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.     

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.     

The elastic stabilization of multiple-height growth ledges during diffusional phase transformations

In diffusional phase transformations that proceed by the terrace-ledge-kink mechanism, both atomic-scale, unit-height ledges and multiple-height ledges are observed. Both types of ledges also have dislocation character, thereby making them disconnections. The dislocation interaction is shown to provide a strong stabilizing force for the multiple ledges and to lead to a capture distance for additional unit ledges that increases with ledge height. Implications for the mechanism of phase transformation are discussed.     

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.)     

On the role of elastic interactions in the nucleation and growth of ledges

Elastic interactions among ledges on transformation interfaces have noticeable consequences when chemical and interfacial tension (capillary) forces are small, namely, near equilibrium. This occurs just at nucleation (unstable equilibrium) or during the slow coarsening regime. When the interface lies perpendicular to the misfit strain (as do the large faces of misfits in Al-Cu alloys), ledges of like sign repel one another, and nucleation of new ledges occurs as far as possible from existing ones. However, when the interface lies parallel to the misfit strain, ledges of like sign attract one another. We then expect the formation of superledges. Essentially, such an interface with ledges is elastically unstable. Expressions are derived for the kinetics of ledge amaleamation. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformation Committee of the Materials Science Division, ASM INTERNATIONAL.     

A two-dimensional analysis of the evolution of coherent precipitates in elastic media

The morphological evolution of coherent inclusions in elastic media is studied in two-dimensions. The inclusions are simple dilations with isotropic surface energy in a system with homogeneous elastic constants of negative anisotropy. The equilibrium sizes at which a circular inclusion transforms to a rectangle or square, and at which a square splits into a doublet or quartet of separated inclusions are computed analytically. A finite-element model is then constructed to simulate the evolution of an arbitrary distribution of inclusions along the minimum-energy path. In the model, the circle evolves into a square, which splits into a doublet by hollowing from its center, or, if this is forbidden, by drawing in a perturbation on its surface. The sizes at which shapes spontaneously transform are compared to the equilibrium values. Finally, the simulation is used to study the evolution of a random distribution of inclusions. The first metastable state assumed by the distribution depends on the elastic interaction, surface energy and areal fraction of the inclusion phase through a single dimensionless parameter that groups these three effects. The results are compared to prior theoretical and experimental work on coarsening patterns in three dimensions.     

Interactions of grain boundaries with coherent precipitates during grain growth

Nimonic PE 16 has been heat treated below the γ′ solvus to initiate grain growth and strong interactions between coherent γ′ and migrating grain boundaries noted. In a small number of cases the γ′ is cut-through by the migrating grain boundary so as to retain coherency with the matrix. More usually very strong pinning of the grain boundary occurred as part of an attempt at by pass. In these cases the γ′ in contact with the grain boundary was found to competitively grow/coarsen at the expense of the surrounding matrix γ′ so that the local volume fraction of the γ′ on the boundary increased more than did its size. Direct measurements from electron micrographs have been used to establish the microstructural parameters characterising the γ′ dispersion and also the nature of the boundary-particle interactions. These measurements have been used to assess the local driving forces for grain boundary migration and the very large pinning effects due to the γ′ particles. The data are interpreted in terms of models proposed by Zener and by Ashby and colleagues.     

Symmetry-breaking transitions in equilibrium shapes of coherent precipitates: Effect of elastic anisotropy and inhomogeneity

We examine the symmetry-breaking transitions in equilibrium shapes of coherent precipitates in two-dimensional (2-D) systems under a plane-strain condition with the principal misfit strain components ε* _xx and ε* _yy . For systems with cubic elastic moduli, we first show all the shape transitions associated with different values of t=ε* _yy /ε* _xx . We also characterize each of these transitions, by studying its dependence on elastic anisotropy and inhomogeneity. For systems with dilatational misfit (t=1) and those with pure shear misfit (t=−1), the transition is from an equiaxed shape to an elongated shape, resulting in a break in rotational symmetry. For systems with nondilatational misfit (−1<t<1; t ≠ 0), the transition involves a break in mirror symmetries normal to the x- and y-axes. The transition is continuous in all cases, except when 0<t<1. For systems which allow an invariant line (−1≤t<0), the critical size increases with an increase in the particle stiffness. However, for systems which do not allow an invariant line (0<t≤1), the critical size first decreases, reaches a minimum, and then starts increasing with increasing particle stiffness; moreover, the transition is also forbidden when the particle stiffness is greater than a critical value.     

The calculation of habit planes for elastic transformations by minimization of their elastic strain energy

By using Eshelby’s method for the determination of the stresses and strains generated in a transformation, the shape and orientation of an ellipsoidal region of transformed phase that minimizes the elastic strain energy accompanying the transformation are calculated together with the change in elastic strain energy. The orientation of minimum total energy describes the habit plane. The applicability of the approach is demonstrated by showing that the habit planes of twins in cubic crystals, and martensitic plates in In-20 pct TI and Fe-31 pct Ni, which have been calculated by other methods, can alternatively be determined by this method. It is then used to calculate that the habit plane for martensitic plates in bulk, high density, oriented orthorhombic polyethylene should be (4.67,1,0) orthorhombic. As a consequence of these calculations it is shown that minimization of the total elastic strain energy must be the dominating factor in the nucleation event of transformed products in most metals.     

The calculation of habit planes for elastic transformations by minimization of their elastic strain energy

By using Eshelby's method for the determination of the stresses and strains generated in a transformation, the shape and orientation of an ellipsoidal region of transformed phase that minimizes the elastic strain energy accompanying the transformation are calculated together with the change in elastic strain energy. The orientation of minimum total energy describes the habit plane. The applicability of the approach is demonstrated by showing that the habit planes of twins in cubic crystals, and martensitic plates in In-20 pct Tl and Fe-31 pct Ni, which have been calculated by other methods, can alternatively be determined by this method. It is then used to calculate that the habit plane for martensitic plates in bulk, high density, oriented orthorhombic polyethylene should be (4.67,1,0) orthorhombic- As a consequence of these calculations it is shown that minimization of the total elastic strain energy must be the dominating factor in the nucleation event of transformed products in most metals.     

Yield strength of overaged alloys containing coherent ordered precipitates

The existing models for yielding of overaged alloys do not account for the observed strain rate sensitivity of the yield stress and predict a sharper decrease in stress as a function of particle size than the experimental observations. A dynamic model has been presented here to account for the yield strength of overaged alloys containing coherent ordered precipitates which takes into consideration the intrinsic characteristics of the precipitates. Concentric glide loops are supposed to form around precipitate particles. Their sizes as a function of applied stress have been computed. The stress on the innermost loop rises with increasing applied stress and when it exceeds the opposing force due to antiphase boundary, the loops shrink. The rate of shrinkage of loops has been calculated for different values of applied stresses. The yielding occurs when the external stress is high enough to account for the applied strain rate. The model has been applied to calculate the yield strength of an austenitic stainless steel containingγ’ precipitates. The antiphase boundary energy of the precipitates was measured by dislocation pair spacing method. The mode of variation of calculated yield stress values with particle size is predicted more accurately than by earlier models. The model qualitatively accounts for the observed sensitivity of yield stress to strain rate and antiphase boundary energy of the precipitates. Formerly Manager Research & Development, Alloy Steels Plant, Durgapur     

Yield strength of overaged alloys containing coherent ordered precipitates

The existing models for yielding of overaged alloys do not account for the observed strain rate sensitivity of the yield stress and predict a sharper decrease in stress as a function of particle size than the experimental observations. A dynamic model has been presented here to account for the yield strength of overaged alloys containing coherent ordered precipitates which takes into consideration the intrinsic characteristics of the precipitates. Concentric glide loops are supposed to form around precipitate particles. Their sizes as a function of applied stress have been computed. The stress on the innermost loop rises with increasing applied stress and when it exceeds the opposing force due to antiphase boundary, the loops shrink. The rate of shrinkage of loops has been calculated for different values of applied stresses. The yielding occurs when the external stress is high enough to account for the applied strain rate. The model has been applied to calculate the yield strength of an austenitic stainless steel containing γ′ precipitates. The antiphase boundary energy of the precipitates was measured by dislocation pair spacing method.The mode of variation of calculated yield stress values with particle size is predicted more accurately than by earlier models. The model qualitatively accounts for the observed sensitivity of yield stress to strain rate and antiphase boundary energy of the precipitates. Formerly Manager Research & Development, Alloy Steels Plant, Durgapur     

The dendritic growth of γ′ precipitates and grain

The growth pattern of γ precipitates in the grains and at the grain boundaries has been investigated in a Ni-24Co-4Al-4Ti-5Cr-5Mo (weight percent) alloy of very small lattice misfit between the precipitate and the matrix phases under varying heat-treatment conditions. When aged at temperatures lower than the solvus temperature (T _s = 1150 °C) by more than 30 °C after direct cooling from the solution-treatment temperature, the nucleation density is high. In this condition, the supersaturation is quickly removed because of the overlapping diffusion fields and the precipitates undergo Ostwald ripening from the early stage. The precipitates then have an equilibrium shape of spheres in the grains and truncated spheres at nearly straight grain boundaries. The precipitates at the grain boundaries are coherent with one of the grains, and their number density is not much larger than that in the grains, apparently because of a large contact angle (about 150 deg) with the grain boundary. Quenching the alloy after the solution treatment and aging at any temperature also produce high precipitate number density and equilibrium shapes. When aged at temperatures just belowT _s (above 1140 °C), the nucleation density is low, the precipitates grow dendritically in the grains, and the grain boundaries become serrated. The observed dendritic growth characteristics do not quantitatively agree with the predictions of Mullins and Sekerka theory, but the discrepancy may be due to the uncertainties in both the observed and calculated quantities. By deeply etching the matrix, it is shown that the grain boundary serration is produced by the precipitates growing preferentially in the direction of the incoherent boundary because of the rapid solute diffusion along the grain boundary. The dendritic growth and grain boundary serration can be obtained also by slowly cooling through the temperature range just belowT _s.     

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.     

The effect of the precipitation of coherent and incoherent precipitates on the ductility and toughness of high-strength steel

The effect of the coexistence of coherent and incoherent precipitates, such as M₂C and NiAl, on the ductility and plane strain fracture toughness of 5 wt pct Ni-2 wt pct Al-based high-strength steels was studied. In order to disperse coherent and incoherent precipitates, the heat treatments were carried out as follows: (a) austenitizing at 1373 K, (b) tempering at 1023 or 923 K for dispersing the incoherent precipitates of M₂C and NiAl, and then (c) aging at 843 K for 2.4 ks to disperse the coherent precipitate of NiAl into the matrix, which contains incoherent precipitates, such as M₂C and NiAl. The results were obtained as follows: (a) when the strengthening precipitates consist of coherent ones, such as M₂C and/or NiAl, the ductility and toughness are extremely low, and (b) when the strengthening precipitates consist of coherent and incoherent precipitates, such as M₂C and NiAl, the ductility and fracture toughness significantly increase with no loss in strength. It is shown that the coexistence of coherent and incoherent precipitates increases homogeneous deformation, thus preventing local strain concentration and early cleavage cracking. Accordingly, the actions of coherent precipitates in strengthening the matrix and of incoherent precipitates in promoting homogeneous deformation can be expected to increase both the strength and toughness of the material.     

Effect of clustering of precipitates on grain growth

The present work shows that clustering of particles promotes deviation in the classical mathematical expressions describing the grain growth control by second-phase particles. On the basis of experimental results and theoretical laws, a semiempirical expression to predict the limiting grain size is presented. The latter expression takes account of agglomeration phenomena and can be extended to large volume fractions of particles, conditions under which classical theories clearly fail. The equation remains valid as far as the nucleation of precipitates takes place at random. From a practical point of view, it is shown that volume fractions larger than 0.12 cannot significatively control the grain size because of the increased probability for clustering.     

Effect of clustering of precipitates on grain growth

The present work shows that clustering of particles promotes deviation in the classical mathematical expressions describing the grain growth control by second-phase particles. On the basis of experimental results and theoretical laws, a semiempirical expression to predict the limiting grain size is presented. The latter expression takes account of agglomeration phenomena and can be extended to large volume fractions of particles, conditions under which classical theories clearly fail. The equation remains valid as far as the nucleation of precipitates takes place at random. From a practical point of view, it is shown that volume fractions larger than 0.12 cannot significatively control the grain size because of the increased probability for clustering.     

The geometry and properties of ledges in interfaces

This paper is an overview of work on the structure and properties of line defects in crystalline interfaces. Emphasis is placed upon the treatment of the dislocation and step nature of the defects in a unified manner and upon the correct accounting for the step nature in topological analyses. It is also shown that consideration of the step character leads to predictions of behavior that are not always consonant with predictions based upon dislocation concepts alone. Much of the previous work on this subject has been developed for grain boundaries in cubic metals, and the means of extension to less simple cases is indicated. It is shown that results often depend upon the choice of a particular reference frame. This paper is based on a presentation made in the symposium “The Role of Ledges in Phase Transformations” presented as part of the 1989 Fall Meeting of TMS-MSD, October 1–5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL.