Criterion for predicting the morphology of crystalline cubic precipitates in a cubic matrix

The present work is concerned with the configuration of precipitates having cubic crystal symmetry in a cubic matrix. The shape and orientation of the precipitates were determined by minimizing the elastic strain energy, while neglecting surface energy effects. Lee, Barnett, and Aaronson7 have shown that only spherical or plate-shaped precipitates are associated with minimum strain energy. Equating the exact expression for the energy of an infinite coherent plate-shaped precipitate with an approximation suggested for the energy of a spherical precipitate, a simple criterion is derived. The criterion enables the prediction of the shape and orientation of the precipitate associated with minimum strain energy, and allows identification of the basic elastic parameters which determine this configuration. When compared to exact numerical results, good agreement was obtained. The criterion predicts that the minimum strain energy is associated with a plate-shaped precipitate, parallel to its 100 planewhen HC ₄₄ C* ₄₄/A* and the anisotropy factor of the precipitateA * > 1, and parallel to its {111} plane whenHC ₄₄ F*(111)C*₄₄/A* andA* 1. In all other cases, a spherical precipitate is associated with minimum strain energy.H is a parameter which depends on the anisotropy of the matrix.F * is an orientation factor which depends on the anisotropy of the precipitate. Sabbatical leave with the Department of Welding Engineering, The Ohio State University, 190 West 19th Avenue, Columbus, OH 43210.     

The shape and orientation of the minimum strain energy of coherent ellipsoidal precipitate in an anisotropic cubic material

The elastic strain energy of perfectly coherent ellipsoid of revolution, which has the cube-cube orientation relationship with the matrix, has been calculated as a function of the orientation of the axis of revolution and of shape factor in anisotropic cubic crystalline materials. The minimum strain energy condition occurs at four different shapes and orientations, i.e. sphere, rod along 〈001〉 axis, disc on 001 plane and disc on 111 plane, depending on the two shear moduli of precipitate, i.e. μ*₁((C*₁₁—C*₁₂)/2) and μ* (C*₄₄). This is true regardless of the elastic property of the matrix phase when its anisotropy factor is larger than 1. The conditions of the occurrence of each shape and orientation are greatly affected by the difference in the misfit accommodation behavior depending on the shape of precipitate. A review of the experimental observations indicates the presence of all four different shapes and orientations in the case of GP zones in Al alloys. The conditions of their appearance are in good agreement with the prediction of the present calculation.     

Strain-induced shape changes of cubic precipitates in a cubic matrix with positive anisotropy

This paper considers the evolution of equilibrium shape during the coarsening of a system of cubic precipitates in a cubic matrix with positive anisotropy (Δ = c₁₁ − c₁₂ − 2c₄₄ > 0). The system is assumed to have homogeneous elastic constants and isotropic surface tension. The low-energy shapes include the sphere, and the octahedron, tetrahedron and plate with {111} faces. Minimization of the sum of the elastic and surface energies shows that the sphere is preferred at arbitrarily small sizes, but ordinarily transforms into a tetrahedron, and finally into a plate as size increases. When Δ > 0.79c₁₁ the octahedron is preferred for a small range of sizes between sphere and tetrahedron. Analytic expressions are given for the equilibrium shape transitions. The results are compared to experimental observations of shape changes during coarsening in (Mg, Y)-ZrO₂. The experimentally observed absence of octahedral shaped precipitates and the size at which tetrahedral shaped precipitates become stable in this system are consistent with the theoretical predictions.     

Precipitate-induced plastic anisotropy: Explicit solutions of the plastic anisotropy due to plate-shaped precipitates

In some aluminum alloys, the observed plastic anisotropy cannot be explained solely by the measured Taylor factor variation. Qualitatively, it has been suggested that this difference results from a secondary effect due to plate-shaped precipitates. Models addressing the effect of plastically-deforming and elastically-deforming precipitates have been previously proposed. In the present article, explicit solutions of the anisotropic strengthening increment are presented for the case of plate-shaped precipitates. These solutions allow a quantitative consideration of the effect of precipitates on different habit planes and of the effect due to stress aging. Generally, in fcc materials, precipitates on {100} habit planes are predicted to minimize the anisotropy due to texture; precipitates on {111} habit planes are predicted to accentuate the anisotropy due to texture; and precipitates on other habit planes are predicted to produce a minor effect resulting from an averaging over a greater number of crystallographcally equivalent habit planes. Stress aging to alter the relative orientation distribution of a single precipitate type is predicted to produce only slight changes in the plastic anisotropy. Larger effects on the yield variation will be observed when stress aging alters the relative volume fractions of two precipitate types on different habit planes.     

The effect of orientation and sense of applied uniaxial stress on the morphology of coherent gamma prime precipitates in stress annealed nickel-base superalloy crystals

The coarsening of coherent γ′[Ni₃(Al, Ti)] precipitates in single crystals of a representative nickel-base superalloy, Udimet-700, is shown to be affected by a uniaxial stress applied during annealing. Depending on the sense of the applied stress and its crystallographic orientation, stress annealing results in oriented cuboidal, plate, or parallelepiped shaped γ′ precipitates. A general thermodynamic analysis of the effect of stress annealing on precipitate morphology is presented that takes into account free energy changes due to changes in bulk precipitation strain, effective modulus, coherency strain energy, and the total interphase boundary area. The analysis correctly predicts the observed γ′ precipitate morphologies as a function of stress axis orientation, stress sense, the lattice misfit of the precipitate phase, and the elastic constants of the matrix and precipitate phases. The analysis also shows that stress induced morphological changes can be completely precluded, as may be desired to optimize mechanical behavior, only if the elastic constants of the matrix and precipitate phases are equal. Changes in morphology due to changes in bulk precipitation strain, which in the case of Udimet-700 is shown to be the dominant effect, can be eliminated by alloying for zero lattice misfit or, in single crystals, by stressing parallel to < 111> . Applications to long-term creep behavior and to the fabrication of composite structures are discussed. Formerly with Advanced Materials Research and Development Laboratory, Pratt & Whitney Aircraft     

The elastic strain energy of coherent ellipsoidal precipitates in anisotropic crystalline solids

The elastic strain energy of coherent ellipsoidal precipitates (ellipsoids of revolution) in anisotropic crystalline solids has been calculated as a function of ellipsoid aspect ratio using the method of Eshelby. When the precipitate is eithermuch softer or harder, elastically, than the matrix, the results are similar to those previously obtained using isotropic elasticity. When this condition is not met, however, anisotropic elasticity can yield quite different results which vary markedly with the orientation relationship between precipitate and matrix. When the precipitate has a non-cubic crystal structure, the elastic strain energy often passes through a maximum or a minimum at shapes which are neither thin discs nor spheres. During this study, the isotropic elasticity result that the strain energy associated with a disc-shaped precipitate is independent of the matrix elastic constants was also shown to hold under the conditions of anisotropic elasticity, and in such circumstances it depends only on the elastic properties of the precipitate in the direction of the principal directions of the disc. Incorporation of the anisotropic elastic strain energy into the calculation of ΔG ^*, the free energy of activation for the formation of a critical nucleus for the basic case of homogeneous nucleation with boundary-orientation independent interfacial energy, showed that the ratio of the strain energy to the volume free energy change must usually be somewhat larger than 3/4 in order to cause the shape of the critical nucleus to differ from that of a sphere.     

Effect of elastic strain energy on the core-shell structures of the precipitates in Al-Sc-Er alloys

The effect of the elastic strain energy on the core-shell structures was studied in an Al-0.06Sc-0.02Er (at.%) alloy. A theoretical model for the calculation of the elastic strain energy caused by core-shell precipitates, which is applicable to materials with weak elastic ani-sotropy, was adopted. It was demonstrated that the partitioning of Er to the precipitate core did not reduce the elastic strain energy as expected in the previous study. The resistance due to the elastic strain energy to form an Al3(Sc0.36Er0.64)-Al3(Sc0.8Er0.2) core-shell precipitate was quite small, and could be easily overcome by the decrease of the total interfacial energy, which was consistent with the previous experimental re-sults. On the other hand, the resistance due to the elastic strain energy to form an Al3Er-Al3Sc core-shell precipitate was much larger than that to form an Al3(Sc0.36Er0.64)-Al3(Sc0.8Er0.2) core-shell precipitate, thus the partitioning of all the Er atoms to the core was strongly hindered by the elastic strain energy and was not observed in the experiment of the previous study.)     

The atomic structure of A1-B2 interfaces in a NiBe alloy

The atomic structure of the interfaces between a Ni-based solid solution with f.c.c.-structure (α-phase) and β-precipitates (NiBe with B2-structure) in a Ni-12 at.% Bi alloy has been investigated by field ion microscopy (FIM) and computer simulation of the images. The precipitates were found to be plate-shaped with an aspect ratio of about 10. Most of the plates were formed roughly but not exactly on (001)-planes of the matrix. The phase boundary is extremely narrow and does not exceed 2–3 atomic layers in thickness. The orientation relationship between α- and β-phases was found to be the Baker-Nutting one, where (001)_α//(001)_β with 〈110〉_α//〈110〉. The observed morphology of the precipitates and the structure of the interfaces were analysed on the basis of an anisotropy of the interfacial energy, the effect of the elastic strains set up during the phase separation and the growth mechanism of the precipitates.     

Elastic interaction energy of two spherical precipitates in an anisotropic matrix

Following the difference method of Eshelby, the elastic interaction energy between two spherical precipitates embedded in an infinite matrix of cubic anisotropy is studied as a function of their distance of separation and alignment direction. When the precipitates are positioned along the [100] direction of the matrix phase, the elastic interaction is found to be attractive and often to exhibit a maximum value at an intercenter distance of two to three radii. For the [110] and [111] alignments, the results depend on the sign of the anisotropic factor,H=2C₄₄+C₁₂−C₁₁, of the matrix phase. When it is positive as in Cu and Ni, the interaction is found to be repulsive. In the reverse case, the situation is substantially different; for the [111] alignment with a Mo matrix, the interaction is found to be of an attractive nature.     

The formation of multi-domain precipitates in a NiW alloy

The formation of twinned Ni₄W precipitates in a NiW alloy has been studied. On prolonged aging the dominant morphology found is one where two perpendicular twin related variants of Ni₄W form plate-shaped precipitates with a near (100)_Ni habit plane. The orientation relationship between the Ni₄W variants and the Ni matrix leads to a macroscopic shape change with the (100) habit plane interface nearly invariant. In addition each domain has a simple shear associated with it, which can be accomodated either elasticity or by arrays of screw dislocations. Elastic accommodation leads to a strain contrast effect which is similar to that observed at screw dislocations lying normal to the foil surface.     

Determination of the elastic moduli of a directionally solidified nickel-based TaC eutectic

Measurements of ultrasonic wave velocities in a polycrystalline directionally solidified nickel based eutectic alloy are used to evaluate the three independent single crystal elastic moduli at temperatures between 298 to 925 K. The C_u and C₄₄ moduli are obtained directly from high frequency wave propagation along the D.S. axis, corresponding to <100>. Evaluation of C₁₂ requires measurements at lower frequencies to obtain (C₁₁ −E _<100>). The elastic anisotropy and temperature dependence of the elastic moduli are almost identical to those reported for pure Ni, indicating that neither TaC fiber reinforcement nor γ′ precipitate have strong effects on elastic properties of Ni based turbine blade alloys.     

A transmission electron microscopy investigation of the early stages of precipitation in an Al-Zn-Mg alloy

The growth of spherical precipitates in an Al-5 wt pct Zn-2 wt pct Mg alloy was monitored by hardness measurements and transmission electron microscopy. The growth of precipitates from 15 to 80Å in diameter was followed for various aging times at temperatures of 30, 70, 100, and 150°C. For short times at low aging temperatures precipitate growth followed at _1/9 law, whereas growth at longer times, or higher temperatures, followed at _1’3 law. Various contrast experiments lead to the conclusion that the precipitates produced at all but the shortest aging times at the lowest aging temperatures have a hexagonal structure and are spherical precursors of the η′ phase. The advantages and disadvantages associated with the various imaging techniques for small precipitates utilized in this study are discussed. Formerly Research Assistant, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA. 02139     

Trace element effects on precipitation processes and mechanical properties in an Al-Cu-Li alloy

A study has been made of how impurities (Na and K) and trace additions of indium, magnesium, and silicon affect the microstructure and related mechanical properties of an Al-Cu-Li alloy. Transmission electron microscopy (TEM) was used to determine the size and distribution of particles in four alloys. Indium and magnesium are both seen to stimulate T ₁ precipitation. Indium also modifies ϑ″ morphology, and magnesium greatly increases the number density of ϑ″ precipitates. Strain localization was observed in underaged Al-Cu-Li-In tensile samples, consistent with observed changes in precipitate structure. No superposition of the effects of indium and magnesium was seen. High-resolution analytical microscopy was used to inspect precipitates for segregation of trace elements during early stages of aging, but no segregation was found within the detection limits of the system. Variations in heat treatment were made in order to study nucleation kinetics and trace element interactions with vacancies. Indium, with a binding energy less than that of lithium, was not seen to interact with quenched-in vacancies, while magnesium, with a binding energy greater than that of lithium, had a strong interaction. Yield anisotropies and fracture toughnesses were measured. Removal of trace impurities of sodium and potassium correlated with improved fracture properties. Magnesium was observed to increase anisotropy, especially in the T8 temper. A model was used to explain the anisotropy data in terms of texture and precipitate distribution.     

The influence of elastic strain energy on the formation of coherent hexagonal phases

The elastic strain energy function,Y (n), of coherent hexagonal phases has been derived for arbitrary directions,n, in a parent phase with arbitrary crystal structure. These calculations indicate that in all casesY (n) exhibits transverse isotropy about thec axis. As a result,Y (n) has a pronounced effect on the morphology of the precipitating structures. In the case of hexagonal inclusions, three possible optimum shapes in reciprocal space are identified: 1) a rod parallel to thec axis, 2) a plate perpendicular to thec axis, and 3) a hollow conical shape with the axis of revolution parallel toc. The precise precipitate shape can be predicted by identifying the directionn _o which minimizes the strain energy function,Y (n). Evaluation ofY (n) for η and ή MgZn2 precipitates in the ternary Al-Mg-Zn system correctly predicts the orientation and morphology of the particles. This method has also been extended to explore the morphology of the microstructure of hexagonal spinodal alloys. It is shown that the gradient energy term is generally anisotropic, and that together with the strain energy function,Y(n), has a strong influence on composition fluctuations. It is predicted that a one-dimensional periodic compositional variation along the [001] direction should be observed whenY [001] is a global minimum. In all other cases, the microstructure is complex and lacks periodicity.     

Precipitation of the δ-Ni3Nb phase in two nickel base superalloys

The precipitation of the equilibrium δ-Ni₃Nb phase has been studied in two niobium bearing nickel base superalloys—INCONEL 718 and INCONEL* 625—both of which are hardenable by the precipitation of the metastableγ″-Ni₃Nb phase. The morphology and the distribution of precipitates have been examined and the crystallographic orientation relationship between the austenite and theδ phases has been determined. The nucleation of theδ phase at stacking faults within pre-existing δ" precipitates has been discussed.     

Precipitation at coherency loss in Cu-0.35 wt pct Cr

An electron microscopy study of the nature of the precipitate in Cu-0.35 wt pct Cr when coherency begins to break down has been carried out. As opposed to previous observations, extra diffraction reflections are seen and are attributed to chromium bec precipitate havinga = 2.8(9)?. The particles appear to be rodlike, but some observations suggest that a plate like shape also exists. The long axes of most precipitates are seen to lie on <110>, <111>, <112> and <113> directions, all common to (110) planes. The position of the major extra reflections due to chromium can be shown to arise from chromium particles having at least two distinct orientations relative to the matrix but in each case having (110)_Cr ‖(111)_Cu, the minimum misfit of low indiced rational planes. Not all the extra reflections can be explained.     

Plastic relaxation of the transformation strain energy of a misfitting spherical precipitate: Ideal plastic behavior

Assuming ideal plastic behavior for an isotropic matrix containing a misfitting spherical precipitate, the total amount of work expended during elasto-plastic deformation is calculated and compared with the total strain energy in the corresponding pure elastic state. For precipitates larger than one micron (μm), the effective yield stress is taken as the macroscopic yield stress while for smaller precipitates, size-dependent yield stresses are obtained from the Ashby-Johnson model. In the case of coherent submicron precipitates, the effective yield stress becomes the theoretical yield strength and thus plastic relaxation is not possible unless the transformation stress is extremely large. For incoherent submicron precipitates, the effective yield stress is approximately inversely proportional to the precipitate radius,r. Hence plastic relaxation again is not possible whenr < 10 nm, but whenr ≃100 nm the strain energy can decrease by 10 ∼ 40 pct at a misfit of 3 pct. For supra-micron particles, however, the ratio of the effective yield stress to the shear modulus becomes 10⁻³ or less, and plastic relaxation can reduce the strain energy by factors of 3 to 15 at misfits of 1 to 3 pct. Under this circumstance, the plastic zone becomes wide, its radius ranging from 3 to 5r.     

A transmission electron microscopy investigation of the early stages of precipitation in an Al-Zn-Mg alloy

The growth of spherical precipitates in an Al-5 wt pct Zn-2 wt pct Mg alloy was monitored by hardness measurements and transmission electron microscopy. The growth of precipitates from 15 to 80Å in diameter was followed for various aging times at temperatures of 30, 70, 100, and 150°C. For short times at low aging temperatures precipitate growth followed at _1/9 law, whereas growth at longer times, or higher temperatures, followed at _1’3 law. Various contrast experiments lead to the conclusion that the precipitates produced at all but the shortest aging times at the lowest aging temperatures have a hexagonal structure and are spherical precursors of the η′ phase. The advantages and disadvantages associated with the various imaging techniques for small precipitates utilized in this study are discussed.     

The influence of annealing in the ferrite-plus-austenite phase field on the stability of vanadium carbide precipitates

The influence of rapid excursions into the ferrite-plus-austenite two-phase field on V₄C₃ precipitates formed by tempering in the ferrite phase was explored. The iron-based alloy studied contained 0.14 carbon and 0.49 vanadium in wt pct. A fine distribution of V₄C₃ particles was obtained through solution treatment followed by quench and tempering at 973 K. Samples were then rapidly heated to 1088 K for various times between 50 and 300 seconds. The plate-shaped V₄C₃ precipitates in regions that maintained their original ferrite matrix exhibited a continuous coarsening and decrease in their aspect ratio with increasing hold times. The V₄C₃ precipitates in regions that transformed to austenite were observed to dissolve by a reversion process if they were below a certain size. If the V₄C₃ particles were above that size, the matrix change caused them to coarsen and reduce their aspect ratio. Based upon the size of V₄C₃ particles that did show reversion, the interphase interfacial surface energy between the V₄C₃ partieles and aus tenite in the nonequilibrium orientation relationships produced in this study was determined to be between 1125 and 1840 mJ/m². When samples where reversion was observed were retempered at 973 K, the original distribution of fine, plate-shaped V₄C₃ particles could be reproduced accompanied by a measurable secondary hardening response.     

Variations in Creep Performance of P92 Steel Within Its Composition Range: Influence of N Solubility

Underspecified composition ranges often lead to alloys with unpredictable mechanical performance. To better understand the changes in microstructure and mechanical performance associated with variations of key elements, three versions of P92 are formulated within, or close to, the specified allowable for N, B, and C ranges. Chromium and Si are also varied to influence N solubility. Different service conditions (i.e., temperature and stress) are explored. It is observed that >80% decrease in creep life occurs at 625 °C and 155 MPa for the highest B and N containing alloy. Multiscale characterization reveals key changes due to the trace element variation. The high B and N containing alloy forms deleterious BN precipitates with morphology that promotes crack nucleation and damage accumulation, but this alloy additionally forms higher fractions of beneficial MX precipitates. The alloy with the lowest B and N concentrations but greater C content shows the best creep performance—a consequence of the refined M₂₃C₆ carbide precipitate population and the absence of large-scale inclusions or BN precipitates. Calculations of creep activation energy reveal that the high B and N containing alloy is more prone to damage accumulation which causes an early onset of accelerating creep and greater minimum creep rate.