On volume and surface densities of dynamical germ-grain models with ellipsoidal growth: a rigorous approach with applications to Materials Science

Many engineering materials of interest are polycrystals: an aggregate of many crystals with size usually below 100 μm. Those small crystals are called the grains of the polycrystal, and are often equiaxed. However, because of processing, the grain shape may become anisotropic; for instance, during recrystallization or phase transformations, the new grains may grow in the form of ellipsoids. Heavily anisotropic grains may result from a process, such as rolling, and they may have most of their interfacial area parallel to the rolling plane. Therefore, to a first approximation, these heavily deformed grains may be approximated by random parallel planes; as a consequence, the nucleation process may be assumed to take place on random parallel planes. The case of nucleation on random parallel planes and subsequent ellipsoidal growth is also possible. In this paper we model such situations employing time dependent germ grain processes with ellipsoidal growth. We provide explicit formulas for the mean volume and surface densities and related quantities. The known results for the spherical growth follow here as a particular case. Although this work has been done bearing applications to Materials Science in mind, the results obtained here may be applied to nucleation and growth reactions in general. Moreover, a generalization of the so called mean value property, crucial in finding explicit analytical formulas in the paper, is also provided as a further result in the Appendix A.