The Wulff Shape of Alumina: IV. Ti4+-Doped Alumina

Controlled-geometry cavities, initially ≈20 μm × 20 μm × 0.5 μm, were introduced into     -plane titanium-doped (≈210 ppma;≈500 wt. ppm) sapphire substrates using photolithographic methods, and subsequently internalized by diffusion bonding. The samples were annealed in air for prolonged periods at 1600° and 1800°C to convert the titanium to the 4+ state and to allow the pore shapes to adjust. Pores with an equivalent spherical radius of ≈3.6 μm reached a quasi-equilibrium shape within 160 h at 1600°C and within 48 h at 1800°C. The Wulff shape was determined using optical microscopy, scanning electron microscopy, and atomic force microscopy. The Wulff shape of Ti⁴⁺-doped alumina includes well-defined c(0001),     , and     facets and smoothly curved sections. The     and a     facets, which are components of the Wulff shape of undoped sapphire, are not discernable. In contrast to undoped alumina, for which the r-plane has the lowest energy, the c-plane is the lowest energy plane in Ti⁴⁺-doped alumina. The surface energy sequence of the stable c, r, and p surfaces differs from that in undoped alumina. The Wulff shape varies with temperature. Samples equilibrated at 1800°C were re-annealed at 1600°C. Pore shape changes were reversible, indicating that the observed pore shapes were close to the equilibrium (Wulff) shape.     

Crystallographic Facetting in Sintered Barium Titanate

A commercial TiO₂-excess BaTiO₃ powder has been sintered and its microstructure analyzed for crystallographic facetting via both scanning and transmission electron microscopy (SEM and TEM). Facetted grain surfaces are developed initially from {111} at a low temperature of 1215°C, which are then altered to {111} and {100} at 1290°C in the presence of a grain-boundary liquid phase. The grain shape is also modified correspondingly from platelike to polygonal. Facetting of the intragranularly located residual pores in BaTiO₃ along the {141} planes further develops on the (quasi-)equilibrium shape after annealing at 1400°C for 100 h from the initially well-characterized {111}, {110}, and {100} in as-sintered samples sintered at the same temperature for 10 h. The Wulff plots derived from the residual pores in as-sintered and annealed samples are constructed for the 〈011〉 zone. Microstructural analysis also suggests that the shape of grains and intragranular residual pores is modified progressively upon annealing. The initial solid–vapor surface energy has become less anisotropic crystallographically. Abnormal grain growth in relation to the surface energy anisotropy is discussed.     

Equilibrium Shape of Internal Cavities in Sapphire

The equilibrium shape of internal cavities in sapphire was determined through the study of submicrometer internal cavities in single crystals. Cavities formed from indentation cracks during annealing at 1600°C. Equilibrium could be reached only for cavities that were smaller than is approximately100 nm. Excessive times were required to achieve equilibrium for cavities larger than is approximately 1μm. Five equilibrium facet planes were observed to bound the cavities: the basal (C) {0001}, rhombohedral (R) {1¯012}, prismatic (A) {12¯10}, pyramidal (P) {112¯3}, and structural rhombohedral (S) {101¯1}. The surface energies for these planes relative to the surface energy of the basal plane were γR = 1.05, γA = 1.12, γP = 1.06, γS = 1.07. These energies were compared with the most recent theoretical calculations of the surface energy of sapphire. The comparison was not within experimental scatter for any of the surfaces, with the measured relative surface energies being lower than the calculated energies. Although the prismatic (M) {101¯0} planes are predicted to be a low-energy surface, facets of this orientation were not observed.     

The Wulff Shape of Alumina: II, Experimental Measurements of Pore Shape Evolution Rates

The rate at which a facetted tetragonal cavity of nonequilibrium shape approaches a cubic equilibrium (Wulff) shape via surface diffusion was modeled. The shape relaxation rate of a facetted "stretched cylinder" was also modeled. For the first geometry, only an approximate solution based on linearizing the mean potential difference between the source and sink facets was obtained. For the stretched cylinder, both an approximate and an exact solution can be obtained; the approximate solution underestimates the evolution rate by a factor of ∼2. To assess the applicability of the models, nonequilibrium shape pores of identical initial geometry (∼20 μm × 20 μm × 0.5 μm) were introduced into (0001), {10[Onemacr]2}, {1120}, and {100} surfaces of sapphire single crystals using microfabrication techniques, ion-beam etching, and hot pressing. The large (∼20 μm × 20 μm) faces of the pore are low-index surfaces whose nature is dictated by the wafer orientation. A series of anneals was performed at 1900°C, and the approach of the pore shape to an equilibrium shape was monitored. The kinetics of shape evolution are highly sensitive to the crystallographic orientation and stability of the low-index surface that dominates the initial pore shape. The measured variations of the pore aspect ratio were compared to those predicted by the kinetic model. The observations suggest that when the initial bounding surface is unstable, shape relaxation may be controlled by diffusion. However, surface-attachment-limited kinetics (SALK) appears to play a major role in determining the pore shape evolution rate in cases where the initial bounding surfaces have orientations that are part of the Wulff shape.     

The Wulff Shape of Alumina: I, Modeling the Kinetics of Morphological Evolution

The rate at which fully facetted nonequilibrium shaped particles and pores approach their equilibrium (Wulff) shape via surface diffusion was modeled, and calculations relevant to alumina were performed to guide experimental studies. The modeling focuses on 2-D features, and considers initial particle/pore shape, size, surface energy anisotropy, and temperature (surface diffusivity) as variables. The chemical potential differences driving the shape change are expressed in terms of facet-to-facet differences in weighted mean curvature. Two approaches to modeling the surface flux are taken. One linearizes the difference in the mean chemical potential of adjacent facets, and assumes the flux is proportional to this difference. The other approach treats the surface chemical potential as a continuous function of position, and relates the displacement rate of the surface to the divergence of the surface flux. When consistent values for the relevant materials parameters are used, the predictions of these two modeling approaches agree to within a factor of 1.5. As expected, the most important parameters affecting the evolution times are the cross-sectional area (volume in 3-D) and the temperature through its effect on the surface diffusivity. Pores of micrometer size are predicted to reach near-equilibrium shapes in reasonable times at temperatures as low as 1600°C. The detailed geometry of the initial nonequilibrium shape and the Wulff shape appear to have relatively minor effects on the times required to reach a near-equilibrium shape.     

High-Temperature Morphological Evolution of Lithographically Introduced Cavities in Silicon Carbide

Internal cavities of controlled geometry and crystallography were introduced in 6 H silicon carbide single crystals by combining lithographic methods, ion-beam etching, and solid-state diffusion bonding. The morphologic evolution of these internal cavities (negative crystals) in response to anneals of up to 128 h duration at 1900°C was examined using optical microscopy. Surface energy anisotropy and faceting had a strong influence on the geometric and kinetic characteristics of evolution. Decomposition of {12     10} cavity edges into {101     x } facets was observed after 16 h anneals, indicating that {12     10} faces are not components of the Wulff shape. The shape evolution kinetics of penny-shaped cavities were also investigated. Experimentally observed evolution rates decreased much more rapidly with those predicted by a model in which surface diffusion was assumed to be rate limiting. This suggested that the development of facets and the associated loss of ledges and terraces during the initial stages of evolution resulted in an evolution process limited by the nucleation rate of attachment/detachment sites (ledges) on the facets.     

Surface Energy Anisotropy of SrTiO3 at 1400°C in Air

Geometric and crystallographic measurements of grain-boundary thermal grooves and surface faceting behavior as a function of orientation have been used to determine the surface energy anisotropy of SrTiO₃ at 1400°C in air. Under these conditions, thermal grooves are formed by surface diffusion. The surface energy anisotropy was determined using the capillarity vector reconstruction method under the assumption that Herring's local equilibrium condition holds at the groove root. The results indicate that the (100) surface has the minimum energy. For surfaces inclined between 0° and 30° from (100), the energy increases with the inclination angle. Orientations inclined by more than 30° from (100) are all about 10% higher in energy and, within experimental uncertainty, energetically equivalent. A procedure for estimating the uncertainties in the reconstructed energies is also introduced. Taken together, the orientation dependence of the surface-facet formation and the measured energy anisotropy lead to the conclusion that the equilibrium crystal shape is dominated by {100}, but also includes {110} and {111} facets. Complex planes within about 15° of {100} and 5° of {110} are also part of the equilibrium shape.     

Faceting of High-Angle Grain Boundaries in Titanium-Excess BaTiO3

The grain boundaries in BaTiO₃ with excess Ti of 0.5, 0.3, and 0.1 at.% sintered at 1300° or 1250°C have been examined by scanning electron microscopy (SEM), electron backscattered diffraction pattern (EBSP), and transmission electron microscopy (TEM). In the 0.1% Ti-excess specimen, large grains growing abnormally form high-angle grain boundaries when they impinge on each other as verified by EBSP. A large fraction of these grain boundaries are faceted with hill-and-valley shapes. In the 0.5% Ti-excess specimen, large grains growing abnormally are elongated in the directions of their {111} double twins. These grains often form flat grain boundaries parallel to their {111} planes with the fine matrix grains, and the grain-boundary segments between the large impinging grains with high misorientation angles are often also parallel to the {111} planes of one of the grains. These grain boundaries are expected to be singular. Most of the grain boundaries between the randomly oriented fine-matrix grains in the 0.3 at.% Ti-excess specimen are also faceted with hill-and-valley shapes at finer scales when observed under TEM. The facet planes are parallel to {111}, {011}, and {012} planes of one of the grain pairs and are also expected to be singular. These high-angle grain boundaries lying on low index planes of one of the grain pairs are similar to those observed in other oxides and metals.     

Liquid Phase Sintering of Alumina, II. Penetration of Liquid Phase into Model Microstructures

Alumina preforms containing artificial pores were sintered at 1630°C in air and vacuum. Glass penetration into the alumina preforms was conducted at 1600°C in air. It was found that the trapped gases in alumina preforms sintered in air caused the random and incomplete filling of the smaller and larger artificial pores. In contrast, the pores in the alumina preform sintered in vacuum were completely filled during glass penetration.     

Plasticity and Creep in Single Crystals of Zirconium Carbide

High-temperature deformation in ZrC single crystals was studied. Seeded crystals were grown by a direct rf-coupling floating-zone process. Yield stresses were measured from 1080° to 2000°C as a function of stress axis orientation. The Burgers vector was shown to be parallel to the 〈110〉 axes by transmission electron microscopy. Slip was observed on {100}, {110}, or {111} planes, depending on the orientation of the stress axis; it always occurred on the most favorably oriented slip system. The dependence of steady-state creep rate on the applied stress indicated that recovery occurred by a dislocation climb mechanism. Examination of the dislocation structure in deformed crystals by transmission electron microscopy supported this conclusion.     

Alpha Particle Irradiation Damage in ThO2

The effect of α-particle irradiation (5.5 MeV, total dose ∼10¹⁵ particles/cm²) on the microstructure of ThO₂ was studied by transmission electron microscopy. The only as-irradiated damage seen consisted of point defect agglomerations visible as small black spots. Subsequent heating to ∼1500°C in the electron microscope caused the spots to grow into loops 50 to 200 Å in diameter, but no voids, bubbles, or pores were observed. The preferred loop planes were {311}, {110}, and {012}, in that order of frequency.     

A geometric model of growth for cubic crystals: Diamond

A mathematical and software implementation of a geometrical model of the morphology of growth in a cubic crystal system, such as diamond, is presented based on the relative growth velocities of four low index crystal planes: {100}, {110}, {111}, and {113}. The model starts from a seed crystal of arbitrary shape bounded by {100}, {110}, {111} and/or {113} planes, or a vicinal (off axis) surface of any of these planes. The model allows for adjustable growth rates, times, and seed crystal sizes. A second implementation of the model nucleates a twinned crystal on a {100} surface and follows the evolution of its morphology. New conditions for the stability of penetration twins on {100} and {111} surfaces in terms of the alpha, beta, and gamma growth parameters are presented.     

Phase Separation in the SiC–AlN Pseudobinary System: The Role of Coherency Strain Energy

SiC–AlN polycrystalline samples of equimolar composition were fabricated by hot-pressing. Single-phase samples were annealed over a range of temperatures between 1600° and 2000°C for up to 1145 h to effect the formation of modulated structures. Electron diffraction from the [2110] zone axis showed four distinct satellite spots around every main diffraction spot with modulations occurring orthogonal to {0112} planes. The direction of modulation experimentally observed is consistent with that determined by the minimization of the coherency strain energy. Computer simulation was conducted to generate the morphology of modulated structures. The simulated microstructure was found to be very similar to that observed experimentally.     

Equilibrium Shape of Internal Cavities in Ruby and the Effect of Surface Energy Anisotropy on the Equilibrium Shape

Cavities formed in ruby (99.46Al₂O₃·0.54Cr₂O₃) by the healing and annealing of indentation cracks at 1600°C are more equiaxed than similar cavities in sapphire. Surface energies for the observed facet planes (R, S, and A) relative to the surface energy of the basal plane, C , were γ_A/C= 1.00 ± 0.03, γ_R/C= 1.05 ± 0.07, and γ_S/C= 1.02 ± 0.04, with the uncertainty representing 95% confidence limits. Thus, the surface energies of all observed facets were statistically indistinguishable. Unlike sapphire, P-plane facets were not observed. The substantial rounding of the cavities in ruby indicated that portions of the Wulff shape were above the roughening transition temperature. Thus, even though Cr₂O₃ and Al₂O₃ form ideal solutions, Cr³⁺ ions are sufficiently surface active to modify the relative free energy of the surfaces.     

Creep Deformation of 0° Sapphire

Creep deformation of 0° sapphire was studied between 1600°and 1800°C at stresses up to 114 MN/m². Microscopical evidence (dislocation structures observed by transmission electron microscopy (TEM) and by etch pits) suggested that Nabarro climb was the predominant deformation mechanism. Both the experimental creep rates and stress exponents were in good agreement with those predicted by this model. Although non-basal dislocations with a ½〈101〉 Burgers vector were present, the good creep resistance of 0° sapphire was attributed to the difficulty of activating pyramidal slip.     

Atomic Structures and Energies of Σ7 Symmetrical Tilt Grain Boundaries in Alumina Bicrystals

Symmetrical Σ7 tilt grain boundaries of alumina (Al₂O₃) were studied using bicrystals. Three types of Σ7 boundaries were successfully fabricated, that is, rhombohedral twin (Σ7{1[Onemacr]02}) and two types of [0001] symmetrical tilt grain boundaries with grain-boundary planes {4[Fivemacr]10} and {2[Threemacr]10} (Σ7{4[Fivemacr]10} and Σ7{2[Threemacr]10}). Their atomic structures and grain-boundary energies were investigated using high-resolution transmission electron microscopy (HRTEM) and a thermal grooving technique, respectively. HRTEM observations showed that the Σ7{1[Onemacr]02} boundary had a completely symmetrical atomic arrangement with respect to the grain-boundary plane. In contrast, Σ7{2[Threemacr]10} and Σ7{4[Fivemacr]10} boundaries exhibited asymmetrical atomic structures, which were confirmed by analyzing the atomic configurations using static lattice calculations. Thermal grooving experiments showed that the grain-boundary energies strongly depended on the properties of the grain-boundary planes.     

Alumina Volatility in Water Vapor at Elevated Temperatures

The volatility of alumina in high-temperature water vapor was determined by a weight loss technique. Sapphire coupons were exposed at temperatures between 1250° and 1500°C, water partial pressures between 0.15 and 0.68 atm in oxygen, a total pressure of 1 atm, and flowing gas velocities of 4.4 cm/s. The water vapor pressure dependence of sapphire volatility was consistent with Al(OH)₃( g ) formation. The enthalpy of reaction to form Al(OH)₃( g ) from sapphire and water vapor was determined to be 210 ± 20 kJ/mol, comparing favorably to other studies. Microstructural examination of tested sapphire coupons revealed surface rearrangement consistent with a volatilization process.     

Relating Grain Boundary Complexion to Grain Boundary Kinetics II: Silica-Doped Alumina

This second paper in a series describes the relationship between grain growth kinetics and grain boundary complexions in silica-doped alumina. Dense high-purity silica-doped alumina samples were annealed for various times in the temperature range of 1300° and 1900°C and their grain growth behavior was quantified. Four different grain boundary complexions were observed in silica-doped alumina, all of which enhanced the kinetics relative to the intrinsic undoped alumina. These complexions included a thick crystallized film that was likely amorphous at high temperatures, a thin intergranular film, multilayer adsorption, and a type of boundary that showed no observable film by high-resolution transmission electron microscopy. A generational change in the population of grains occurred at 1500°C where all of the abnormal grains impinged and reestablished a new normal distribution. At higher temperatures a new set of abnormal grains containing different complexions formed in the microstructure. The activation energy of the normal and abnormal grains was approximately the same. The effects of varying dopant concentration were analyzed. The results for silica-doped alumina are compared with previous results for calcia-doped alumina in order to draw some generalized conclusions about the effect of complexions on grain growth.     

Deformation Microstructure in (001) Single Crystal Strontium Titanate by Vickers Indentation

Recent interests on the plastic deformation of strontium titanate (SrTiO₃) are derived from its unusual ductile-to-brittle-to-ductile transition (DBDT). The transition is divided into three regimes (A, B, and C) corresponding to the temperature range of 113–1053 K (−160° to 780°C), 1053 to ∼1503 K (780° to ∼1230°C), and ∼1503–1873 K (∼1230° to 1600°C), discovered by Sigle and colleagues in the MPI-Stuttgart. We report the dislocation substructures in (001) single crystal SrTiO₃ deformed by Vickers indentation at room temperature, studied by scanning and transmission electron microscopy. Dislocation dipoles of screw and edge character are observed and confirmed by inside–outside contrast using ± g -vector by weak-beam dark field imaging. They are formed by edge trapping, jog dragging, and cross slip pinching-off. Similar to dipole breaking off in deformed sapphire (α-Al₂O₃) at 1200°C and γ-TiAl intermetallic at room temperature, the dipoles pinch off at one end, and emit a string of loops at trail. Two sets of slip systems {110}〈     〉 and {100}〈011〉 are activated under both 100 g and 1 kg load. The suggestion is that plastic deformation has reached the stage II work hardening, which is characterized by multiplication of dislocations through cross slip, interactions between dislocations, and operating of multiple slip systems.     

Singular Grain Boundaries in Alumina and Their Roughening Transition

The shapes and structures of grain boundaries formed between the basal (0001) surface of large alumina grains and randomly oriented small alumina grains are shown to depend on the additions of SiO₂, CaO, and MgO. If a sapphire crystal is sintered at 1620°C in contact with high-purity alumina powder, the grain boundaries formed between the (0001) sapphire surface and the small alumina grains are curved and do not show any hill-and-valley structure when observed under transmission electron microscopy (TEM). These observations indicate that the grain boundaries are atomically rough. When 100 ppm (by mole) of SiO₂ and 50 ppm of CaO are added, the (0001) surfaces of the single crystal and the elongated abnormal grains form flat grain boundaries with most of the fine matrix grains as observed at all scales including high-resolution TEM. These grain boundaries, which maintain their flat shape even at the triple junctions, are possible if and only if they are singular corresponding to cusps in the polar plots of the grain boundary energy as a function of the grain boundary normal. When MgO is added to the specimen containing SiO₂ and CaO, the flat (0001) grain boundaries become curved at all scales of observation, indicating that they are atomically rough. The grain boundaries between small matrix grains also become defaceted and hence atomically rough.