Stereographic Triangle Research Articles

Orientation distribution and deviation behaviors of dislocation loops in a quenched Al-Cu alloy

Dislocation loops in quenched aluminum alloys are usually seen deviating from their nominal habit planes, while the orientation distribution and deviation mechanism are far from understanding. Based on the method of tomographic crystallography of dislocations and correlative in-situ heating experiments in transmission electron microscopy (TEM), we revisit the three-dimensional (3D) characteristics and deviation behaviors of dislocation loops in a quenched Al-Cu alloy with an aim to fully unravel the relationship between dislocation crystallography and deviation behaviors. The results show that all the dislocation loops exhibit approximately circular shapes and have a/2 < 110>-type Burgers vectors, with loop orientations deviating from their pure edge orientation {110} of 0°-17.8° and distributing mainly close to the [101]-[001] and [101]-[111] edges of the standard stereographic triangle. Three types of deviation behaviors of dislocation loops are then defined, which are characterized by loop rotation around 〈100〉, 〈110〉 and irregular axes, respectively. Theoretical calculations of strain energy variation with increasing deviation angle indicate that the relatively low energy barrier may account for a high frequency of loop deviation in the low angle range. Detailed in-situ TEM heating experiments and correlative tomographic crystallography analyses strongly demonstrate the specific roles of dislocation gliding and climbing at elevated temperatures in the shrinkage, polygonization and deviation processes of dislocation loops, and the competitive and cooperative motion of dislocation segments close to slip planes {111} are the fundamental reasons for the formation of three types of deviated dislocation loops.

Stereographic Projection of Theoretical Orientation Relationships Between Crystals of Any Types in Phase Transformation and Precipitation

AbstractStereographic projection is a classic technology to represent the angular relationships of lattice planes and directions. In some cases, it is necessary to project arbitrary lattice planes or directions of a crystal onto an arbitrary lattice plane of another crystal, e.g., when representing orientation relationships (ORs) in phase transformation or precipitation. Commercial EBSD (electron backscatter diffraction) software cannot illustrate theoretical orientation relationships based purely on the parallelism conditions. This work presents a generalized and unified formulation for stereographic projection of any lattice planes or directions onto any lattice planes of any crystals in the 32 point groups, which is utilized to represent the theoretical orientation relationships between any crystal types. Examples are given to illustrate the correlations of common orientation relationships in both cubic and hexagonal crystals. A procedure is also established to color code and simultaneously project all low‐index Miller/Bravais planes or directions of a crystal onto an arbitrary lattice plane of the same crystal, which can be utilized to visualize the crystal symmetry and determine the standard stereographic triangle (SST). All these illustrations are realized in a computer program written in C++ using the OpenGL graphic libraries.

Analysis of slip transfer behavior of tantalum during quasi in-situ compression

This paper reports a quasi in-situ study on the plastic deformation behavior of high purity tantalum, particularly the slip activation and slip transfer, using the electron backscatter diffraction method. We found that the rotation degree of grains is in the range of 0–15°; the rotation axes of slip-activated grains are mainly concentrated in the three corners, while the axes of slip-transfer ones are concentrated near {111} corner in the stereographic triangle. Microstructural parameters including grain size and grain aspect ratio were found to have no apparent relationships with the slip activation or slip transfer process. Most m' factors for the grains underwent slip-transfer are larger than 0.27, and most slip systems involved in the slip-transfer observations are associated with relatively high Schmid factors (SFs), indicating that slip would more easily transfer through the boundaries with large m' factors. In addition, according to the distribution of geometrical necessary dislocation density and the SF of slipping activated grains, we found the slip transfer process is affected by a combination of factors, including the SF, m' factor, and the strain concentration in local regions.

Origin of variable propensity for anomalous slip in body-centered cubic metals

Many transition metals crystalizing in the body-centered cubic (bcc) structure exhibit anomalous slip on low-stressed planes at low homologous temperatures, which cannot be reconciled with the Schmid law. Specifically, for uniaxial loading in the center of the stereographic triangle, this is manifested by and screw dislocations moving on low-stressed planes. While the anomalous slip is often attributed to non-planar cores of screw dislocations or to the tendency for their networks to glide easily, it remains unclear why it dominates the plastic deformation in some bcc metals, whereas it is weak or even absent in others. Using molecular statics simulations at 0 K, we demonstrate that the anomalous slip in bcc metals is intimately linked with the stability of screw junctions between two intersecting screw dislocations under stress (for example, and screws giving rise to the [100] junction). Our atomic-level studies show that in nearly all bcc metals of the 5th and 6th groups these junctions cannot be broken by the applied stress and the three dislocations can only move on the common plane (in the above example on the plane). On the other hand, these junctions are found to be unstable in alkali metals, tantalum, and iron, where the application of stress results in unzipping of the two dislocations and their further glide on the planes predicted for isolated dislocations. These results also suggest that the experimentally observed increased propensity for the anomalous slip in further stages of plastic deformation may be explained by reduced curvatures of screw dislocations in dense networks.

Screening Surface Structure–Electrochemical Activity Relationships of Copper Electrodes under CO2 Electroreduction Conditions

Understanding how crystallographic orientation influences the electrocatalytic performance of metal catalysts can potentially advance the design of catalysts with improved efficiency. Although single crystal electrodes are typically used for such studies, the one-at-a-time preparation procedure limits the range of secondary crystallographic orientations that can be profiled. This work employs scanning electrochemical cell microscopy (SECCM) together with co-located electron backscatter diffraction (EBSD) as a screening technique to investigate how surface crystallographic orientations on polycrystalline copper (Cu) correlate to activity under CO2 electroreduction conditions. SECCM measures spatially resolved voltammetry on polycrystalline copper covering low overpotentials of CO2 conversion to intermediates, thereby screening the different activity from low-index facets where H2 evolution is dominant to high-index facets where more reaction intermediates are expected. This approach allows the acquisition of 2500 voltammograms on approximately 60 different Cu surface facets identified with EBSD. The results show that the order of activity is (111) < (100) < (110) among the Cu primary orientations. The collection of data over a wide range of secondary orientations leads to the construction of an “electrochemical–crystallographic stereographic triangle” that provides a broad comprehension of the trends among Cu secondary surface facets rarely studied in the literature, [particularly (941) and (741)], and clearly shows that the electroreduction activity scales with the step and kink density of these surfaces. This work also reveals that the electrochemical stripping of the passive layer that is naturally formed on Cu in air is strongly grain-dependent, and the relative ease of stripping on low-index facets follows the order of (100) > (111) > (110). This allows a procedure to be implemented, whereby the oxide is removed (to an electrochemically undetectable level) prior to the kinetic analyses of electroreduction activity. SECCM screening allows for the most active surfaces to be ranked and prompts in-depth follow-up studies.

Development of a method for calculating the LTEC anisotropy of Zr alloys

The accuracy of determining the LTEC with respect to metals and alloys with an HCP lattice based on inverse pole figures (IPF) is analyzed depending on the number of experimental points on the IPF using three averaging options: (1) taking into account the irregular arrangement of reflections on the stereographic triangle according to Morris; (2) by the multiplicity factor and (3) with the same weight of each orientation. It is shown that when evaluating the LTEC for semi-finished products with a basal texture, 17 reflections on the IPF are sufficient to provide an error of <1% when using Morris averaging and the multiplicity factor; in the case of a prismatic texture, an error of <1% is provided by all three averaging options, while Morris averaging is minimal.

Crystallographic anisotropy of the superelastic and mechanical properties of the Ti–20Zr–3Mo–3Sn alloy evidenced by nanoindentation at the grain scale

Nanoindentation was combined with electron backscatter diffraction (EBSD) to investigate the crystallographic anisotropy of indentation responses of individual grains in the superelastic Ti2033 alloy. The high statistic of measurements from grains oriented over a large range of crystallographic directions facilitated a fundamental study covering the entire stereographic triangle of the β phase. For a highlighted visual effect, results were presented as inverse pole figure distribution maps of depth recovery, work recovery, indentation modulus and hardness. Results showed that the pronounced anisotropy behaviors of indentation modulus, depth and work recovery ratios resulted from the anisotropy in Young’s modulus and in compressive lattice distortion between the bcc structure of β phase and the orthorhombic structure of the stress-induced α’’ martensite phase, respectively. However, orientation dependence was gone when plasticity occurred. The orientation-independency in indentation hardness was due to the various slip systems that might be activated within bcc crystals.

Ab Initio molecular dynamics study of threshold displacement energy in Zirconium Nitride

• The threshold displacement energy ( E d ) in ZrN was evaluated by the AIMD method. • The weighted average values of E d were 33 eV and 29 eV for Zr and N atoms, respectively, from the several crystallographic directions investigated in this study. • The values of E d and the collision process were significantly dependent on the type of atom and crystallographic direction. • The [110] direction exhibited the minimum value of E d , which is associated with sequential replacement collisions. • The preferable configurations for Zr and N interstitials were the <111> split-type and the tetrahedral sites, respectively, which influence the difference in the values of E d . Zirconium nitride (ZrN) is a promising matrix candidate for advanced nuclear fuels and transmutation of minor actinides. This study investigates the displacement process induced by low-energy recoils in ZrN using ab initio molecular dynamics (AIMD) simulations to evaluate the threshold displacement energy ( E d ). Observations of the collision processes of primary knock-on atoms (PKAs) for both Zr and N atoms were performed for seven different directions: [100], [110], [111], [210], [211], [221], and [321], which cover most regions of the stereographic triangle. The values of E d ranged from 15 eV to 50 eV, and the collision processes were dependent on the crystallographic orientations. The weighted average values of E d evaluated from the PKA directions investigated in this study were 33 eV and 29 eV for the Zr and N atoms, respectively. Anti-site defects were not formed for either Zr or N PKAs. Sequential replacement collisions along the [110] atomic row played an important role in the collision process, providing a lower value of E d in the crystallographic directions. Furthermore, configurations of interstitials were different between the Zr and N atoms, which also influenced E d values.

Crystal plasticity modeling of non-Schmid yield behavior: from Ni3Al single crystals to Ni-based superalloys

A crystal plasticity finite element (CPFE) framework is proposed for modeling the non-Schmid yield behavior of L12 type Ni3Al crystals and Ni-based superalloys. This framework relies on the estimation of the non-Schmid model parameters directly from the orientation- and temperature-dependent experimental yield stress data. The inelastic deformation model for Ni3Al crystals is extended to the precipitate (γ′) phase of Ni-based superalloys in a homogenized dislocation density based crystal plasticity framework. The framework is used to simulate the orientation- and temperature-dependent yield of Ni3Al crystals and single crystal Ni-based superalloy, CMSX-4, in the temperature range 260–1304 K. Model predictions of the yield stress are in general agreement with experiments. Model predictions are also made regarding the tension–compression asymmetry and the dominant slip mechanism at yield over the standard stereographic triangle at various temperatures for both these materials. These predictions provide valuable insights regarding the underlying (orientation- and temperature-dependent) slip mechanisms at yield. In this regard, the non-Schmid model may also serve as a standalone analytical model for predicting the yield stress, the tension–compression asymmetry and the underlying slip mechanism at yield as a function of orientation and temperature.

Plastic deformation of magnetically isotropic Cr single crystals compressed at 77 K

A series of compression tests at 77 K was carried out on chromium single crystals in magnetically isotropic state to investigate the slip activity during the first 2–4% of plastic strain. Macroscopic samples of 3 mm thickness were polished mechanically and electrolytically and their orientations determined before and after deformation by electron backscatter diffraction to assess the magnitudes and directions of crystal rotations. The crystals compressed in the direction close to [001] deformed primarily by slip on the most highly stressed (1‾01)[111] and (101)[1‾11] systems. Loading the crystals in the directions close to the [011]−[1‾11] edge of the stereographic triangle resulted in preferential twinning on 112111 systems. For loading in the center-triangle region, diffraction contrast imaging in TEM reveals the presence of [100] screw junctions between intersecting 1/2111 and 1/2[11¯1¯] screw dislocations. Using molecular statics simulations, we show that this junction does not break under the applied stress and moves together with the two dislocations on their common (01‾1) plane. In contrast to the prediction of the Schmid law, the slip morphology for center-triangle orientations is thus dominated by anomalous slip on the low-stressed (01‾1) plane.

Crystallographic orientation dependence of mechanical properties in the superelastic Ti-24Nb-4Zr-8Sn alloy

Nanoindentation and electron backscattered diffraction techniques are used to study the dependence of crystallographic orientation on mechanical properties in individual grains of the superelastic polycrystalline Ti-24Nb-4Zr-8Sn alloy. Strain recovery, elastic modulus, and hardness are evaluated from the force-displacement curves using spherical and Berkovich indenters. Experimental data are reported in a standard stereographic triangle covering all possible crystallographic directions of loading in the bcc-\ensuremath{\beta} structure of the Ti-24Nb-4Zr-8Sn alloy. Our experiments show that both spherical and Berkovich indenters are suitable for probing anisotropy of elastic modulus whereas spherical indenter is more appropriate for probing anisotropy of superelastic response. The highest indentation strain recovery is measured along $\ensuremath{\langle}001{\ensuremath{\rangle}}_{\ensuremath{\beta}}$ and the lowest one is observed along $\ensuremath{\langle}111{\ensuremath{\rangle}}_{\ensuremath{\beta}}$. Our inverse pole figure distribution of indentation strain recoveries is in good agreement with inverse pole figure distribution of compressive lattice distorsions calculated from the crystallographic model of martensitic transformation in \ensuremath{\beta} titanium alloys. This established pattern in compression is very different from that seen during tensile deformation, and both our experiments and calculations confirm the strong tension-compression asymmetry of the strain response in superelastic \ensuremath{\beta} titanium alloys. The indentation modulus also shows significant crystallographic anisotropy: ${E}_{[001]\ensuremath{\beta}}&lt;{E}_{[101]\ensuremath{\beta}}&lt;{E}_{[111]\ensuremath{\beta}}$. In contrast, orientation dependence is lost when plasticity has set in, and hardness becomes independent of crystallographic orientation due to the fact that multiple directions are prone to generate slip systems in the bcc-\ensuremath{\beta} structure.

Atomistic simulations of energies for arbitrary grain boundaries. Part II: Statistical analysis of energies for tilt and twist grain boundaries

The orientation-dependency of grain boundary (GB) energy in Al is investigated using a cutoff sphere molecular dynamics model presented in Part I, which is applicable to arbitrary GBs. A GB energy dataset is generated using the model with deletion of over-close atoms and without rigid body translations in building the candidate configurations for a given GB. The dataset includes 12,062 tilt GBs (TIGBs) and twist GBs (TWGBs) with a broad range of misorientation angle/axis (θ/O) and GB-plane orientation (n). Statistical analyses of the results show that the energies overall increase with θ in the low-θ range and then level off for higher θ, and that the energies for non-coincident site lattice GBs also follow the increasing trend in the low-θ range. The energies of TIGBs decrease with the variation of O from the central regions to the edge and then corners of the stereographic triangle, and the O-dependency of GB energy becomes more severe for TWGBs. The current energy dataset suggests that the n-dependency of GB energy is best correlated with the orientation-dependency of surface energy following the broken-bond model and it does not support the existing perspective of low energies for GBs terminated by low-index or dense planes. For both types of GBs, the energy suffers a stronger influence from θ than n.

Origin of an unusual systematic variation in the heteroepitaxy of Ag on Ni – The roles of twinning and step alignment

A systematic variation in the orientation relationship (OR) of Ag films grown on Ni substrates previously discovered by a combinatorial approach is analyzed using concepts of grain boundaries, surface science and phase transformations. On roughly half of all Ni substrate orientations, Ag adopts a “special” OR that varies systematically from a twin OR, which develops on substrates lying along the (111)–(210) line of the standard stereographic triangle (SST), to the so-called oct-cube OR, which arises exclusively on (100) substrates. On the other half of Ni substrate orientations, Ag adopts the standard cube-on-cube OR. The special ORs are modeled by a) linear interpolation, b) 1D edge-to-edge matching and c) 2D transformation strains, using the twin relationship as a reference. The 1D model shows that the systematic variation in the special ORs can be understood in terms of a growth mechanism by attachment at steps and a lattice rotation due to the difference between the step heights in the substrate and the film. The 2D model explains why this mechanism does not apply to substrates on which Ag displays the cube-on-cube OR.

Impact of non-Schmid stress components present in the yield criterion for bcc metals on the activity of [formula omitted] slip systems

It has been found in a number of computer simulations that the yield criterion for the onset of crystal plasticity of body-centered cubic (bcc) metals generally involves stress components other than the shear stress parallel to the slip direction. This is a consequence of the non-planar cores of 1/2〈111〉 screw dislocations that control the plastic deformation of bcc crystals. The significance of these non-Schmid stress components differs from material to material and thus their contributions to the yield criterion requires either extensive atomistic simulations of isolated 1/2[111] screw dislocations or their estimates from experiments. In this paper, we derive a set of constraints imposed on the parameters of the yield criterion that control the importance of different stress components so that the orientation dependence of the critical resolved shear stress and its dependence on the shear stress perpendicular to the slip direction have trends agreeing with atomistic computer simulations. Using the allowed combinations of these parameters, we identify the primary and secondary {110}〈111〉 slip systems for loadings in tension and compression along three specifically oriented loading axes within the standard stereographic triangle. In this context we show that the origin of the tension-compression asymmetry is not related purely to the well-known twinning-antitwinning asymmetry of the yield stress but to a combination of this asymmetry with the effect of the shear stresses perpendicular to the slip direction. Additionally, we demonstrate that our yield criterion, which is based only on the shear stresses parallel and perpendicular to the slip direction, is fully equivalent to more complex yield criteria that contain explicit dependences on normal stresses perpendicular and parallel to the dislocation line. These additional stress terms are shown to emerge due to projection of the applied stress tensor into an orthogonal basis that does not conform with the three-fold screw symmetry of the 〈111〉 rotation axis implicit in bcc crystal structures.

Effect of orientation and mode of loading on deformation behaviour of Cu nanowires

Molecular dynamics simulations have been performed to understand the variations in deformation mechanisms of Cu nanowires as a function of orientation and loading mode (tension or compression). Cu nanowires of different crystallographic orientations distributed uniformly on the standard stereographic triangle have been considered under tensile and compressive loading. The simulation results indicate that under compressive loading, the orientations close to <100> corner deform by twinning mechanism, while the remaining orientations deform by dislocation slip. On the other hand, all the nanowires deform by twinning mechanism under tensile loading. Further, the orientations close to <110> and <111> corner exhibit tension-compression asymmetry in deformation mechanisms. In addition to deformation mechanisms, Cu nanowires also display tension-compression asymmetry in yield stress. The orientations close to <001> corner exhibits higher yield stress in tension than in compression, while the opposite behaviour (higher yield stress in compression than in tension) has been observed in orientations close to <110> and <111> corners. For the specific orientation of <102>, the yield stress asymmetry has not been observed. The tension-compression asymmetry in deformation mechanisms has been explained based on the parameter αM, defined as the ratio of Schmid factors for leading and trailing partial dislocations. Similarly, the asymmetry in yield stress values has been attributed to the different Schmid factor values for leading partial dislocations under tensile and compressive loading.

Optimizing Surface Modifications for Quantum Dot Labelled DNA SAMs Using Electrochemistry Coupled Fluorescence Imaging

Electrochemical biosensors are being developed for detecting specific sequences of nucleic acids using DNA modified gold surfaces[1]. A single crystal gold bead electrode was used to study the arrangement on DNA on different gold surfaces[2]. The gold surface was prepared with a mercaptoundecanol (MUDOL) self-assembled monolayer (SAM). Using a specific desorption potential of -0.850 V/SCE, the MUDOL was reductively desorbed from the Au(111) facet. The defects in the SAM were passivated with a shorter alkylthiol, mercaptohexanol. Thiol modified DNA which is also labeled with a fluorophore was allowed to populate the 111 facet specifically. This was studied using fluorescence microscopy coupled with electrochemical control. A fluorescence image of the gold bead (Figure a) showed that only the 111 facets were modified. The density of the DNA modification was also evaluated using the reorientation of DNA based on the electrode charge density. At positive potentials the DNA was positioned close to the electrode and the fluorescence was quenched. At negative potentials the DNA was repelled and the fluorescence increased. This change in fluorescence is indicative of the density of DNA on the surface. The same approach will be used to assemble DNA modified Quantum Dots (QDs) onto the Au(111) facets. A complementary ssDNA was attached to a glutathione capped quantum dot vis a his-tag peptide. The QD labelled ssDNA was allowed to hybridize with the ssDNA on the gold electrode. Figure (b) is the cartoon depiction of such a layer. Recent research results will be described on the characterization of QD labelled DNA SAMs. The QDs due to their broad absorption and narrow emission spectrum also offer the advantage of building more complicated multiplexed systems in future. (1) Hsieh, K.; Ferguson, B. S.; Eisenstein, M.; Plaxco, K. W.; Soh, H. T. Integrated Electrochemical Microsystems for Genetic Detection of Pathogens at the Point of Care. Acc. Chem. Res. 2015, 48, 911–920. (2) Yu, Z. L.; Casanova-Moreno, J.; Guryanov, I.; Maran, F.; Bizzotto, D. Influence of Surface Structure on Single or Mixed Component Self-Assembled Monolayers via in Situ Spectroelectrochemical Fluorescence Imaging of the Complete Stereographic Triangle on a Single Crystal Au Bead Electrode. J. Am. Chem. Soc 2015, 137, 276–288. Figure 1

Dependence of Crystallographic Orientation on Pitting Corrosion Behavior of Ni-Fe-Cr Alloy 028

The influence of crystallographic orientation on the pitting corrosion behavior of Ni-Fe-Cr alloy 028 was studied using a combination of X-ray diffraction (XRD), electron backscatter diffraction (EBSD), potentiodynamic polarization technique, and atomic force microscopy (AFM). The results show that there is anisotropy of pitting corrosion that strongly depends on crystallographic orientation of the surface plane. The distribution of pit density in a standard stereographic triangle indicates that the crystallographic planes close to {100} are more prone to pitting corrosion compared to planes {110} and {111}. The surface energy calculation of (001) and (111) shows that the plane with a high atomic packing density has a low surface energy with concomitant strong resistance to pitting corrosion. A correlation function between crystallographic orientation and pitting corrosion susceptibility suggests a method that not only predicts the pitting resistance of known textured materials, but also could help to improve corrosion resistance by controlling material texture.

Mesoscale plastic texture in body-centered cubic metals under uniaxial load

We develop a minimal continuum dislocation dynamics model for slip in bcc metals that accounts explicitly for non-Schmid behavior of screw dislocations in these materials. The dislocation substructure is represented by a continuum distribution of $1/2[111]$ screw dislocations moving on three possible ${110}$ planes according to the flow rule that describes the response of isolated screw dislocations to external loads. The cross-slip of dislocations is assessed using the master equation which takes into account different energy barriers for dislocations moving on the three slip planes. To demonstrate the performance of the model, we study the buildup of plastic strain at 77 K in both tension and compression for a number of loading directions covering the entire area of the standard stereographic triangle. The non-Schmid behavior of screw dislocations is shown to persist to the continuum level, whereby interactions between dislocations merely affect the rate of plastic flow.

Methodological aspects of the quantitative texture analysis of HCP Alloy (Ti, Zr) sheet semiproducts

The specific features of the existing methods of quantitative texture analysis of semiproducts and products made of titanium and zirconium alloys are analyzed. A technique is proposed to determine the Kearns coefficients (f parameters) for sheets made of hcp-metal-based alloys with allowance for the weighting factor of each reflection in the standard stereographic triangle. The f parameters calculated for sheets made of a zirconium alloy and various titanium alloys using different techniques are compared. The accuracy of measuring the Kearns coefficients by the inverse pole figure method is estimated.

Computational simulation of threshold displacement energies of GaAs

Abstract