Influence of faceting upon the equilibrium shape of nuclei at grain boundaries-II. Three-dimensions
Based on the exact equilibrium shapes of nuclei at a grain boundary in the presence of a facet at one orientation of the nucleus-matrix boundary determined in the two-dimensional case, three-dimensional nucleus shapes are studied under the same conditions. These shapes are investigated under the assumptions that the grain boundary is constrained to remain planar and that the grain boundary puckers in order to allow equilibrium junction angles with the nucleus to be more closely approached. At planar grain boundaries, the 'upper' portion of the nucleus is taken to be a faceted spherical cap, as delineated by a modified version of the Wulff construction. For the shape of the 'lower' or unfaceted portion of the nucleus, a second-order non-linear elliptic partial differential equation was derived and numerically solved. In the case of the pucker mechanism, the entire nucleus is considered as a faceted spherical cap. A catenoid surface, which is a minimal surface, is used to calculate the additional grain boundary area introduced by the pucker. Calculation of the free energy of activation for nucleation. ΔG*, indicates that planar boundary mechanism is favored when φ, the tilt angle between the facet and the grain boundary plane, exceeds ≈ 18° +φc1 (where φ = smallest φ at which the facet touches the grain boundary), while the pucker mechanism has a lower ΔG* when φ is smaller. ΔG* increases rapidly with φ, particularly when the relative specific interfacial energy of the facet is small. Hence nucleation at a disordered grain boundary should normally occur with pronounced preference parallel to only one of n crystallographically equivalent habit planes. © 1975.
Influence of faceting upon the equilibrium shape of nuclei at grain boundaries-II. Three-dimensions.
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/5184