Stereographic Triangle Research Articles

Lattice orientation effects on void growth and coalescence in fcc single crystals

Void growth and coalescence in fcc single crystals were studied using crystal plasticity under uniaxial and biaxial loading conditions and various orientations of the crystalline lattice. A 2D plane strain unit cell with one and two cylindrical voids was employed using three-dimensional 12 potentially active slip systems. The results were compared to five representative orientations of the tensile axis on the stereographic triangle. For uniaxial tension conditions, the void volume fraction increase under the applied load is strongly dependent on the crystallographic orientation with respect to the tensile axis. For some orientations of the tensile axis, such as [1 0 0] or [1 1 0], the voids exhibited a growth rate twice as fast compared with other orientations ([1 0 0], [2 1 1]). Void growth and coalescence simulations under uniaxial loading indicated that during deformation along some orientations with asymmetry of the slip systems, the voids experienced rotation and shape distortion, due mainly to lattice reorientation. Coalescence effects are shown to diminish the influence of lattice orientation on the void volume fraction increase, but noteworthy differences are still present. Under biaxial loading conditions, practically all differences in the void volume fraction for different orientations of the tensile axes during void growth vanish. These results lead to the conclusion that at microstructural length scales in regions under intense biaxiality/triaxiality conditions, such as crack tip or notched regions, the plastic anisotropy due to the initial lattice orientation has only a minor role in influencing the void growth rate. In such situations, void growth and coalescence are mainly determined by the stress triaxiality, the magnitude of accumulated strain, and the spatial localization of such plastic strains.

Correlation Between Grain‐Boundary Segregation and Grain‐Boundary Plane Orientation in Nb‐Doped TiO2

Grain‐boundary (GB) segregation has been measured in Nb‐doped TiO2 (rutile) by Auger electron spectroscopy analyses of the two surfaces exposed by inter‐granular fracture. In addition, the orientations of the exposed GB planes on both sides of the fracture surface have been determined. A strong anisotropy of GB composition is found. Measurements on 166 individual GB fracture surfaces show that boundary planes with low and high levels of segregation tend to be clustered along the (110)–(010) and (001)–(011) regions of the stereographic triangle, respectively. The matching halves of GB planes with either high or low segregation also tend to be terminated by planes in the same respective regions of orientation space. High and low levels of segregation occur in regions of GB plane orientation space estimated to correspond to high and low GB energy, respectively. This trend is consistent with qualitative arguments based on Gibbsian adsorption.

Ultrasonic characterization of point defects induced by cross slip under pure shear

Longitudinal wave velocity is used to characterize the point defects in crystalline solids. High purity Al single crystal was selected for both the finite element analysis and experimental work. Since the jog motions of dislocations caused by intersected slides such as cross slips induce point defects, the total amount of cross slips was calculated instead of calculating directly from the point defects. The effect of crystal orientations on total amount of cross slips under pure shear was also investigated via the finite element method. The result suggest that if the initial shear stress direction is located at the inner side of stereographic triangle, only single slip activities occurred at the beginning of plastic deformation and no effects due to point defects were present. However, as the shear stress direction rotates along the slip direction, point defects are induced by cross slips between primary and secondary slip systems due to work-hardening. This phenomenon was then examined by measuring longitudinal wave velocity changes propagating in Al single crystal subjected to the combination loads of equi-biaxial tension and compression (a pure shear state). Good qualitative agreement between the finite element result and measured data suggest that the longitudinal wave velocity can be used as an index to characterize point defects in crystalline materials.

Orientation dependence behavior of self-assembled (In,Ga)As quantum structures on GaAs surface

We report on the formation of (In,Ga)As self-assembled quantum structures grown on different orientations of GaAs along one side of the stereographic triangle between (1 0 0) and (1 1 1)A. The samples were characterized by atomic force microscopy. A systematic transition from zero-dimensional (In,Ga)As quantum dots to one-dimensional quantum wires was observed as the substrate was varied along the side of the triangle between (1 0 0) and (1 1 1)A. An explanation for the role of the substrate in determining the size of the nanostructure is proposed.

Lattice rotations of individual bulk grains Part II: correlation with initial orientation and model comparison

The lattice rotations of 95 individual bulk grains during 6% tensile elongation of aluminium have been analysed for correlations between the initial grain orientation and the rotation behaviour. Four distinct regions in orientation space with different rotation behaviour have been identified. The analysis shows that the grain orientation is the main factor controlling the rotations. Some variation within each region shows that other factors have a secondary effect, also depending on the grain orientation. The Taylor model using the solution to the ambiguity problem which maximises primary slip predicts the overall rotations reasonably well in some of the regions, however large discrepancies occur, especially in the 〈1 0 0〉 corner and in the middle of the stereographic triangle. Self-consistent modelling predicted the observed large variation in the 〈1 0 0〉 corner but did not predict the correct rotation directions.

Controlling (In, Ga)As quantum structures on high index GaAs surfaces

AbstractWe investigate the formation of (In, Ga)As self assembled quantum structures grown on different orientations of a GaAs substrate along one side of the stereographic triangle between (100) and (111)A surfaces. The samples were grown by Molecular Beam Epitaxy, monitored by Reflection High-Energy Electron Diffraction during the growth and characterized by in-situ Scanning Tunneling Microscopy and Atomic Force Microscopy. A systematic transition from zero dimensional (In, Ga)As quantum dots to one dimensional quantum wires was observed as the substrate was varied along the side of the triangle within 25° miscut from the (100) toward (111)A, which includes several high index surfaces. We propose an explanation for the role of the substrate in determining the type of the nanostructure that is formed.

Controlling (In,Ga)As quantum structures on high index GaAs surfaces

ABSTRACTWe investigate the formation of (In, Ga) As self assembled quantum structures grown on different orientations of a GaAs substrate along one side of the stereographic triangle between (100) and (111)A surfaces. The samples were grown by Molecular Beam Epitaxy, monitored by Reflection High-Energy Electron Diffraction during the growth and characterized by in-situ Scanning Tunneling Microscopy and Atomic Force Microscopy. A systematic transition from zero dimensional (In, Ga) As quantum dots to one dimensional quantum wires was observed as the substrate was varied along the side of the triangle within 25° miscut from the (100) toward (111)A, which includes several high index surfaces. We propose an explanation for the role of the substrate in determining the type of the nanostructure that is formed.

Controlling (In, Ga)As quantum structures on high index GaAs surfaces

ABSTRACTWe investigate the formation of (In, Ga)As self assembled quantum structures grown on different orientations of a GaAs substrate along one side of the stereographic triangle between (100) and (111)A surfaces. The samples were grown by Molecular Beam Epitaxy, monitored by Reflection High-Energy Electron Diffraction during the growth and characterized by in-situ Scanning Tunneling Microscopy and Atomic Force Microscopy. A systematic transition from zero dimensional (In, Ga)As quantum dots to one dimensional quantum wires was observed as the substrate was varied along the side of the triangle within 25° miscut from the (100) toward (111)A, which includes several high index surfaces. We propose an explanation for the role of the substrate in determining the type of the nanostructure that is formed.

Electrochemical reduction of carbon dioxide on kinked stepped surfaces of platinum inside the stereographic triangle

Structural effects on the rate of CO 2 reduction are studied on kinked stepped surfaces of Pt inside the stereographic triangle. The surfaces examined are Pt(431), Pt(531) and Pt(532), which consist of two or three atomic rows of (111) terraces and (111) or (100) steps incorporated with kink atoms. The rate of CO 2 reduction is enhanced with the increase of the kink atom density. The kinked stepped surfaces have a higher activity for CO 2 reduction than the analogous stepped surfaces that have similar terrace and step structures without kink sites. Introduction of kink atoms to Pt(S)-[3(111)×(100)] surface changes the potential dependence of the reduction rates significantly.

Cyclic deformation mechanisms in precipitated NiTi shape memory alloys

Results are presented on the cyclic deformation of single crystal NiTi containing Ti 3Ni 4 precipitates of various sizes. Mechanical cycling experiments reveal that the cyclic degradation resistance of NiTi is strongly dependent on crystallographic orientation. Under compression, orientations approaching the [100] pole of the stereographic triangle possess the highest fatigue resistance. Orientations approaching the [111] pole of the stereographic triangle demonstrate the lowest fatigue resistance. Aging to produce small coherent Ti 3Ni 4 precipitates (10 nm) improves the fatigue resistance of NiTi compared to the other heat treatments (solutionized or overaged) for nearly all orientations. NiTi with 10 nm Ti 3Ni 4 precipitates consistently showed stabilized martensite due to mechanical cycling, and an absence of dislocation activity. Samples with large incoherent Ti 3Ni 4 precipitates (500 nm) consistently showed significant dislocation activity due to mechanical cycling in addition to stabilized martensite colonies. The first cycle stress–strain hysteresis was found to correlate to the fatigue resistance of the material. Samples demonstrating large inherent hysteresis, with different heat treatments and orientations, showed poor fatigue performance. Rational for the observed behaviors is discussed in terms of operant deformation mechanisms and ramifications on modeling the cyclic deformation of NiTi are presented.

Structure of high-index GaAs surfaces - the discovery of the stable GaAs (2 5 11) surface

We present a brief overview of surface structures of high-index GaAs surfaces, putting emphasis on recent progress in our own laboratory. By adapting a commercial scanning tunneling microscope (STM) to our molecular beam epitaxy and ultra high vacuum analysis chamber system, we have been able to atomically resolve the GaAs(\(\bar{1}\)\(\bar{1}\)\(\bar{3}\))B (8 ×1), (114)Aα2(2×1), (137), (3 7 15), and (2 5 11) surface structures. In cooperation with P. Kratzer and M. Scheffler from the Theory Department of the Fritz-Haber Institute we determined the structure of some of these surfaces by comparing total-energy calculations and STM image simulations with the atomically resolved STM images. We present the results for the {112}, {113}, and {114} surfaces. Then we describe what led us to proceed into the inner parts of the stereographic triangle and to discover the hitherto unknown stable GaAs (2 5 11) surface.

Cyclic Twins in High-Purity Titanium Single Crystals / Zyklische Zwillingsbildung in hochreinen Titan-Einkristallen

The titanium single crystal with special orientation was prepared and the microstructures formed during the cyclic deformation were examined by transmission electron microscope (TEM). Four important twinning systems, {1011} and {1121} (both with parallelsides shape), {1012} (with lens-shape), {1122} (with short and narrow shape), were observed in the microstructure of titanium single crystals after cyclic loading. It is found that the deformation mechanisms of titanium single crystals vary with their orientations. With the orientation of the titanium single crystals changing from the center ([0001] direction) to the [1010]-[1120] border of the stereographic projection unit triangle, the deformation mode varies from twinning predominant to slipping predominant.

Surface structure of GaAs(2 5 11)

GaAs samples with orientations vicinal to (2 5 11) within $1\ifmmode^\circ\else\textdegree\fi{}$ were prepared by molecular beam epitaxy and analyzed in situ by scanning tunneling microscopy, low-energy electron diffraction, and reflection high-energy electron diffraction. In addition, first-principles electronic structure calculations were carried out. GaAs(2 5 11) is a stable surface whose orientation is located within the stereographic triangle. For a wide range of As-rich conditions a $(1\ifmmode\times\else\texttimes\fi{}1)$ reconstruction forms that is characterized by an inclined series of three As dimers and that fulfills the electron counting rule. The terrace size is limited only by the macroscopic off-orientation of the samples. The surface is perturbed by thin stripes of the nearby orientation (3 7 15). While the dangling bond densities of GaAs(2 5 11) and GaAs(3 7 15) are almost equal, GaAs(3 7 15) violates the electron counting rule. The analysis of this perturbation suggests that, in general, on semiconductor surfaces the gain in stability arising from the minimization of the number of dangling bonds is significantly greater than the gain arising from reaching a semiconducting ground state. Upon annealing of the samples in ultrahigh vacuum, a fairly rough surface structure develops whose mean orientation is different from (2 5 11).

GaAs(2 5 11): a new stable surface within the stereographic triangle.

The atomic structure of GaAs(2 5 11), a hitherto unknown stable surface, has been determined by in situ scanning tunneling microscopy and first-principles electronic structure calculations. This orientation is located within the stereographic triangle, i.e., far away from all low-index surfaces. A low-energy ( 1x1) reconstruction containing arsenic dimers forms on the surface. The analysis of the surface structure shows that, for semiconductor surfaces, the gain in stability due to minimization of the number of dangling bonds is more important than the gain from rendering a semiconducting ground state.

In-situX-ray diffraction study of single-slip-conditioned copper single crystals during uniaxial deformations

Abstract The broadening of X-ray peak profiles and movements of tensile axes were obtained simultaneously in single-slip-conditioned copper single crystals. Two types of sample showed different tensile axis movements and slip system activities. These results are in accord with the observation of slip lines with scanning electron microscopy. The broadening of the X-ray peak profiles is highly dependent on the absolute value of g·b despite the same amount of deformation. The slope between shear stress and βdisorder obtained from broadening generally shows similar trends for samples A and B. As soon as the tensile axis leaves its original stereographic triangle, βdisorder stops increasing in sample B. This phenomenon indicates the change of activation of (111)[101] and (1 11)[011] slip systems.

In-situ X-ray diffraction study of single-slip-conditioned copper single crystals during uniaxial deformations

Abstract The broadening of X-ray peak profiles and movements of tensile axes were obtained simultaneously in single-slip-conditioned copper single crystals. Two types of sample showed different tensile axis movements and slip system activities. These results are in accord with the observation of slip lines with scanning electron microscopy. The broadening of the X-ray peak profiles is highly dependent on the absolute value of g·b despite the same amount of deformation. The slope between shear stress and βdisorder obtained from broadening generally shows similar trends for samples A and B. As soon as the tensile axis leaves its original stereographic triangle, βdisorder stops increasing in sample B. This phenomenon indicates the change of activation of (111)[101] and (1 11)[011] slip systems.

Dislocation structures in cyclically deformed nickel polycrystals

The effect of the grain orientation and the plastic strain amplitude ε pa on the saturated dislocation structure was studied on individual grains of cyclically deformed nickel polycrystals by means of scanning electron microscopy using the electron back scattering pattern technique and the channelling contrast of back scattered electrons. The main features of the dislocation configuration in a grain were found to be essentially determined by the crystallographic axial orientation of the grain. A labyrinth-like dislocation pattern is typical for grains with axial orientations near [001], a patch pattern exists in grains with a loading axis (LA) near [011] and fragmented dislocation walls are dominant in grains with LA near [1̄11]. Grains with axial orientations in the central part of the stereographic standard triangle contain a bundle arrangement of dislocation structures. All four types of dislocation structures, but mostly the bundle type, can occur together with the ladder structure of persistent slip bands. Cell patterns were found to be a result of a modification of the bundle and patch configuration at high deformation amplitudes. The mesoscopic dimensions of the dislocation patterns turned out to depend on ε pa in the same way for all grain orientations: while the thickness of regions with high dislocation density is reduced with increasing ε pa, the width of regions with low dislocation density remains roughly constant.

Anisotropic creep in CMSX-4 in orientations distant from 〈001〉

Constant load creep tests have been performed on the single crystal nickel base superalloy CMSX-4 in crystallographically unstable orientations distant from the 〈001〉 pole of the standard stereographic triangle at 1023 K (750°C) and varying stress levels. Orientations distant from the 〈001〉–〈111〉 boundary had high creep rates and relatively low rupture lives, while the orientations closer to it had relatively low initial creep rates and longer rupture lives. Electron back scatter diffraction patterns, cross section shape change measurements on creep tested specimens, and optical microscopy indicate that the operating slip system is of the type {111}〈112〉 in orientations far from the 〈001〉–〈111〉 boundary. The results are explained based on the resolved shear stress acting on the operating slip system taking into consideration the rotations and changes in cross section. A theory based on the ease of generation of superlattice intrinsic stacking faults is proposed to account for the observed creep rates in various orientations. A finite element model based on the slip system approach is used to demonstrate the validity of the experimental results.

Survey of plateau behaviour in the cyclic stress-strain curve of copper single crystals

Experimental results relating to the plateau behaviour in the cyclic stress-strain (CSS) curve of copper single crystals located on different sides of the stereographic triangle are summarized. Unlike the situation for single-sliporiented crystals, the crystallographic orientation has a strong effect on the plateau behaviour in the CSS curves of double- and multiple-slip-oriented crystals. The existence or non-existence of a plateau in the CSS curves, as well as the corresponding plateau stress amplitude, depend not only on the modes and intensities of dislocation interactions among slip systems operating in the crystals but also on the slip deformation characteristics associated with crystal orientations. The plateau region in CSS curve disappears only when multiple slip plays a determining role during cyclic deformation.

Effect of texture and ramp loading on the cyclic response and dislocation substructure of copper polycrystals

Fatigue tests were carried out on copper polycrystals that had a texture with strong components along directions inside the standard stereographic triangle (single slip orientations in monocrystals). The experiments showed that this “single slip” texture, ramp loading as a mechanical pretreatment and extended cycling at a given stress to insure saturation provide the conditions to obtain a plateau in the Cyclic Stress–Strain Curve (CSSC) of copper polycrystals in a reproducible fashion. Transmission Electron Microscopy (TEM) studies revealed the presence of numerous Persistent Slip Bands (PSB's) in samples deformed within the plateau regime.