the Escaig stress. We found that τp quasi-periodically oscillates and the oscillation gradually decreases with the increase of Escaig stress. This quasi-periodic variation of τp can be mathematically described by the combination of a sinusoidal and an exponential function, and further accounted for by the variation of the stacking fault width (SFW) between two partials during their movement under applied stress. For the maximum τp, SFW is about integral multiples of the Peierls period. For the minimum τp, SFW is around half-integral multiples of Peierls period. The variation of τp is associated with the oscillation magnitude of SFW from half-integral multiples to integral multiples of the Peierls period and then back to integral multiples of Peierls period caused by the Escaig stress. Molecular dynamics (MD) simulations further examined quasi-periodic variation of τp, validating the SVPN model's capability of predicting sophisticated behavior of dislocation under applied stress.

Peierls stress in face-centered-cubic metals predicted from an improved semi-discrete variation Peierls-Nabarro model

In order to quantitatively predict Peierls stress, a semi-discrete variational Peierls-Nabarro model is improved by incorporating an additional gradient energy term into the energy functional. This gradient energy term is designed to effectively represent the influence of both the discreteness of atoms and the quick variations of the displacement profile in the dislocation core. Using face-centered-cubic metals as a model system for validation, we obtain a more accurate prediction of the displacement profile across the slip plane, and consequent precise Peierls stress, within a few times the prediction from molecular dynamics calculations.

Can vacancies lubricate dislocation motion in aluminum?

The interaction of vacancy with dislocations in Al is studied using the semidiscrete variational Peierls-Nabarro model with ab initio determined gamma surface. For the first time, we confirm theoretically the so-called vacancy lubrication effect on dislocation motion in Al, a discovery that can settle a long-standing controversy in dislocation theory for fcc metals. We provide insights into the lubrication effect by exploring the connection between dislocation mobility and its core width. We predict an increased dislocation splitting in the presence of vacancy. We find that on average there is a weak repulsion between vacancies and dislocations which is independent of dislocation character.