Dislocation Energy Research Articles

A maximum-principle approach to the minimisation of a nonlocal dislocation energy

In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies Iα defined on probability measures in $\mathbb{R}^2$. The purely nonlocal term in Iα is of convolution type, and is isotropic for α = 0 and anisotropic otherwise. The cases α = 0 and α = 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of Iα have been characterised by the same authors in an earlier paper, by exploiting some formal similarities with the Euler equation, and by means of complex-analysis techniques. We here propose a different approach, that we believe can be applied to more general energies.

The key role played by dislocation core radius and energy in hydrogen interaction with dislocations

It is generally believed that the H-induced reduction in dislocation energy plays a key role in modifying dislocation behaviors in the process of hydrogen embrittlement. Here, we examine the factors that lead to H reducing the line energies of the edge and screw dislocations in bcc Fe by atomistic simulations. Grand canonical Monte Carlo simulations are conducted to obtain the distribution of H around the dislocations. We find that H mainly aggregates at the dislocation cores and the H concentration in the elastic field of dislocations is extremely low. The direct consequences of such a distribution pattern of H are as follows. (i) In contrast with previous studies, H induces no change in the shear modulus of the systems containing dislocations. (ii) H increases the core radii and decreases the core energies of the dislocations, which are the only factors leading to the reduction of dislocation line energy by H. (iii) H brings little effect on the stress field of either the edge or screw dislocation, implying that H induces almost no stress-shielding effect on dislocations. A linear relation between the critical shear stress for homogeneous dislocation nucleation and logarithmic bulk H concentration is thus revealed, based on the atomistic result of the H-induced increase in the core radius and decrease in the core energy of the dislocations. The present results indicate that the dislocation-dislocation interaction in the presence of H, which is the key ingredient for the H-enhanced localized plasticity mechanism for hydrogen embrittlement, can be easily evaluated by the linear elastic theory of dislocations if the core radius and energy of dislocations are properly described.

Atomistic dislocation core energies and calibration of non-singular discrete dislocation dynamics

The total energy of an atomistic dislocation includes contributions from the inelastic/large-distortion ‘core’ region. Capturing this inelastic ‘core’ energy is important, especially for dislocations with a curvature in the 10–100 nm scale. Current implementations of discrete dislocation dynamics (DDD) mesoscale simulations either approximate or neglect the core energy and so do not provide consistency with fully-atomistic studies. Using established interatomic potentials for FCC metals, the total dislocation energy is computed directly in atomistic simulations of straight dislocations and a core energy at any desired cut-off core radius is obtained as a function of dislocation character. A proper introduction of the atomistic core energy into the ParaDiS DDD code that uses a non-singular theory (Cai et al 2006 J. Mech. Phys. Solids 54 561–87) is then presented. The resulting atomistically-informed ParaDiS DDD is used to simulate the periodic bow-out of edge and screw dislocations in near-elastically-isotropic aluminum at various length and stress, with comparisons to fully-atomistic simulations. Generally good agreement is obtained between DDD and atomistics, with the best agreement achieved using a non-singular regularization parameter in the range of a = 5 – 10b. The analysis is then extended to compute the core energy of the Shockley partial dislocations that arise in the dissociation of perfect dislocations in fcc metals.

Mechanical properties of single-phase Al1−xInxN films across the compositional range (0 ≤ x ≤ 0.7) grown by radio-frequency magnetron sputtering

Aluminum indium nitride (Al1−xInxN) films across the compositional range (0 ≤ x ≤ 0.7) were grown on silicon substrate by radio-frequency magnetron sputtering. The In composition was tuned by varying the number of In blocks (nIn) covered on the Al target. We further investigated the dependence of structures, morphologies and mechanical properties on In composition x. It has been found that the as-grown Al1−xInxN films are pure hexagonal phase without phase separation and the (0 0 2) orientation gradually enhances with increasing nIn. While nIn is more than 3, the preferred orientation of Al1−xInxN films would change from (1 0 1) to (0 0 2) plane. Compared with pure AlN film, the Al1−xInxN films exhibit flatter and smoother surface morphology. In addition, the increased nIn leads to a decrease of root-mean-squared (RMS) surface roughness and grain size. All ternary Al1−xInxN films exhibit better mechanical properties than pure AlN film. The hardness and modulus of as-grown Al1−xInxN films tend to decrease when the In composition × is larger than 0.3. The phenomenon can be attributed to the lower density of dislocation and weaker bond energy in Al1−xInxN alloys.

Impact of Molecular Dynamics Simulations on Research and Development of Semiconductor Materials

ABSTRACTAtomic scale defects critically limit performance of semiconductor materials. To improve materials, defect effects and defect formation mechanisms must be understood. In this paper, we demonstrate multiple examples where molecular dynamics simulations have effectively addressed these issues that were not well addressed in prior experiments. In the first case, we report our recent progress on modelling graphene growth, where we found that defects in graphene are created around periphery of islands throughout graphene growth, not just in regions where graphene islands impinge as believed previously. In the second case, we report our recent progress on modelling TlBr, where we discovered that under an electric field, edge dislocations in TlBr migrate in both slip and climb directions. The climb motion ejects extensive vacancies that can cause the rapid aging of the material seen in experiments. In the third case, we discovered that the growth of InGaN films on (0001) surfaces suffers from a serious polymorphism problem that creates enormous amounts of defects. Growth on ($11\bar{2}0$) surfaces, on the other hand, results in single crystalline wurtzite films without any of these defects. In the fourth case, we first used simulations to derive dislocation energies that do not possess any noticeable statistical errors, and then used these error-free methods to discover possible misuse of misfit dislocation theory in past thin film studies. Finally, we highlight the significance of molecular dynamics simulations in reducing defects in the design space of nanostructures.

Mechanism of grain boundary serration during hot deformation of AZ31 alloy: Role of grain boundary dislocations and grain boundary sliding

Despite the industrial significance of grain boundary serration for improving creep resistance and retarding crack propagation, the mechanism of grain boundary serration during hot deformation is still unclear. The results of several studies indicated that increased inhomogeneity of the stored energy under high-stress deformation suppressed grain boundary serration. Thus, traditional strain-induced boundary migration, in which the stored energy difference between adjacent grains acts as the driving force for grain boundary serration, must be modified for hot deformation. The experimental results of this study suggested that dissociated grain boundary dislocations from lattice dislocations, rather than the elastic energy of the lattice dislocations, cause the nucleation of serrated grain boundaries. Grain boundary sliding on the serrated grain boundary asymmetrically generated an elastic energy field and an accommodated serrated grain boundary. The presence of an elastic energy field was predicted by dislocation climbing and statically distributed dislocations near the elastic energy field. Since the asymmetrically deformed serrated grain boundary exhibited sub-migration behavior in the elastic energy field, it was concluded that the applied shear stress (applied by grain boundary sliding) on the serrated grain boundary dominated the growth of serrated grain boundaries until dynamic recrystallization occurred.

Modeling of atomistic scale shear failure of Ag/MgO interface with misfit dislocation network

Metal/ceramic interfaces have broad applications and misfit dislocation network (MDN) is a prominent feature of the equilibrium metal/ceramic interfaces. As one main failure mode, interface shear failure is strongly affected by the motion of MDN. In this work, we investigate the equilibrium interface structure and shear failure of Ag/MgO interface via atomistic simulation method. Periodically distributed in-plane strain field caused by MDN and severe strain concentration at dislocation node regions are found by strain analysis. During interface shearing, these dislocation nodes act as strong pinning points to the gliding motion of MDN, which leads to bending of dislocation lines. Besides, energy analysis shows the interface shear stress is largely dependent on the variation of misfit dislocation energy. To understand interface shear failure under more complex conditions, we study the effect of model thickness and shear direction further. Due to transformation of nodal structure, the shear strength of thick model eventually decreases by almost a quarter; shear failure along the direction of Burgers vector is found to be energetically favored, with the lowest interface shear strength. This work reveals the crucial role of MDN in interface shear process, and the theoretical understanding gives some hints to metal/ceramic interface design.

Effects of Calcium on Strength and Microstructural Evolution of Extruded Alloys Based on Mg-3Al-1Zn-0.3Mn

A family of alloys based on the Mg-Al-Zn-Ca-Mn system (Mg-3Al-1Zn-1Ca-0.3Mn, Mg-3Al-1.5Zn-0.5Ca-0.3Mn, and Mg-3Al-1Ca-0.3Mn, wt pct) was developed for extrusion. Their mechanical properties were evaluated by tensile testing at room temperature, and compared to those of the benchmark Mg-alloy Mg-3Al-1Zn-0.3Mn (AZ31). The microstructures of the extruded alloys were characterized in detail in order to reveal the effect of Ca on microstructural evolution, and consequently the alloy strength. The addition of Ca to the AZ31 stifles dynamic recrystallization and grain growth, with only ~30 pct recrystallization and a recrystallized grain size of ~480 nm. In contrast, the benchmark alloy is essentially completely recrystallized with an average grain size of ~2.3 μm. A high density of low-angle grain boundaries (LAGBs) and dislocations were observed in Ca-containing alloys, and were identified as a major factor in the observed strengthening. Such LAGBs form cellular subgrains predominantly along initial grain boundaries, or newly formed boundaries that are closely spaced (~ 600 nm) and nearly parallel to the extrusion direction. The subgrains have an ultrafine size of 100 to 400 nm, and difficult to convert to recrystallized grains. Solute segregation to grain boundaries was also observed. It is hypothesized that it is the Ca segregation to dislocation cores along LAGBs that decreases the dislocation mobility and stabilizes LAGBs, by thermodynamically decreasing the dislocation energy and/or kinetically imposing a solute drag effect.

The Key Role Played by Dislocation Core Radius and Energy in Hydrogen Interaction with Dislocations

It is generally believed that the H-induced reduction in dislocation energy plays a key role in modifying dislocation behaviors in the process of hydrogen embrittlement. Here, we examine the factors that lead to H reducing the line energies of the edge and screw dislocations in bcc Fe by atomistic simulations. Grand canonical Monte Carlo simulations are conducted to obtain the distribution of H around the dislocations. We find that H mainly aggregates at the dislocation cores and the H concentration in the elastic field of dislocations is extremely low. The direct consequences of such a distribution pattern of H are as follows. (i) In contrast with previous studies, H induces no change in the shear modulus of the dislocation systems. (ii) H increases the core radii and decreases the core energies of the dislocations, which are the only factors leading to the reduction of dislocation line energy by H. (iii) H brings little effect on the stress field of either the edge or screw dislocation, implying that H induces almost no stress-shielding effect on dislocations. A linear relation between the critical shear stress for homogeneous dislocation nucleation and logarithmic bulk H concentration is thus revealed, based on the atomistic result of the H-induced increase in the core radius and decrease in the core energy of the dislocations. The present results indicate that the dislocation-dislocation interaction in the presence of H, which is the key ingredient for the H-enhanced localized plasticity mechanism for hydrogen embrittlement, can be easily evaluated by the linear elastic theory of dislocations if the core radius and energy of dislocations are properly described.

V-pit pinning at the interface of high and low-temperature gallium nitride growth

We have performed a detailed study on the development of V-pits in low temperature GaN, varying the temperature for different precursors and carrier gases, showing that the growth conditions can lead to two growth regimes. From this, and considering surface energies, we have built a theoretical framework for interpreting these data. We can then consider that for one growth regime, the V-pit size is proportional to the thickness of the grown layer, implying that the V-pits are essentially pinned to the high temperature–low temperature interface, and the pit continues to grow larger as the film becomes thicker. In the other regime, the increased surface energy of the V-pit balances the energy of the threading dislocation, and the V-pit maintains a constant size with increasing thickness. These stabilized V-pits are considered unpinned, or “floating”. We have shown that increasing temperature and the use of hydrogen rather than nitrogen for the carrier gas favors unpinned V-pits, and can therefore be considered to lower the energy of the (0002) plane versus the preferred inclined plane of {10–11}. These unpinned V-pits are much smaller than those that are pinned, and this is therefore the favored regime for high quality growth interfaces and surface morphology.

Combined elastic-plastic energy driving criteria of rafting behavior for nickel-based single crystal superalloys

The rafting of the microstructure of Nickel-based single crystal superalloys is related to the mechanics of the precipitated and matrix phases flowing together. Thus, based on the distribution of the driving force for two-phase movement, an elastic-plastic energy driving criteria of rafting behavior have been proposed to explain its mechanics and to predict the type of rafting. The microscopic cell finite element model that was developed introduced the rafting criteria to predict the morphology and type of rafting during creep under uniaxial tensile and compressive loads, respectively. The prediction is consistent with experimental observation. The results show that the drop in the interfacial dislocation energy density between the three <001> crystal directions affects the direction of the matrix phase movement, and the disturbance of the interfacial elastic strain energy changes the growth direction of the precipitated phase.

Dependence of Dislocation Energy on Interatomic Distance

Dislocations span over distances that are much larger than the interatomic distance . Therefore, an asymptotic expansion of dislocation energy as is of interest. In this paper, such asymptotic expansion is constructed. It has a simple form of , where and are the functionals of the dislocation line (which do not depend on ) and is the dislocation length. Analytical formulas for and are obtained.

Size effect of layer thickness on stress fields due to interface core-spreading dislocation arrays in multilayers

The elastic fields due to interface dislocation arrays with spreading core in multilayers are derived analytically by means of the superposition of Green function of individual compact dislocations. The numerical results for Cu-Nb multilayers with dislocation arrays demonstrate that: (1) There exists a critical layer thickness (CLT) in describing interface shear stress. The maximum interface shear stress (MISS) increases with the layer thickness as it exceeds CLT; while the MISS decreases rapidly with the layer thickness as it is thinner than CLT. (2) Both the density and core width of interface dislocations have salient effect on the stress fields. The overall stress field is confined by high density of dislocations, and the stress singularity is released by core spreading. (3) A new dislocation prefers to glide in the middle region between the two adjacent interface dislocations by means of Peach-Kohler (P-K) force acting on the dislocation and elastic energy of dislocations. These findings would provide foundations to deepen the understanding of microscopic plastic deformation mechanisms and their related macroscopic mechanical properties of metallic anisotropic multilayers.

Study of Geometrically Necessary Dislocations of a Partially Recrystallized Aluminum Alloy Using 2D EBSD.

During recrystallization, the growth of fresh grains initiated within a deformed microstructure causes dramatic changes in the dislocation structure and density of a heavily deformed matrix. In this paper, the microstructure of a cold rolled and partially recrystallized Al-Mg alloy (AA5052) was studied via electron backscattered diffraction (EBSD) analysis. The structure and density of the geometrically necessary dislocations (GNDs) were predicted using a combination of continuum mechanics and dislocation theory. Accordingly, the Nye dislocation tensor, which determines the GND structure, was estimated by calculation of the lattice curvature. To do so, five components of the Nye dislocation tensor were directly calculated from the local orientation of surface points of the specimen, which was determined by two-dimensional EBSD. The remaining components of GNDs were determined by minimizing a normalized Hamiltonian equation based on dislocation energy. The results show the elimination of low angle boundaries, lattice curvature, and GNDs in recrystallized regions and the formation of low angle boundaries with orientation discontinuities in deformed grains, which may be due to static recovery.

Structure and energy of the 〈11¯〉 screw dislocation in silicon using generalized Peierls theory

For the ⟨11¯0⟩ screw dislocation (nondissociated) in silicon, there are three types of core structures that are, respectively, referred to as the shuffle-set dislocation (A-core), glide-set dislocation (C-core), and mix-set dislocation (B-core). Each type of core further displays a planar or nonplanar (fourfold) configuration. In this paper, in the context of the generalized Peierls theory, the energy functional and equation of equilibrium of a screw dislocation with a nonorthogonal fourfold structure is derived and is used to investigate the structure and energy of the ⟨11¯0⟩ screw dislocation in silicon. We find that the energy of the shuffle-set dislocation with a fourfold core is considerably lower than that of the other types of cores. Thus, the shuffle-set dislocation with the fourfold core is the most stable. The glide-set dislocation with a planar core has the highest energy and is the most unstable. As the most stable structure, the shuffle-set screw dislocation with the fourfold core was responsible for wavy slip traces observed at low temperature.

Plasticity as the $${\Gamma}$$ Γ -Limit of a Two-dimensional Dislocation Energy: The Critical Regime Without the Assumption of Well-Separateness

In this paper, a strain-gradient plasticity model is derived from a mesoscopic model for straight parallel edge dislocations in an infinite cylindrical crystal. The main difference to existing work is that in this work the well-separateness of disloactions is not assumed. In order to prove meaningful lower bounds the ball construction technique, which was developed in the context of Ginzburg-Landau by Jerrard and Sandier, is adapted and modified. To overcome the difficulty of a loss of rigidity on thin annuli during the ball construction a combination of combinatorial arguments and local modifications of the occurring elastic strains is presented.

PNADIS: An automated Peierls–Nabarro analyzer for dislocation core structure and slip resistance

Dislocation is one of the most critical and fundamental crystal defects that dominate the mechanical behavior of crystalline solids, however, a quantitative determination of its character and property in experiments is quite challenging and limited so far. In this paper, a fully automated Peierls–Nabarro (P–N) analyzer named PNADIS is presented; a complete set of the character and property of dislocation can be automatically derived, including the dislocation core structure, Peierls energy and stress, pressure field around dislocation core, solute/dislocation interaction energy, as well as the energy barrier and increase in critical-resolved shear stress at 0 K for solid solution strengthening. Furthermore, both one-dimensional (1D) and two-dimensional (2D) P–N models are implemented to meet the demand to analyze the character and property of dislocation for not only simple FCC and HCP structures but also complex crystals. The implementation of this code has been critically validated by a lot of evaluations and tests including 1D P–N model for complex crystals, 2D P–N model for FCC and HCP metals, pressure field around dislocation core, and solid solution strengthening for alloys. We expect that the automated feature of this code would provide a high-efficiency solution for determining the character and property of dislocation. Program summaryProgram title: PNADISProgram Files doi:http://dx.doi.org/10.17632/whk6wdy3nn.1Licensing provisions: GNU General Public License 3Programming language: MATLABNature of problem: To determine automatically the character and property of dislocation, including dislocation core structure, Peierls stress, pressure field around dislocation core and solid solution strengthening, for not only FCC and HCP structures but also complex crystals.Solution method: The generalized stacking fault energy is firstly fitted by Fourier expansion, and meanwhile an appropriate trial function of disregistry vector is chosen. Afterwards, a least square minimization of the difference between elastic resistance and restoring force for one-dimensional Peierls–Nabarro model, or a global minimization of the total dislocation energy via particle swarm optimization or genetic algorithm for two-dimensional Peierls–Nabarro model, will be performed to determine the dislocation core structure of complex crystals, or FCC and HCP structures. Finally, the Peierls stress, pressure field around dislocation core and solid solute strengthening are derived from the calculated dislocation core structure.

Response of Hydrogen Desorption and Hydrogen Embrittlement to Precipitation of Nanometer-Sized Copper in Tempered Martensitic Low-Carbon Steel

This work is aimed at studying the hydrogen–copper precipitate interaction in a martensitic steel. Analysis of hydrogen thermal desorption revealed that precipitation of copper particles enhances the hydrogen trapping capability of tempered copper-containing martensitic steel. Moreover, precipitation of copper could make hydrogen retain longer in the steel, indicating a retarded diffusion of hydrogen. Copper precipitates as a hydrogen trapping site were observed to preserve an activation energy of 35.6 kJ mol−1 by Choo-and-Lee method after release for 4 h at room temperature. This value is higher than the activation energy of dislocation. Moreover, tempered steel with copper particles displayed better resistance to hydrogen embrittlement in notched, slow-strain-rate tensile tests.

Effect of mechanical activation on the structure of pulverized coal and iron ore powder

Morphologies and structures of pulverized coal and iron ore powder after mechanical activation were studied by SEM, XRD, FTIR and laser particle sizer. The microcrystalline structure of coal was found to be destroyed by mechanical activation via reducing the pile height and number of layers, and the organic structure of coal was altered through the destruction of the ether bond. Mechanical activation led to distortions and dislocations of the crystal lattice of iron ore, decreasing crystallite size, increasing the grain boundary area, and producing an amorphous phase. These increased the Gibbs free energies of dislocations and grain boundaries as well as the surface Gibbs free energy and the amorphization Gibbs free energy, which would eventually increase the energy stored in iron ore called mechanical storage energy. The longer the mechanical activation process, the higher mechanical storage energy for the iron ore will be saved. The amorphization Gibbs free energy is the biggest among the four kinds of Gibbs free energy, accounting for 94.8% ∼ 87.1% of the total storage energy in the mechanical activated iron ore.

Strain-Gradient Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

In this paper, we derive by means of $\Gamma$-convergence a macroscopic strain-gradient plasticity theory from a semidiscrete model for edge dislocations in an infinite cylindrical crystal. We consider an energy with subquadratic growth close to the dislocations. This allows us to treat the stored elastic energy without the need to introduce an ad hoc cut-off radius. As the main tool to prove a complementing compactness statement, we present a generalized version of the geometric rigidity result for fields with nonvanishing $\operatorname{curl}$. One of the main ingredients for the proof is a fine decomposition result for $L^1$-functions whose divergence is in certain critical Sobolev spaces.