Determining the three-dimensional morphology of γ-particles in γ-γ′ superalloys

We propose qualitative and quantitative methods for determining the three-dimensional morphology of second-phase particles in Ni-based superalloys in the late stages of coarsening. Qualitatively, we determine a way to identify the three-dimensional shape of a particle from its (111) cross section. Quantitatively, we derive a method that uses stereological analysis on a single (111) section to compute the average shape of a particle in three dimensions. For cases where the average shape does not necessarily reflect the particles' true morphology, we derive another method based on sectioning probability to compute the shape of individual particles. We are also able to determine a particle's orientation in three dimensions by examining its (111) cross section. The methods were tested by using computer-generated (111) sections of three-dimensional arrays of rectangular particles. We conclude that (111) sections can be used to provide an accurate calculation of the interfacial area per unit volume (S_v) of the structure. Finally, we illustrate the efficacy of using (111) sections to determine particle morphology by examining (111) transmission electron microscopy (TEM) micrographs of particles undergoing splitting.