Generalizing Ellipsoidal Growth

Abstract

This paper generalizes previous work of Rios and Villa on spherical growth. The generalized equation applies to nucleation of ellipsoids according to an inhomogeneous Poisson point process. Microstructural evolution in three dimensions of nucleation and growth transformations of ellipsoids is simulated using the causal cone method. In the simulation, nuclei are located in space according to an inhomogeneous Poisson point process. The transformed regions grow with prolate and oblate ellipsoidal shapes. The ellipsoids have their corresponding axes parallel. The simulation and the exact analytical solution are in excellent agreement. Microstructures generated by the computer simulation are displayed. From these generated microstructures one can obtain the contiguity. In the contiguity against volume fraction plot, data from the sphere and all ellipsoids fall on the same curve. The contiguity curve for nucleation according to an inhomogeneous Poisson point process falls above the contiguity curve for nucleation according to a homogeneous Poisson point process. This behavior indicates that nucleation according to an inhomogeneous Poisson point process introduced a nucleus clustering effect.

Analytical Modeling and Computer Simulation of the Transformation of Ellipsoids Nucleated on Random Parallel Planes

“…The problem is the same for ellipsoidal growth nucleated on nuclei located in space according to a Poisson point process. It is well-known [15][16][17] that when ellipsoids are aligned, that is, their equivalent axes are parallel; there is an exact expression, see, for example, refs. 17,21 .…”

“…Figure 3 Visual inspection shows that the fully transformed microstructures generated by the oblate ellipsoids have a significant proportion of their interfacial area parallel to the plane 3 x 0 = . Compare, for instance, with the microstructure generated by ellipsoids nucleated in space according to a homogeneous Poisson point process from our previous work 17 .…”

“…One can found that the contiguity has been used in several computer simulation papers, some of which are listed here. 16,17,26,27,[29][30][31] The contiguity, C ββ , is defined as…”

“…Moreover, as demonstrated by Ventura et al 26 , if the position of the nucleation sites are arranged periodically or approaching a periodic arrangement, the impingement takes place later, and the contiguity lies below the dashed line 26 . When the growing regions are ellipsoids, their contiguity still falls on the dashed line provided that their nucleation sites nuclei are located in space according to a homogeneous Poisson point process 16,17 .…”

“…The assumption of spherical growth is standard. Nonetheless, new regions do not always grow as spheres 4,[15][16][17] . Bradley et al demonstrated in a series of papers [18][19][20] that a grain boundary nucleated ferrite allotriomorph is best described by an oblate ellipsoid.…”